constrained optimization

The End of Theory: Financial Crises, the Failure of Economics, and the Sweep of Human Interaction
by Richard Bookstaber
Published 1 May 2017

Well, not really optimizing—we don’t really want to rank all ten thousand possible Starbucks combinations every morning—so it must be that people are operating under cognitive constraints. People must act as if they are doing a constrained optimization, looking at the cost or limits to search or to available information. There’s an irony to adding constraints to reflect cognitive limits—constraints just make the problem worse. It is even more difficult to solve a constrained optimization. If people can’t optimize because that exercise is too difficult, then they certainly cannot work their way through a constrained optimization. It might be that we don’t fit into an optimization because we are constrained and cannot pull off the task, or it could be that we are in a world where things change suddenly and wholly unexpectedly, where the past does not give us a window into what is happening, and where what is happening could run off the rails and down the chasm, so the conditions required for following the mathematically rational path of optimization are no longer really optimal.

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It might be that we don’t fit into an optimization because we are constrained and cannot pull off the task, or it could be that we are in a world where things change suddenly and wholly unexpectedly, where the past does not give us a window into what is happening, and where what is happening could run off the rails and down the chasm, so the conditions required for following the mathematically rational path of optimization are no longer really optimal. And so it becomes just an academic exercise, because people actually do not behave in a way that is “as if” they were optimizing or even executing a constrained optimization. They display various behavioral flaws that cannot be consistent with a constrained optimization. Economics has a roundabout answer for this, too. People now are said to behave as if they are doing a constrained optimization with a utility function that essentially zigs and zags to accommodate their irrational behavior, behaving as if they are both constrained and have a weird utility function that adds more and more adjustable parameters.

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Then, recognizing that people cannot always do so, they step back to concede that people will solve the optimization problem subject to constraints, such as limited time, information, and computational ability. If computational ability is an issue, then moving into a constrained optimization is moving in the wrong direction, because a constrained optimization problem is generally more difficult to solve than an unconstrained one. But given the axioms, what else can you do? It doesn’t take much familiarity with humans—even human mathematicians—to realize that we don’t actually solve these complex, and often unsolvable, problems.

Elements of Mathematics for Economics and Finance
by Vassilis C. Mavron and Timothy N. Phillips
Published 30 Sep 2006

Within the locality of a maximum point x = x0 , y = y0 (that is, for any x and y sufficiently close to these values), the value f (x, y) of f attains a maximum when x = x0 and y = y0 . Similarly for a minimum point. 9. Optimization 193 3. The problem stated in Example 9.5 above was initially a constrained optimization problem in three variables. But, upon substitution for z, it became an unconstrained optimization problem in the remaining two variables, x and y. 9.3 Constrained Optimization Optimization of a quantity in economic models, or indeed in many practical situations, is rarely unconstrained. Usually there are constraints involving some or all of the variables. For instance, in considering ways to maximize, say, output, there will be constraints due to costs or of the available labour.

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Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 9.2 Unconstrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 9.3 Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 9.3.1 Substitution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 9.3.2 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9.3.3 The Lagrange Multiplier λ: An Interpretation . . . . . . . . . . 201 9.4 Iso Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 10.

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A pilot may fly an aircraft so as to cover the maximum possible air miles when the total fuel cost is stipulated. Consumers try to maximize utility subject to a given budget. In this chapter, we will describe techniques of optimization when there are no constraints specified (unconstrained optimization) and subject to a constraint (constrained optimization). 185 186 Elements of Mathematics for Economics and Finance 9.2 Unconstrained Optimization The optimization of functions of one variable was discussed in Chapter 7. Optimization there meant finding the stationary (or critical or turning) points of the function and then testing to see whether they were maxima or minima.

A Primer for the Mathematics of Financial Engineering
by Dan Stefanica
Published 4 Apr 2008

., infinitely many times differentiable. We want to find the extrema of j (x) subject to m constrains given by g(x) = 0, where 9 : U -----7 ~m is a smooth function, i.e., Find Xo E U such that max g(x) = 0 xEU j(x) = j(xo) or min g(x) = 0 xEU j(x) j(xo). (8.1) Problem (8.1) is called a constrained optimization problem. For this problem to be well posed, a natural assumption is that the number of constraints is smaller than the number of degrees of freedom, i.e., m < n. Another way to formalize problem (8.1) is to introduce the set S of points satisfying the constraint g( x) = 0, and find extrema for the restriction of the function j (x) to the space S, i.e., for j Is. 235 236 CHAPTER 8.

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The point Xo E U c IRn is called a constrained extremum of the function f : U -----+ IR with respect to the constraint g( x) = 0 if and only if Xo is an extremum point of fls, where S = {x E U such that g(x) = o}. where Vf(x) and Vg(x), the gradients of f : U -----+ IR and 9 : U -----+ IRm are given by v f(x) To solve the constrained optimization problem (8.1), let A = (Ai)i=l:m be a vector of the same size, m, as the number of constraints; A is called a Lagrange multipliers vector. Let F : U x IRm -----+ IR given by F(x, A) = f(x) + be the Lagrangian function of f and g. If ::n (x, >.)) ; ~;~ (x) ~;~ (x) ... ~;~ (x) ) Vg(x) (8.2) At g(x) (:;, (x, >.) ... 237 Og2 (x) Og2 (x) '" OXl.

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The reason is that if a problem is known to have a unique constrained extremum, and if the Lagrangian has exactly one critical point, then that critical point must be the constrained extremum point. For this line of thinking to be sound we must know that the constrained extremum problem has a unique solution. In general, showing that a problem has a unique solution is not straightforward. The steps required to solve a constrained optimization problem using Lagrange multipliers can be summarized as follows: Step 1: Check that rank(\lg(x)) = m, for all xES. q( v) = (V2 - 2V3? + 2v~ + 3v~ - 2(V2 - 2V3)V2 + 4V2V3 = v~ + 4V2V3 + 7v~. Note that Vred = ( ~~ ). Step 2: Find (xo, Ao) E U X ffi.m such that \l(x,)..)F(xo, Ao) Step 3.1: Compute q(v) + 4V2V3 + 7v~. 0 (8.15) Whether the point Xo is a constrained extremum for f (x) will depend on the quadratic form qred being either positive definite2 , i.e., = v D2 Fo(xo) v, where Fo(x) = f(x) + Ab9(x).

Hands-On Machine Learning With Scikit-Learn and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems
by Aurélien Géron
Published 13 Mar 2017

The resulting vector p will contain the bias term b = p0 and the feature weights wi = pi for i = 1, 2, ⋯, m. Similarly, you can use a QP solver to solve the soft margin problem (see the exercises at the end of the chapter). However, to use the kernel trick we are going to look at a different constrained optimization problem. The Dual Problem Given a constrained optimization problem, known as the primal problem, it is possible to express a different but closely related problem, called its dual problem. The solution to the dual problem typically gives a lower bound to the solution of the primal problem, but under some conditions it can even have the same solutions as the primal problem.

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Each constraint (in this case just one) is subtracted from the original objective, multiplied by a new variable called a Lagrange multiplier. Joseph-Louis Lagrange showed that if is a solution to the constrained optimization problem, then there must exist an such that is a stationary point of the Lagrangian (a stationary point is a point where all partial derivatives are equal to zero). In other words, we can compute the partial derivatives of g(x, y, α) with regards to x, y, and α; we can find the points where these derivatives are all equal to zero; and the solutions to the constrained optimization problem (if they exist) must be among these stationary points. In this example the partial derivatives are: When all these partial derivatives are equal to 0, we find that , from which we can easily find that , , and .

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It implies that either or the ith instance lies on the boundary (it is a support vector). Note that the KKT conditions are necessary conditions for a stationary point to be a solution of the constrained optimization problem. Under some conditions, they are also sufficient conditions. Luckily, the SVM optimization problem happens to meet these conditions, so any stationary point that meets the KKT conditions is guaranteed to be a solution to the constrained optimization problem. We can compute the partial derivatives of the generalized Lagrangian with regards to w and b with Equation C-2. Equation C-2. Partial derivatives of the generalized Lagrangian When these partial derivatives are equal to 0, we have Equation C-3.

Market Risk Analysis, Quantitative Methods in Finance
by Carol Alexander
Published 2 Jan 2007

Once we have found all possible local maxima and minima, the global maximum (if it exists) is the one of the local maxima where the function takes the highest value and the global minimum (if it exists) is the one of the local minima where the function takes the lowest value. More generally, a constrained optimization problem takes the form max fx such that x hx ≤ 0 (I.1.47) where hx ≤ 0 is a set of linear or non-linear equality or inequality constraints on x. Examples of constrained optimization in finance include the traditional portfolio allocation problem, i.e. how to allocate funds to different types of investments when the investor has constraints such as no short sales and/or at least 50% of his capital must be in UK bonds.

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To Walter Ledermann Contents List of Figures xiii List of Tables xvi List of Examples xvii Foreword xix Preface to Volume I I.1 Basic Calculus for Finance I.1.1 Introduction I.1.2 Functions and Graphs, Equations and Roots I.1.2.1 Linear and Quadratic Functions I.1.2.2 Continuous and Differentiable Real-Valued Functions I.1.2.3 Inverse Functions I.1.2.4 The Exponential Function I.1.2.5 The Natural Logarithm I.1.3 Differentiation and Integration I.1.3.1 Definitions I.1.3.2 Rules for Differentiation I.1.3.3 Monotonic, Concave and Convex Functions I.1.3.4 Stationary Points and Optimization I.1.3.5 Integration I.1.4 Analysis of Financial Returns I.1.4.1 Discrete and Continuous Time Notation I.1.4.2 Portfolio Holdings and Portfolio Weights I.1.4.3 Profit and Loss I.1.4.4 Percentage and Log Returns I.1.4.5 Geometric Brownian Motion I.1.4.6 Discrete and Continuous Compounding in Discrete Time I.1.4.7 Period Log Returns in Discrete Time I.1.4.8 Return on a Linear Portfolio I.1.4.9 Sources of Returns I.1.5 Functions of Several Variables I.1.5.1 Partial Derivatives: Function of Two Variables I.1.5.2 Partial Derivatives: Function of Several Variables xxiii 1 1 3 4 5 6 7 9 10 10 11 13 14 15 16 16 17 19 19 21 22 23 25 25 26 27 27 viii Contents I.1.5.3 Stationary Points I.1.5.4 Optimization I.1.5.5 Total Derivatives I.1.6 Taylor Expansion I.1.6.1 Definition and Examples I.1.6.2 Risk Factors and their Sensitivities I.1.6.3 Some Financial Applications of Taylor Expansion I.1.6.4 Multivariate Taylor Expansion I.1.7 Summary and Conclusions I.2 Essential Linear Algebra for Finance I.2.1 Introduction I.2.2 Matrix Algebra and its Mathematical Applications I.2.2.1 Basic Terminology I.2.2.2 Laws of Matrix Algebra I.2.2.3 Singular Matrices I.2.2.4 Determinants I.2.2.5 Matrix Inversion I.2.2.6 Solution of Simultaneous Linear Equations I.2.2.7 Quadratic Forms I.2.2.8 Definite Matrices I.2.3 Eigenvectors and Eigenvalues I.2.3.1 Matrices as Linear Transformations I.2.3.2 Formal Definitions I.2.3.3 The Characteristic Equation I.2.3.4 Eigenvalues and Eigenvectors of a 2 × 2 Correlation Matrix I.2.3.5 Properties of Eigenvalues and Eigenvectors I.2.3.6 Using Excel to Find Eigenvalues and Eigenvectors I.2.3.7 Eigenvalue Test for Definiteness I.2.4 Applications to Linear Portfolios I.2.4.1 Covariance and Correlation Matrices I.2.4.2 Portfolio Risk and Return in Matrix Notation I.2.4.3 Positive Definiteness of Covariance and Correlation Matrices I.2.4.4 Eigenvalues and Eigenvectors of Covariance and Correlation Matrices I.2.5 Matrix Decomposition I.2.5.1 Spectral Decomposition of a Symmetric Matrix I.2.5.2 Similarity Transforms I.2.5.3 Cholesky Decomposition I.2.5.4 LU Decomposition I.2.6 Principal Component Analysis I.2.6.1 Definition of Principal Components I.2.6.2 Principal Component Representation I.2.6.3 Case Study: PCA of European Equity Indices I.2.7 Summary and Conclusions 28 29 31 31 32 33 33 34 35 37 37 38 38 39 40 41 43 44 45 46 48 48 50 51 52 52 53 54 55 55 56 58 59 61 61 62 62 63 64 65 66 67 70 Contents I.3 Probability and Statistics I.3.1 Introduction I.3.2 Basic Concepts I.3.2.1 Classical versus Bayesian Approaches I.3.2.2 Laws of Probability I.3.2.3 Density and Distribution Functions I.3.2.4 Samples and Histograms I.3.2.5 Expected Value and Sample Mean I.3.2.6 Variance I.3.2.7 Skewness and Kurtosis I.3.2.8 Quantiles, Quartiles and Percentiles I.3.3 Univariate Distributions I.3.3.1 Binomial Distribution I.3.3.2 Poisson and Exponential Distributions I.3.3.3 Uniform Distribution I.3.3.4 Normal Distribution I.3.3.5 Lognormal Distribution I.3.3.6 Normal Mixture Distributions I.3.3.7 Student t Distributions I.3.3.8 Sampling Distributions I.3.3.9 Generalized Extreme Value Distributions I.3.3.10 Generalized Pareto Distribution I.3.3.11 Stable Distributions I.3.3.12 Kernels I.3.4 Multivariate Distributions I.3.4.1 Bivariate Distributions I.3.4.2 Independent Random Variables I.3.4.3 Covariance I.3.4.4 Correlation I.3.4.5 Multivariate Continuous Distributions I.3.4.6 Multivariate Normal Distributions I.3.4.7 Bivariate Normal Mixture Distributions I.3.4.8 Multivariate Student t Distributions I.3.5 Introduction to Statistical Inference I.3.5.1 Quantiles, Critical Values and Confidence Intervals I.3.5.2 Central Limit Theorem I.3.5.3 Confidence Intervals Based on Student t Distribution I.3.5.4 Confidence Intervals for Variance I.3.5.5 Hypothesis Tests I.3.5.6 Tests on Means I.3.5.7 Tests on Variances I.3.5.8 Non-Parametric Tests on Distributions I.3.6 Maximum Likelihood Estimation I.3.6.1 The Likelihood Function I.3.6.2 Finding the Maximum Likelihood Estimates I.3.6.3 Standard Errors on Mean and Variance Estimates ix 71 71 72 72 73 75 76 78 79 81 83 85 85 87 89 90 93 94 97 100 101 103 105 106 107 108 109 110 111 114 115 116 117 118 118 120 122 123 124 125 126 127 130 130 131 133 x Contents I.3.7 Stochastic Processes in Discrete and Continuous Time I.3.7.1 Stationary and Integrated Processes in Discrete Time I.3.7.2 Mean Reverting Processes and Random Walks in Continuous Time I.3.7.3 Stochastic Models for Asset Prices and Returns I.3.7.4 Jumps and the Poisson Process I.3.8 Summary and Conclusions I.4 Introduction to Linear Regression I.4.1 Introduction I.4.2 Simple Linear Regression I.4.2.1 Simple Linear Model I.4.2.2 Ordinary Least Squares I.4.2.3 Properties of the Error Process I.4.2.4 ANOVA and Goodness of Fit I.4.2.5 Hypothesis Tests on Coefficients I.4.2.6 Reporting the Estimated Regression Model I.4.2.7 Excel Estimation of the Simple Linear Model I.4.3 Properties of OLS Estimators I.4.3.1 Estimates and Estimators I.4.3.2 Unbiasedness and Efficiency I.4.3.3 Gauss–Markov Theorem I.4.3.4 Consistency and Normality of OLS Estimators I.4.3.5 Testing for Normality I.4.4 Multivariate Linear Regression I.4.4.1 Simple Linear Model and OLS in Matrix Notation I.4.4.2 General Linear Model I.4.4.3 Case Study: A Multiple Regression I.4.4.4 Multiple Regression in Excel I.4.4.5 Hypothesis Testing in Multiple Regression I.4.4.6 Testing Multiple Restrictions I.4.4.7 Confidence Intervals I.4.4.8 Multicollinearity I.4.4.9 Case Study: Determinants of Credit Spreads I.4.4.10 Orthogonal Regression I.4.5 Autocorrelation and Heteroscedasticity I.4.5.1 Causes of Autocorrelation and Heteroscedasticity I.4.5.2 Consequences of Autocorrelation and Heteroscedasticity I.4.5.3 Testing for Autocorrelation I.4.5.4 Testing for Heteroscedasticity I.4.5.5 Generalized Least Squares I.4.6 Applications of Linear Regression in Finance I.4.6.1 Testing a Theory I.4.6.2 Analysing Empirical Market Behaviour I.4.6.3 Optimal Portfolio Allocation 134 134 136 137 139 140 143 143 144 144 146 148 149 151 152 153 155 155 156 157 157 158 158 159 161 162 163 163 166 167 170 171 173 175 175 176 176 177 178 179 179 180 181 Contents I.4.6.4 Regression-Based Hedge Ratios I.4.6.5 Trading on Regression Models I.4.7 Summary and Conclusions xi 181 182 184 I.5 Numerical Methods in Finance I.5.1 Introduction I.5.2 Iteration I.5.2.1 Method of Bisection I.5.2.2 Newton–Raphson Iteration I.5.2.3 Gradient Methods I.5.3 Interpolation and Extrapolation I.5.3.1 Linear and Bilinear Interpolation I.5.3.2 Polynomial Interpolation: Application to Currency Options I.5.3.3 Cubic Splines: Application to Yield Curves I.5.4 Optimization I.5.4.1 Least Squares Problems I.5.4.2 Likelihood Methods I.5.4.3 The EM Algorithm I.5.4.4 Case Study: Applying the EM Algorithm to Normal Mixture Densities I.5.5 Finite Difference Approximations I.5.5.1 First and Second Order Finite Differences I.5.5.2 Finite Difference Approximations for the Greeks I.5.5.3 Finite Difference Solutions to Partial Differential Equations I.5.6 Binomial Lattices I.5.6.1 Constructing the Lattice I.5.6.2 Arbitrage Free Pricing and Risk Neutral Valuation I.5.6.3 Pricing European Options I.5.6.4 Lognormal Asset Price Distributions I.5.6.5 Pricing American Options I.5.7 Monte Carlo Simulation I.5.7.1 Random Numbers I.5.7.2 Simulations from an Empirical or a Given Distribution I.5.7.3 Case Study: Generating Time Series of Lognormal Asset Prices I.5.7.4 Simulations on a System of Two Correlated Normal Returns I.5.7.5 Multivariate Normal and Student t Distributed Simulations I.5.8 Summary and Conclusions 185 185 187 187 188 191 193 193 195 197 200 201 202 203 I.6 Introduction to Portfolio Theory I.6.1 Introduction I.6.2 Utility Theory I.6.2.1 Properties of Utility Functions I.6.2.2 Risk Preference I.6.2.3 How to Determine the Risk Tolerance of an Investor I.6.2.4 Coefficients of Risk Aversion 225 225 226 226 229 230 231 203 206 206 207 208 210 211 211 212 213 215 217 217 217 218 220 220 223 xii Contents I.6.2.5 I.6.2.6 I.6.2.7 I.6.3 I.6.4 I.6.5 I.6.6 Some Standard Utility Functions Mean–Variance Criterion Extension of the Mean–Variance Criterion to Higher Moments Portfolio Allocation I.6.3.1 Portfolio Diversification I.6.3.2 Minimum Variance Portfolios I.6.3.3 The Markowitz Problem I.6.3.4 Minimum Variance Portfolios with Many Constraints I.6.3.5 Efficient Frontier I.6.3.6 Optimal Allocations Theory of Asset Pricing I.6.4.1 Capital Market Line I.6.4.2 Capital Asset Pricing Model I.6.4.3 Security Market Line I.6.4.4 Testing the CAPM I.6.4.5 Extensions to CAPM Risk Adjusted Performance Measures I.6.5.1 CAPM RAPMs I.6.5.2 Making Decisions Using the Sharpe Ratio I.6.5.3 Adjusting the Sharpe Ratio for Autocorrelation I.6.5.4 Adjusting the Sharpe Ratio for Higher Moments I.6.5.5 Generalized Sharpe Ratio I.6.5.6 Kappa Indices, Omega and Sortino Ratio Summary and Conclusions 232 234 235 237 238 240 244 245 246 247 250 250 252 253 254 255 256 257 258 259 260 262 263 266 References 269 Statistical Tables 273 Index 279 List of Figures A linear function The quadratic function fx = 4x2 + 3x + 2 I.1.3 The reciprocal function I.1.4 The inverse of a function I.1.5 The exponential function I.1.6 The natural logarithmic function I.1.7 Definition of the first derivative I.1.8 Two functions I.1.9 The definite integral I.1.10 The h-period log return is the sum of h consecutive one-period log returns I.1.11 Graph of the function in Example I.1.8 I.2.1 A matrix is a linear transformation I.2.2 A vector that is not an eigenvector I.2.3 An eigenvector I.2.4 Six European equity indices I.2.5 The first principal component I.3.1 Venn diagram I.3.2 Density and distribution functions: (a) discrete random variable; (b) continuous variable I.3.3 Building a histogram in Excel I.3.4 The effect of cell width on the histogram shape I.3.5 Two densities with the same expectation I.1.1 I.1.2 4 5 6 7 8 I.3.6 9 I.3.8 10 12 15 I.3.9 24 27 I.3.7 I.3.10 I.3.11 I.3.12 I.3.13 48 I.3.14 49 50 67 I.3.15 I.3.16 69 75 I.3.17 77 78 I.3.18 78 I.3.19 I.3.20 but different standard deviations (a) A normal density and a leptokurtic density; (b) a positively skewed density The 0.1 quantile of a continuous random variable Some binomial density functions A binomial tree for a stock price evolution The standard uniform distribution Two normal densities Lognormal density associated with the standard normal distribution A variance mixture of two normal densities A skewed, leptokurtic normal mixture density Comparison of Student t densities and standard normal Comparison of Student t density and normal with same variance Comparison of standardized empirical density with standardized Student t density and standard normal density The Excel t distribution function Filtering data to derive the GEV distribution A Fréchet density 80 83 84 86 87 89 90 93 95 97 98 99 99 100 102 103 xiv List of Figures I.3.21 Filtering data in the peaks-over-threshold model I.3.22 Kernel estimates of S&P 500 returns I.3.23 Scatter plots from a paired sample of returns: (a) correlation +075; (b) correlation 0; (c) correlation −075 I.3.24 Critical regions for hypothesis tests I.3.25 The dependence of the likelihood on parameters I.3.26 The likelihood and the log likelihood functions I.3.27 FTSE 100 index I.3.28 Daily prices and log prices of DJIA index I.3.29 Daily log returns on DJIA index I.4.1 Scatter plot of Amex and S&P 500 daily log returns I.4.2 Dialog box for Excel regression I.4.3 Unbiasedness and efficiency I.4.4 Distribution of a consistent estimator I.4.5 Billiton share price, Amex Oil index and CBOE Gold index I.4.6 Dialog box for multiple regression in Excel I.4.7 The iTraxx Europe index and its determinants I.4.8 Residuals from the Billiton regression I.5.1 Method of bisection I.5.2 Setting Excel’s Goal Seek I.5.3 Newton–Raphson iteration I.5.4 104 107 I.5.5 I.5.6 I.5.7 I.5.8 I.5.9 113 I.5.10 125 I.5.11 130 I.5.12 131 133 I.5.13 I.5.14 137 138 I.5.15 I.5.16 145 I.5.17 153 I.5.18 156 I.5.19 157 I.5.20 162 I.6.1 164 I.6.2 172 I.6.3 178 187 189 189 I.6.4 I.6.5 Convergence of Newton–Raphson scheme Solver options Extrapolation of a yield curve Linear interpolation on percentiles Fitting a currency smile A cubic spline interpolated yield curve FTSE 100 and S&P 500 index prices, 1996–2007 Sterling–US dollar exchange rate, 1996–2007 Slope of chord about a point Discretization of space for the finite difference scheme A simple finite difference scheme A binomial lattice Computing the price of European and American puts Simulating from a standard normal distribution Possible paths for an asset price following geometric Brownian motion A set of three independent standard normal simulations A set of three correlated normal simulations Convex, concave and linear utility functions The effect of correlation on portfolio volatility Portfolio volatility as a function of portfolio weight Portfolio risk and return as a function of portfolio weight Minimum variance portfolio 190 191 193 195 197 200 204 204 206 209 210 210 216 218 220 221 222 229 239 241 242 243 I.6.6 I.6.7 I.6.8 Solver settings for Example I.6.9 The opportunity set and the efficient frontier Indifference curves of risk averse investor List of Figures xv Indifference curves of risk loving investor I.6.10 Market portfolio I.6.11 Capital market line I.6.12 Security market line 249 251 251 253 I.6.9 246 247 248 List of Tables I.1.1 Asset prices I.1.2 Portfolio weights and portfolio value I.1.3 Portfolio returns I.2.1 Volatilities and correlations I.2.2 The correlation matrix of weekly returns I.2.3 Eigenvectors and eigenvalues of the correlation matrix I.3.1 Example of the density of a discrete random variable I.3.2 Distribution function for Table I.3.1 I.3.3 Biased and unbiased sample moments I.3.4 The B(3, 1/6) distribution I.3.5 A Poisson density function I.3.6 A simple bivariate density I.3.7 Distribution of the product I.3.8 Calculating a covariance I.3.9 Sample statistics I.4.1 Calculation of OLS estimates I.4.2 Estimating the residual sum of sqaures and the standard error of the regression I.4.3 Estimating the total sum of squares I.4.4 Critical values of t3 I.4.5 Some of the Excel output for the Amex and S&P 500 model 18 18 26 56 68 68 75 75 82 86 88 110 110 111 127 147 149 150 152 154 I.4.6 ANOVA for the Amex and S&P 500 model I.4.7 Coefficient estimates for the Amex and S&P 500 model I.4.8 ANOVA for Billiton regression I.4.9 Wald, LM and LR statistics I.5.1 Mean and volatility of the FTSE 100 and S&P 500 indices and the £/$ FX rate I.5.2 Estimated parameters of normal mixture distributions I.5.3 Analytic vs finite difference Greeks I.5.4 Characteristics of asset returns I.6.1 Two investments (outcomes as returns) I.6.2 Two investments (utility of outcomes) I.6.3 Returns characteristics for two portfolios I.6.4 Two investments I.6.5 Sharpe ratio and weak stochastic dominance I.6.6 Returns on an actively managed fund and its benchmark I.6.7 Statistics on excess returns I.6.8 Sharpe ratios and adjusted Sharpe ratios I.6.9 Kappa indices 154 154 164 167 205 205 208 221 227 228 237 258 259 261 262 262 264 List of Examples I.1.1 I.1.2 I.1.3 I.1.4 I.1.5 I.1.6 I.1.7 I.1.8 I.1.9 I.1.10 I.1.11 I.2.1 I.2.2 I.2.3 I.2.4 I.2.5 I.2.6 I.2.7 I.2.8 I.2.9 I.2.10 I.2.11 Roots of a quadratic equation Calculating derivatives Identifying stationary points A definite integral Portfolio weights Returns on a long-short portfolio Portfolio returns Stationary points of a function of two variables Constrained optimization Total derivative of a function of three variables Taylor approximation Finding a matrix product using Excel Calculating a 4 × 4 determinant Finding the determinant and the inverse matrix using Excel Solving a system of simultaneous linear equations in Excel A quadratic form in Excel Positive definiteness Determinant test for positive definiteness Finding eigenvalues and eigenvectors Finding eigenvectors Using an Excel add-in to find eigenvectors and eigenvalues Covariance and correlation matrices 5 12 14 16 18 20 25 28 30 31 32 40 42 43 45 45 46 47 51 53 54 56 I.2.12 Volatility of returns and volatility of P&L I.2.13 A non-positive definite 3 × 3 matrix I.2.14 Eigenvectors and eigenvalues of a 2 × 2 covariance matrix I.2.15 Spectral decomposition of a correlation matrix I.2.16 The Cholesky matrix of a 2 × 2 matrix I.2.17 The Cholesky matrix of a 3 × 3 matrix I.2.18 Finding the Cholesky matrix in Excel I.2.19 Finding the LU decomposition in Excel I.3.1 Building a histogram I.3.2 Calculating moments of a distribution I.3.3 Calculating moments of a sample I.3.4 Evolution of an asset price I.3.5 Normal probabilities I.3.6 Normal probabilities for portfolio returns I.3.7 Normal probabilities for active returns I.3.8 Variance and kurtosis of a zero-expectation normal mixture I.3.9 Probabilities of normal mixture variables I.3.10 Calculating a covariance I.3.11 Calculating a correlation I.3.12 Normal confidence intervals 57 59 60 61 62 63 63 64 77 81 82 87 90 91 92 95 96 110 112 119 xviii List of Examples I.3.13 One- and two-sided confidence intervals I.3.14 Confidence interval for a population mean I.3.15 Testing for equality of means and variances I.3.16 Log likelihood of the normal density I.3.17 Fitting a Student t distribution by maximum likelihood I.4.1 Using the OLS formula I.4.2 Relationship between beta and correlation I.4.3 Estimating the OLS standard error of the regression I.4.4 ANOVA I.4.5 Hypothesis tests in a simple linear model I.4.6 Simple regression in matrix form I.4.7 Goodness-of-fit test in multiple regression I.4.8 Testing a simple hypothesis in multiple regression I.4.9 Testing a linear restriction I.4.10 Confidence interval for regression coefficient I.4.11 Prediction in multivariate regression I.4.12 Durbin–Watson test I.4.13 White’s heteroscedasticity test I.5.1 Excel’s Goal Seek I.5.2 Using Solver to find a bond yield I.5.3 Interpolating implied volatility I.5.4 Bilinear interpolation I.5.5 120 I.5.6 123 I.5.7 127 131 I.5.8 I.5.9 132 147 147 148 150 151 160 I.5.10 I.6.1 I.6.2 I.6.3 I.6.4 I.6.5 I.6.6 164 165 165 168 I.6.7 I.6.8 I.6.9 169 177 I.6.10 I.6.11 177 188 I.6.12 191 I.6.13 I.6.14 194 194 I.6.15 Fitting a 25-delta currency option smile Interpolation with cubic splines Finite difference approximation to delta, gamma and vega Pricing European call and put options Pricing an American option with a binomial lattice Simulations from correlated Student t distributed variables Expected utility Certain equivalents Portfolio allocations for an exponential investor Higher moment criterion for an exponential investor Minimum variance portfolio: two assets Minimum variance portfolio on S&P 100 and FTSE 100 General formula for minimum variance portfolio The Markowitz problem Minimum variance portfolio with many constraints The CML equation Stochastic dominance and the Sharpe ratio Adjusting a Sharpe ratio for autocorrelation Adjusted Sharpe ratio Computing a generalized Sharpe ratio Omega, Sortino and kappa indices 196 198 208 212 215 222 227 228 235 236 241 242 244 245 246 252 258 260 261 263 264 Foreword How many children dream of one day becoming risk managers?

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Another application of differentiation is to the optimal allocation problem for an investor who faces certain constraints, such as no short sales and/or at least 30% of his funds must be invested in US equities. The investor’s problem is to choose his portfolio weights to optimize his objective whilst respecting his constraints. This falls into the class of constrained optimization problems, problems that are solved using differentiation. Risk is the uncertainty about an expected value, and a risk-averse investor wants to achieve the maximum possible return with the minimum possible risk. Standard measures of portfolio risk are the variance of a portfolio and its square root which is called the portfolio volatility.6 The portfolio variance is a quadratic function of the portfolio weights.

The Master Algorithm: How the Quest for the Ultimate Learning Machine Will Remake Our World
by Pedro Domingos
Published 21 Sep 2015

Or, equivalently, we can minimize the weights under the constraint that all examples have a given margin, which could be one—the precise value is arbitrary. This is what SVMs usually do. Constrained optimization is the problem of maximizing or minimizing a function subject to constraints. The universe maximizes entropy subject to keeping energy constant. Problems of this type are widespread in business and technology. For example, we may want to maximize the number of widgets a factory produces, subject to the number of machine tools available, the widgets’ specs, and so on. With SVMs, constrained optimization became crucial for machine learning as well. Unconstrained optimization is getting to the top of the mountain, and that’s what gradient descent (or, in this case, ascent) does.

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Unconstrained optimization is getting to the top of the mountain, and that’s what gradient descent (or, in this case, ascent) does. Constrained optimization is going as high as you can while staying on the road. If the road goes up to the very top, the constrained and unconstrained problems have the same solution. More often, though, the road zigzags up the mountain and then back down without ever reaching the top. You know you’ve reached the highest point on the road when you can’t go any higher without driving off the road; in other words, when the path to the top is at right angles to the road. If the road and the path to the top form an oblique angle, you can always get higher by driving farther along the road, even if that doesn’t get you higher as quickly as aiming straight for the top of the mountain.

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If the road and the path to the top form an oblique angle, you can always get higher by driving farther along the road, even if that doesn’t get you higher as quickly as aiming straight for the top of the mountain. So the way to solve a constrained optimization problem is to follow not the gradient but the part of it that’s parallel to the constraint surface—in this case the road—and stop when that part is zero. In general, we have to deal with many constraints at once (one per example, in the case of SVMs). Suppose you wanted to get as close as possible to the North Pole but couldn’t leave your room. Each of the room’s four walls is a constraint, and the solution is to follow the compass until you bump into the corner where the northeast and northwest walls meet.

Hands-On Machine Learning With Scikit-Learn, Keras, and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems
by Aurelien Geron
Published 14 Aug 2019

The resulting vector p will contain the bias term b = p0 and the feature weights wi = pi for i = 1, 2, ⋯, n. Similarly, you can use a QP solver to solve the soft margin problem (see the exercises at the end of the chapter). However, to use the kernel trick we are going to look at a different constrained optimization problem. The Dual Problem Given a constrained optimization problem, known as the primal problem, it is possible to express a different but closely related problem, called its dual problem. The solution to the dual problem typically gives a lower bound to the solution of the primal problem, but under some conditions it can even have the same solutions as the primal problem.

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However, if we also want to avoid any margin violation (hard margin), then we need the decision function to be greater than 1 for all positive training instances, and lower than –1 for negative training instances. If we define t(i) = –1 for negative instances (if y(i) = 0) and t(i) = 1 for positive instances (if y(i) = 1), then we can express this constraint as t(i)(wT x(i) + b) ≥ 1 for all instances. We can therefore express the hard margin linear SVM classifier objective as the constrained optimization problem in Equation 5-3. Equation 5-3. Hard margin linear SVM classifier objective Note We are minimizing wT w, which is equal to ∥ w ∥2, rather than minimizing ∥ w ∥. Indeed, ∥ w ∥2 has a nice and simple derivative (it is just w) while ∥ w ∥ is not differentiable at w = 0. Optimization algorithms work much better on differentiable functions.

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We now have two conflicting objectives: making the slack variables as small as possible to reduce the margin violations, and making wT w as small as possible to increase the margin. This is where the C hyperparameter comes in: it allows us to define the tradeoff between these two objectives. This gives us the constrained optimization problem in Equation 5-4. Equation 5-4. Soft margin linear SVM classifier objective Quadratic Programming The hard margin and soft margin problems are both convex quadratic optimization problems with linear constraints. Such problems are known as Quadratic Programming (QP) problems.

Solutions Manual - a Primer for the Mathematics of Financial Engineering, Second Edition
by Dan Stefanica
Published 24 Mar 2011

Using Chain Rule and (7.31) we find that 一兰 θσimp(K) 2 θ-y- = 在 + e一叩 (d 2 ) ， dry = In (去)十 (r - q)T 一切 (K)VT h .. (j imp(K)VT 2' ~ 8.1 Solutions to Chapter 8 Exercises Problem 1: Find the maximum and mi山num ofthe function f(Xl' X2 ， 句)= 4X2 - 2X3 subject to the constraints 2X1 - X2 - X3 = 0 and xI + x~ = 13. Solution: We reformulate the problem as a constrained optimization problem. Let f : JR3 • JR and 9 : JR3 • JR be defined as follows: 削 = 4X2 - 2叫仰) = (专JZ二3) where x = (X l, X2 , X3). We want to 五nd the maximum and minimum of f (x) on JRδsubject to the constraint g(叫 =0. We 自rst check that ra出(飞7 g(x)) = 1 for any x such that g(x) = O. Note that \1 g(x) = (2;，二 en It is easy to see that rank( \7 g( x)) = 2, unless Xl = X2 = 0, in which case g(x) 并 O.

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However , the following relationship between the weights must hold true: si口ce Pi ,4 ω1 十 ω2 十 ω3+ω4 = 1. (8.6) (i) We are looki吨 for a portfolio with given expected rate of return E[R] = 0.12 and minimal variance of the rate of return. Using (8 .4-8.6) , we obtain that this problem can be written as the following constrained optimization problem: 且ndω° such that gErof(ω) = f(ω0) ， + 6X2 + 13 and 2 D 几位 ) E[R] = ω1μl 十 ω2μ2+ω3μ3 十 ω4μ4; var(R) = ω?σ? 十 ω2σ~+ω;σ; 十 ωiσ~ I D 2 凡 (x) 工 181 (i) Find the asset allocation for a mi山nal variance portfolio with 12% expected rate of return; 000020·7 。; hL ，，" ， 一­ 4EA diU2' 们向 Cο.

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σ1σ2ρ口内 σ3P1 ，3 月 (ω ) = 21σ1σ2P1， 2 作 σ2匀2，3 飞 is the following (row) vector: 183 is the vector of the expected values of the rates of return of the four assets. The problem of 五日ding a portfolio with m以imum expected rate of return and standard deviation of the rate of return equal to σp can be formulated as a constrained optimization problem as follows: find Wo such that ~~n_ f(ω) = f(ω。)， g(ω)=0 (8.14) 184 CHAPTER 8. LAGRANGE MULTIPLIERS. NEWTON'S METHOD. 8.1. where ω = (Wi)仨吐 However , it is easy to see that f(ω) (ωtkLz)7 σp (8.16) We can now proceed with finding the portfolio with maximum expected return using the Lagrange multipliers method.

Optimization Methods in Finance
by Gerard Cornuejols and Reha Tutuncu
Published 2 Jan 2006

Decision variables, the objective function, and constraints are three essential elements of any optimization problem. Some problems may lack constraints so that any set of decision variables (of appropriate dimension) are acceptable as alternative solutions. Such problems are called unconstrained optimization problems, while others are often referred to as constrained optimization problems. There are problem instances with no objective functions–the so-called feasibility problems, and others with multiple objective functions. Such problems are often addressed by reduction to a single or a sequence of single-objective optimization problems. If the decision variables in an optimization problem are restricted to integers, or to a discrete set of possibilities, we have an integer or discrete optimization problem.

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If (x̂, κ̂) is the solution to (5.7), then x∗ = κ̂x̂ . This last problem can be solved using the techniques we discussed for convex quadratic programming problems. 5.3. RETURNS-BASED STYLE ANALYSIS 5.3 63 Returns-Based Style Analysis In two ground-breaking articles, Sharpe described how constrained optimization techniques can be used to determine the effective asset mix of a fund using only the return time series for the fund and a number of carefully chosen asset classes [13, 14]. Often, passive indices or index funds are used to represent the chosen asset classes and one tries to determine a portfolio of these funds/indices whose returns provide the best match for the returns of the fund being analyzed.

The Inner Lives of Markets: How People Shape Them—And They Shape Us
by Tim Sullivan
Published 6 Jun 2016

Suppose you want to help people commute safely and efficiently across the East River, from Brooklyn to Manhattan. Traditional economics is the equivalent of assuming that the only two ways of doing so are bridge or ferry. Mechanism design imagines the broad set of possibilities—zip line, catapult, people mover—then figures out which will work best. It’s what, in technical terms, is called constrained optimization (the same technique Vickrey deployed to determine that the best way to get to work at Columbia was on roller skates). Mechanism designers consider the restrictions imposed by laws, human nature, our sense of right and wrong, and the strategizing that kidney patients, school applicants, and others may engage in to get a better organ or education.

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., 169–170, 172 Camp, Garrett, 170 candle auctions, 82 capitalism, free-market, 172–173 car service platform, 169–171 cash-back bonus, 116 cash-for-sludge transactions, 167–169 See also Summers, Larry centralized clearinghouses, 140–141 Champagne fairs, 105–106, 126–128 Changi POW camp, 175–177 Le Chatelier, Henry Louis, 29 Le Chatelier’s principle, 29 cheap talk, 62–66, 69 chess, difference between Cold War and, 26 See also poker, bluffing in child labor, 180 cigarettes, as currency in German POW camp, 8–9 Clarke, Edward, 93 Clavell, James, 175 clerkship offers, with federal judges, 140 coat hook, 151–152, 174 Codes of the Underworld (Gambetta), 68 Cold War, difference between chess and, 26 See also poker, bluffing in Collectible Supplies, 128–129 “College Admissions and the Stability of Marriage” (Gale and Shapley), 137 commitment, signs of, 62–63, 69–71, 72–75 community game, 178–179 competition models of, 35, 166, 172–173 platform, 124–126 unethical conduct with, 180–181 “Competition is for Losers” (Thiel), 173 competitive equilibrium, existence of, 29, 31–34, 36–37, 40, 45, 76 competitive markets, 35, 124–126, 172–174, 180–181 See also platforms competitive signaling, 70–71 congestion pricing model, 86, 94 constrained optimization, 85–86, 133 contractorsfromhell.com, 120 copycat competitors, 172–173 corporate philanthropy, 72–75 Cowles, Alfred, 25, 27 Cowles Commission for Research in Economics, 25, 27, 31, 134 “creative destruction,” 50 credit card platforms, 113–116, 123–124 criminal organizations, informational challenges of, 68 currency, at Stalag VII-A POW camp, 8–9 customer feedback, 52, 74–75 Davis, Harry, 154, 157 Debreu, Gérard, 20, 24, 25, 32–33, 36–37 decentralized match, 139–140 deferred acceptance algorithm, 137–141, 145–149 Delmonico, Frank, 164 descending price auctions, 81–82 design, auction, 14, 101–102 Digital Dealing (Hall), 94 Discover card, 115–116 distribution of income, 22 Domar, Evsey, 36–37 Dorosin, Neil, 142–144 Douglas Aircraft Company, 25 Dow, Bob, 1–2 Dow, Edna, 1–2 Drèze, Jacques, 85–86 dumping toxic waste, transactions for, 167–169 Dutch auctions, 81–82 dysfunction, market, 36, 75–77, 143 eBay adverse selection on, 51–55, 57 auction listings, 94–97 concerns on model for, 43, 46, 48 on seller motivation for giving to charities, 73–75 start of, 39–41 as two-sided market, 109, 119 e-commerce, 41–43, 52–55 “The Economic Organization of a P.O.W.

Lean Analytics: Use Data to Build a Better Startup Faster
by Alistair Croll and Benjamin Yoskovitz
Published 1 Mar 2013

A machine can find the optimal settings for something, but only within the constraints and problem space of which it’s aware, in much the same way that the water in a mountainside lake can’t find the lowest possible value, just the lowest value within the constraints provided. To understand the problem with constrained optimization, imagine that you’re given three wheels and asked to evolve the best, most stable vehicle. After many iterations of pitting different wheel layouts against one another, you come up with a tricycle-like configuration. It’s the optimal three-wheeled configuration. Data-driven optimization can perform this kind of iterative improvement.

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cohort analysis about, Segments, Cohorts, A/B Testing, and Multivariate Analysis, A/B and Multivariate Testing engagement tunnel and, Content Creation and Interaction Columbo (TV show), How to Avoid Leading the Witness communications apps, Notification Effectiveness company vision, How Rally Builds New Features with a Lean Approach Concierge Minimum Viable Product (MVP), Poking a Hole in Your Reality Distortion Field Confronting Reality (Bossidy and Charan), Span of Control and the Railroads constrained optimization, Data-Driven Versus Data-Informed consulting organizations, Inspiration content creation metric, Model Five: User-Generated Content, Wrinkles: Passive Content Creation content quality, Spam and Bad Content content sharing metric, Model Five: User-Generated Content, Value of Created Content content upload success metric, Content Upload Success content/advertising balance metric, Model Four: Media Site, Ad Rates Continuum agency, Stars, Dogs, Cows, and Question Marks convergent problem interviews, Convergent and Divergent Problem Interviews, Convergent and Divergent Problem Interviews conversion funnels about, The Long Funnel e-commerce business model, Model One: E-commerce positioning leading indicators in, What Makes a Good Leading Indicator?

Misbehaving: The Making of Behavioral Economics
by Richard H. Thaler
Published 10 May 2015

That is, we choose on the basis of what economists call “rational expectations.” If people starting new businesses on average believe that their chance of succeeding is 75%, then that should be a good estimate of the actual number that do succeed. Econs are not overconfident. This premise of constrained optimization, that is, choosing the best from a limited budget, is combined with the other major workhorse of economic theory, that of equilibrium. In competitive markets where prices are free to move up and down, those prices fluctuate in such a way that supply equals demand. To simplify somewhat, we can say that Optimization + Equilibrium = Economics.

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(Lamont and Thaler), 250 capital asset pricing model (CAPM), 226–29, 348 “CAPM is Wanted, Dead or Alive, The” (Fama and French), 228 Car Talk, 32 Case, Chip, 235 Case-Shiller Home Price Index, 235 cashews, 21, 24, 42, 85–86, 92, 100, 102–3, 107n casinos, 49n cautious paternalism, 323 Census Bureau, 47 Center for Research in Security Prices (CRSP), 208, 221 charity, 66, 129 cheap stocks, 219–21 Checklist Manifesto, The (Gawande), 356 Chen, Nai-fu, 243 Chetty, Raj, 320, 357–58 Chicago, University of, 255–56 behavioral economics conference at, 159–64, 167–68, 169, 170, 205 conference on 1987 crash at, 237 debate on behavioral economics at, 159–63, 167–68, 169, 170, 205 finance studied at, 208 offices at, 270–76, 278 Chicago Bulls, 19 Chicago police department, 260 chicken (game of), 183 choice: number of, 21, 85, 99–103 preferences revealed by, 86 choice architecture, 276, 326–27, 357 Choices, Values, and Frames, xiv Chrysler, 121, 123, 363 Cialdini, Robert, 180, 335, 336 Clegg, Nick, 333 Clinton, Hillary, 22 closed-end funds, 238–39, 239, 240 puzzles of, 240–43, 244, 250 coaches, 292–93 Coase, Ronald, 261 Coase theorem, 261–62, 264–65, 264, 267–68 Cobb, David, 115 Cobb, Michael, 115, 116, 117, 118n, 119, 120, 123 Coca-Cola, 134–35 cognitive dissonance, 178 commitment strategies, 100, 102–3, 106–7 compliance (medical), 189–90 COMPUSTAT, 221 computing power, 208 concert tickets, 18–19, 66 conditional cooperators, 146, 182, 335n “Conference Handbook, The” (Stigler), 162–63 confirmation bias, 171–72 Conservative Party, U.K., 330–33 constrained optimization, 5–6, 8, 27, 43, 161, 207, 365 “Consumer Choice: A Theory of Economists’ Behavior” (Thaler), 35 consumers, optimization problem faced by, 5–6, 8, 27, 43, 161, 207, 365 consumer sovereignty, 268–69 consumer surplus, 59 consumption function, 94–98, 106, 309 “Contrarian Investment, Extrapolation, and Risk” (Lakonishok, Shleifer and Vishny), 228 cooperation, 143–47 conditional, 146, 182, 335n Prisoner’s Dilemma and, 143–44, 145, 301–5, 302 Copernican revolution, 169 Cornell University, 42, 43, 115, 140–43, 153–55, 157 Costco, 63, 71–72 Council of Economic Advisors, 352 coupons, 62, 63, 67–68, 120 credit cards, 18, 74, 76–77 late fees for, 360 crime, 265 Daily Mail, 135 Daily Show, The, 352 Dallas Cowboys, 281 data: financial, 208 collection and recording of, 355–56 Dawes, Robyn, 146 Deal or No Deal, 296–301, 297, 303 path dependence on, 298–300 deals, 61–62 De Bondt, Werner, 216–18, 221, 222–24, 226n, 233, 278 debt, 78 default investment portfolio, 316 default option, 313–16, 327 default saving rate, 312, 316, 319, 357 delayed gratification, 100–102 De Long, Brad, 240 Demos, 330 Denmark, 320, 357–58 descriptive, 25, 30, 45, 89 Design of Everyday Things, The (Norman), 326 Diamond, Doug, 273, 276 Diamond, Peter, 323 Dictator Game, 140–41, 142, 160, 182, 301 diets, 342 diminishing marginal utility, 106 of wealth, 28, 30 diminishing sensitivity, 30–34 discount, surcharge vs., 18 discounts, returns and, 242–43 discounted utility model, 89–94, 99, 110, 362 discretion, 106 Ditka, Mike, 279, 280 dividends, 164–67, 365 present value of, 231–33, 231, 237 Dodd, David, 219 doers, planners vs., 104–9 Donoghue, John, 265n “Do Stock Prices Move Too Much to be Justified by Subsequent Changes in Dividends?”

Artificial Intelligence: A Modern Approach
by Stuart Russell and Peter Norvig
Published 14 Jul 2019

Random restarts and simulated annealing are often helpful. High-dimensional continuous spaces are, however, big places in which it is very easy to get lost. A final topic is constrained optimization. An optimization problem is constrained if solutions must satisfy some hard constraints on the values of the variables. For example, in our airport-siting problem, we might constrain sites to be inside Romania and on dry land (rather than in the middle of lakes). The difficulty of constrained optimization problems depends on the nature of the constraints and the objective function. The best-known category is that of linear programming problems, in which constraints must be linear inequalities forming a convex set4 and the objective function is also linear.

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Preference constraints can often be encoded as costs on individual variable assignments—for example, assigning an afternoon slot for Prof. R costs 2 points against the overall objective function, whereas a morning slot costs 1. With this formulation, CSPs with preferences can be solved with optimization search methods, either path-based or local. We call such a problem a constrained optimization problem, or COP. Linear programs are one class of COPs. 5.2Constraint Propagation: Inference in CSPs An atomic state-space search algorithm makes progress in only one way: by expanding a node to visit the successors. A CSP algorithm has choices. It can generate successors by choosing a new variable assignment, or it can do a specific type of inference called constraint propagation: using the constraints to reduce the number of legal values for a variable, which in turn can reduce the legal values for another variable, and so on.

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The primary conference venue is the International Conference on Principles and Practice of Constraint Programming, often called CP. 1We have been using the term “edge” rather than “arc,” so it would make more sense to call this “edge-consistent,” but the name “arc-consistent” is historical. 2Local search can easily be extended to constrained optimization problems (COPs). In that case, all the techniques for hill climbing and simulated annealing can be applied to optimize the objective function. 3A careful cartographer or patriotic Tasmanian might object that Tasmania should not be colored the same as its nearest mainland neighbor, to avoid the impression that it might be part of that state. 4Sadly, very few regions of the world have tree-structured maps, although Sulawesi comes close.

Mastering Machine Learning With Scikit-Learn
by Gavin Hackeling
Published 31 Oct 2014

It is necessary to normalize the functional margins as they can be scaled by using Z , which is problematic for training. When Z is a unit vector, the geometric margin is equal to the functional vector. We can now formalize our definition of the best decision boundary as having the greatest geometric margin. The model parameters that maximize the geometric margin can be solved through the following constrained optimization problem: min 1 w, w n subject to : yi ( w, xi + b ) ≥ 1 A useful property of support vector machines is that this optimization problem is convex; it has a single local minimum that is also the global minimum. While the proof is beyond the scope of this chapter, the previous optimization problem can be written using the dual form of the model to accommodate kernels as follows: W (α ) = ∑ α i − i 1 ∑ α iα j yi yi K ( xi , x j ) 2 i, j n subject to : ∑ yiα i = 0 i =1 subject to : α i ≥ 0 Finding the parameters that maximize the geometric margin subject to the constraints that all of the positive instances have functional margins of at least 1 and all of the negative instances have functional margins of at most -1 is a quadratic programming problem.

Algorithms to Live By: The Computer Science of Human Decisions
by Brian Christian and Tom Griffiths
Published 4 Apr 2016

Being a circuit lawyer meant literally making a circuit—moving through towns in fourteen different counties to try cases, riding hundreds of miles over many weeks. Planning these circuits raised a natural challenge: how to visit all the necessary towns while covering as few miles as possible and without going to any town twice. This is an instance of what’s known to mathematicians and computer scientists as a “constrained optimization” problem: how to find the single best arrangement of a set of variables, given particular rules and a scorekeeping measure. In fact, it’s the most famous optimization problem of them all. If it had been studied in the nineteenth century it might have become forever known as “the prairie lawyer problem,” and if it had first come up in the twenty-first century it might have been nicknamed “the delivery drone problem.”

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See also language; networking; storytelling confirmation priors and community-supported agriculture (CSA) Comparison Counting Sort comparison-shopping websites complexity penalizing computation, defined by Turing computational kindness confidence interval confirmation congestion avoidance of price of anarchy and Connection Machine constant-time (O(1)) constrained optimization problems constrained problem, preferences for Constraint Relaxation construction projects content distribution networks (CDNs) context switching continuous optimization problems Continuous Relaxation control without hierarchy Cooper, Martin cooperation Copernican Principle Copernicus, Nicolaus corporate marketing cost-benefit analysis Cramer, Jim Cravath system creativity crêpe stand queue Cross-Validation cryptography customer service hold times Darwin, Charles data.

Statistical Arbitrage: Algorithmic Trading Insights and Techniques
by Andrew Pole
Published 14 Sep 2007

This new element, risk, complicates the goal, which now becomes twofold: Maximize expected return and maintain the risk of achieving that return below a certain tolerance. So far so good. Going from foresight to forecast we exchange certainty for uncertainty; we move from guaranteed optimization to constrained optimization of a best guess. However, in practice matters are not quite as straightforward as that sentence seems to imply. The first obstacle is precisely specifying the notion of risk—or, at least, its practical implementation. Risk arises because there is no guarantee that a particular forecast will be borne out in reality.

Lean In: Women, Work, and the Will to Lead
by Sheryl Sandberg
Published 11 Mar 2013

The very concept of having it all flies in the face of the basic laws of economics and common sense. As Sharon Poczter, professor of economics at Cornell, explains, “The antiquated rhetoric of ‘having it all’ disregards the basis of every economic relationship: the idea of trade-offs. All of us are dealing with the constrained optimization that is life, attempting to maximize our utility based on parameters like career, kids, relationships, etc., doing our best to allocate the resource of time. Due to the scarcity of this resource, therefore, none of us can ‘have it all,’ and those who claim to are most likely lying.”1 “Having it all” is best regarded as a myth.

Cogs and Monsters: What Economics Is, and What It Should Be
by Diane Coyle
Published 11 Oct 2021

In other words, the computer agent teaches itself from certain sensory inputs—such as the location of pixels on screen in a game—and its own experience of whether what happens next adds to its own score. The agents were designed to make decisions like homo economicus, rational actors in a classic economic model of constrained optimization, in other words maximising their score subject to the availability of apples, interacting with each other over time as the game played out. Each formed part of the environment to which it had to respond. All was harmony when apples were plentiful. But when they became scarcer, the AIs became more aggressive, ultimately attacking each other.

Finding Alphas: A Quantitative Approach to Building Trading Strategies
by Igor Tulchinsky
Published 30 Sep 2019

In the case of position concentration, this can be easily achieved (assuming there are no constraints on short positions) by subtracting the group mean from the individual positions, by orthogonalizing the position vector to the factor vector, or by subtracting beta times the factor. Dollar-neutral or industry-neutral positions are achieved by hard neutralization. Soft neutralization consists of capping the exposure to the given risk, either by subtracting a portion of the exposure or by using a constrained optimization method to produce the positions. Hedging consists of using one instrument or set of instruments as a hedge against the risk incurred by other instruments or sets of instruments. For instance, one can hedge the market beta of an equity portfolio via S&P 500 futures or exchange-traded funds, or the currency risk of a global bond portfolio via currency spots or futures.

Python for Finance
by Yuxing Yan
Published 24 Apr 2014

Our major function would start from Step 3 as shown in the following code: # Step 3: generate a return matrix (annul return) n=len(ticker) # number of stocks x2=ret_annual(ticker[0],begdate,enddate) for i in range(1,n): x_=ret_annual(ticker[i],begdate,enddate) x2=pd.merge(x2,x_,left_index=True,right_index=True) # using scipy array format R = sp.array(x2) print('Efficient porfolio (mean-variance) :ticker used') print(ticker) [ 216 ] Chapter 8 print('Sharpe ratio for an equal-weighted portfolio') equal_w=sp.ones(n, dtype=float) * 1.0 /n print(equal_w) print(sharpe(R,equal_w)) # for n stocks, we could only choose n-1 weights w0= sp.ones(n-1, dtype=float) * 1.0 /n w1 = fmin(negative_sharpe_n_minus_1_stock,w0) final_w = sp.append(w1, 1 - sum(w1)) final_sharpe = sharpe(R,final_w) print ('Optimal weights are ') print (final_w) print ('final Sharpe ratio is ') print(final_sharpe) From the following output, we know that if we use a naïve equal-weighted strategy, the Sharpe ratio is 0.63. However, the Sharpe ratio for our optimal portfolio is 0.67: Constructing an efficient frontier with n stocks Constructing an efficient frontier is always one of the most difficult tasks for finance instructors since the task involves matrix manipulation and a constrained optimization procedure. One efficient frontier could vividly explain the Markowitz Portfolio theory. The following Python program uses five stocks to construct an efficient frontier: from matplotlib.finance import quotes_historical_yahoo import numpy as np import pandas as pd [ 217 ] Statistical Analysis of Time Series from numpy.linalg import inv, pinv # Step 1: input area begYear,endYear = 2001,2013 stocks=['IBM','WMT','AAPL','C','MSFT'] # Step 2: define a few functions # function 1 def ret_monthly(ticker): x = quotes_historical_yahoo(ticker,(begYear,1,1),(endYear,12,31),asob ject=True,adjusted=True) logret=log(x.aclose[1:]/x.aclose[:-1]) date=[] d0=x.date for i in range(0,size(logret)): date.append(''.join([d0[i].strftime("%Y"),d0[i].strftime("%m")])) y=pd.DataFrame(logret,date,columns=[ticker]) return y.groupby(y.index).sum() # function 2: objective function def objFunction(W, R, target_ret): stock_mean=np.mean(R,axis=0) port_mean=np.dot(W,stock_mean) # portfolio mean cov=np.cov(R.T) # var-cov matrix port_var=np.dot(np.dot(W,cov),W.T) # portfolio variance penalty = 2000*abs(port_mean-target_ret)# penalty 4 deviation return np.sqrt(port_var) + penalty # objective function # Step 3: Generate a return matrix R R0=ret_monthly(stocks[0]) # starting from 1st stock n_stock=len(stocks) # number of stocks for i in xrange(1,n_stock): # then merge with other stocks x=ret_monthly(stocks[i]) R0=pd.merge(R0,x,left_index=True,right_index=True) R=np.array(R0) # Step 4: estimate optimal portfolio for a given return out_mean,out_std,out_weight=[],[],[] stockMean=np.mean(R,axis=0) for r in np.linspace(np.min(stockMean), np.max(stockMean), num=100): [ 218 ] Chapter 8 W = ones([n_stock])/n_stock # starting from equal weights b_ = [(0,1) for i in range(n_stock)] # bounds, here no short c_ = ({'type':'eq', 'fun': lambda W: sum(W)-1. })#constraint result=sp.optimize.minimize(objFunction,W,(R,r),method='SLSQP',constr aints=c_, bounds=b_) if not result.success: # handle error raise BaseException(result.message) out_mean.append(round(r,4)) # 4 decimal places std_=round(np.std(np.sum(R*result.x,axis=1)),6) out_std.append(std_) out_weight.append(result.x) # Step 4: plot the efficient frontier title('Efficient Frontier') xlabel('Standard Deviation of the porfolio (Risk))') ylabel('Return of the portfolio') figtext(0.5,0.75,str(n_stock)+' stock are used: ') figtext(0.5,0.7,' '+str(stocks)) figtext(0.5,0.65,'Time period: '+str(begYear)+' ------ '+str(endYear)) plot(out_std,out_mean,'--') The output graph is presented as follows: [ 219 ] Statistical Analysis of Time Series Understanding the interpolation technique Interpolation is a technique used quite frequently in finance.

The Deep Learning Revolution (The MIT Press)
by Terrence J. Sejnowski
Published 27 Sep 2018

Kuhl, “Birdsong and Human Speech: Common Themes and Mechanisms,” Annual Review of Neuroscience 22 (1999): 567–631. 306 Notes 22. G. Turrigiano, “Too Many Cooks? Intrinsic and Synaptic Homeostatic Mechanisms in Cortical Circuit Refinement,” Annual Review of Neuroscience 34 (2011):89– 103. 23. L. Wiskott and T. J. Sejnowski, “Constrained Optimization for Neural Map Formation: A Unifying Framework for Weight Growth and Normalization,” Neural Computation 10, no. 3 (1998): 671–716. 24. A. J. Bell, “Self-Organization in Real Neurons: Anti-Hebb in ‘Channel Space’?” Advances in Neural Information Processing Systems 4 (1991): 59–66; M. Siegel, E.

The Blockchain Alternative: Rethinking Macroeconomic Policy and Economic Theory
by Kariappa Bheemaiah
Published 26 Feb 2017

Each individual represents a point in a search space and a possible solution. The individuals in the population are then made to go through a process of evolution. GAs are used for searching through large and complex data sets as they are capable of finding reasonable solutions to complex issues by solving unconstrained and constrained optimization issues. They are used to solve problems that are not well suited for standard optimization algorithms, including problems in which the objective function is discontinuous, nondifferentiable, stochastic, or highly nonlinear. GAs also work particularly well on mixed (continuous and discrete) combinatorial problems, as they are less susceptible to getting ‘stuck’ at local optima than classical gradient search methods.

What to Think About Machines That Think: Today's Leading Thinkers on the Age of Machine Intelligence
by John Brockman
Published 5 Oct 2015

There are other AI algorithms, but most fall into categories Minsky wrote about. One example is running Bayesian probability algorithms on search trees or graphs. They have to grapple with exponential branching or some related form of the curse of dimensionality. Another example is convex or other nonlinear constrained optimization for pattern classification. Italian mathematician Joseph-Louis Lagrange found the general solution algorithm we still use today. He came up with it in 1811. Clever tricks and tweaks will always help. But progress here depends crucially on running these algorithms on ever faster computers.

The Art of SQL
by Stephane Faroult and Peter Robson
Published 2 Mar 2006

Take note that the good plan only accesses indexes, not tables: select o.id_outstanding, ap.cde_portfolio, ap.cde_expense, ap.branch_code, to_char(sum(ap.amt_book_round + ap.amt_book_acr_ad - ap.amt_acr_nt_pst)), to_char(sum(ap.amt_mnl_bk_adj)), o.cde_outstd_typ from accrual_port ap, accrual_cycle ac, outstanding o, deal d, facility f, branch b where ac.id_owner = o.id_outstandng and ac.id_acr_cycle = ap.id_owner and o.cde_outstd_typ in ('LOAN', 'DCTLN', 'ITRLN', 'DEPOS', 'SLOAN', 'REPOL') and d.id_deal = o.id_deal and d.acct_enabl_ind = 'Y' and (o.cde_ob_st_ctg = 'ACTUA' or o.id_outstanding in (select id_owner from subledger)) and o.id_facility = f.id_facility and f.branch_code = b.branch_code and b.cde_tme_region = 'ZONE2' group by o.id_outstanding, ap.cde_portfolio, ap.cde_expense, ap.branch_code, o.cde_outstd_typ having sum(ap.amt_book_round + ap.amt_book_acr_ad - ap.amt_acr_nt_pst) <> 0 or (sum(ap.amt_mnl_bk_adj) is not null and sum(ap.amt_mnl_bk_adj) <> 0) Execution Plan ---------------------------------------------------------- 0 SELECT STATEMENT Optimizer=CHOOSE 1 0 FILTER 2 1 SORT (GROUP BY) 3 2 FILTER 4 3 HASH JOIN 5 4 HASH JOIN 6 5 HASH JOIN 7 6 INDEX (FAST FULL SCAN) OF 'XDEAUN08' (UNIQUE) 8 6 HASH JOIN 9 8 NESTED LOOPS 10 9 INDEX (FAST FULL SCAN) OF 'XBRNNN02' (NON-UNIQUE) 11 9 INDEX (RANGE SCAN) OF 'XFACNN05' (NON-UNIQUE) 12 8 INDEX (FAST FULL SCAN) OF 'XOSTNN06' (NON-UNIQUE) 13 5 INDEX (FAST FULL SCAN) OF 'XACCNN05' (NON-UNIQUE) 14 4 INDEX (FAST FULL SCAN) OF 'XAPONN05' (NON-UNIQUE) 15 3 INDEX (SKIP SCAN) OF 'XBSGNN03' (NON-UNIQUE) The addition of indexes to the smaller database leads nowhere. Existing indexes were initially identical on both databases, and creating different indexes on the smaller database brings no change to the execution plan. Three weeks after the problem was first spotted, attention is now turning to disk striping, without much hope. Constraining optimizer directives are beginning to look unpleasantly like the only escape route. Before using directives, it is wise to have a fair idea of the right angle of attack. Finding the proper angle, as you have seen in Chapters 4 and 6, requires an assessment of the relative precision of the various input criteria, even though in this case the reasonably large result set (of some 40,000 rows on the larger database and a little over 3,000 on the smaller database) gives us little hope of seeing one criterion coming forward as the key criterion.

Culture and Prosperity: The Truth About Markets - Why Some Nations Are Rich but Most Remain Poor
by John Kay
Published 24 May 2004

He should do just the amount of calculation needed to find the best strategy in the light of his knowledge that every second devoted to calculation increases the chances ofbeing caught by the bear. 18 Borrowing Herbert Simon's term (but for a very different concept), Oliver Williamson calls this optimization under constraints-bounded rationality. 19 In this vein, transactions costs economics often degenerates into a Panglossian view of the world: institutions that exist must be the solution to some constrained-optimization problem. Economists even have a word-recoverability-for deducing the maximization problem to which observed behavior is the answer. But this version of bounded rationality confronts a fundamental problem. How could the economist know when to stop calculating { 220} John Kay when he cannot know the benefits of further calculation?

Commodity Trading Advisors: Risk, Performance Analysis, and Selection
by Greg N. Gregoriou , Vassilios Karavas , François-Serge Lhabitant and Fabrice Douglas Rouah
Published 23 Sep 2004

The results illustrate a useful idea: If we are concerned about event risk, we may wish to define our objective function as one that has the least number of negative returns during the investment horizon, with the constraint that the correlation at first decile should be the lowest. This could be a useful framework to carry out constrained optimization of portfolio returns. CONCLUSION Our results indicate that the risk-adjusted returns as measured by Sharpe and Sortino ratios are always higher in CTA strategies than in most traditional asset classes for the entire sample period under study. Unlike hedge funds, the correlation coefficients of the CTAs with the equity markets are negative during bad times (worst performance period of the equity markets).

Economic Origins of Dictatorship and Democracy
by Daron Acemoğlu and James A. Robinson
Published 28 Sep 2001

To determine these prices, we examine the cost-minimization problem of a firm choosing input demands to minimize the cost of production. Formally, a firm solves the problem: min { p K YK + p L YL + p N Y N } YK ,YL ,Y N subject to: Y = (YK + σ YL )θ Y N1−θ Here, p K YK + p L YL + p N Y N is the total cost of using the three intermediate goods. This is a simple constrained-optimization problem. To solve it, we form the Lagrangean function: L = p K YK + p L YL + p N Y N − λ (YK + σ YL )θ Y N1−θ − Y A Model of an Open Economy 327 and derive the first-order conditions with respect to the three choice variables YK , YL , and Y N . These are: λθ (YK + σ YL )θ−1 Y N1−θ = p K (10.3) λθ σ (YK + σ YL )θ−1 Y N1−θ = p L λ(1 − θ) (YK + σ YL )θ Y N−θ = p N From these, we derive: YN θ pK = pN 1 − θ YK + σ YL pK 1 = pL σ and (10.4) where the first follows from dividing the first and third equations in (10.3), and the second follows from dividing the first two equations in (10.3).

Never Let a Serious Crisis Go to Waste: How Neoliberalism Survived the Financial Meltdown
by Philip Mirowski
Published 24 Jun 2013

Geanakoplos likes to phrase his commentary in terms of “cycles,” but in fact, general equilibrium forces him to posit separate and unconnected sequential states of equilibrium, driven in a Rube Goldberg fashion by the arbitrarily sequenced arrival of asset issuance→“news”→valuation. Each state along the way is in “equilibrium,” as defined by constrained optimization and market clearing. Because everything is always already in equilibrium, the only reason anything changes is the deus ex machina imposed from outside by the model builder. Worse, the sequence itself is completely arbitrary, dictated more by the math than by anything that happens in the world: for instance, each player has only one chance to issue assets, is proscribed from trading them on secondary markets, and consumption can happen only at the initial issuance and at the end of time.108 It seems that, however proud he may be of hewing to Walrasian heuristics, the suite of models he has constructed rarely do much to actually illuminate the actual crisis and aftermath.

Money and Government: The Past and Future of Economics
by Robert Skidelsky
Published 13 Nov 2018

Marx, K. (1909 (1894)), Capital: A Critique of Political Economy (III) The Process of Capitalist Production as a Whole. Chicago, Ill.: Charles H. Kerr & Co. Marx, K. and Engels, F. (1962), Selected Works. London: Lawrence & Wishart. Marx, K. and Engels, F. (1967), The Communist Manifesto (intro. A. J. P. Taylor). Harmondsworth: Penguin. Masch, V. (2010), An application of risk-constrained optimization (RCO) to a problem of international trade. International Journal of Operations and Quantitative Management, 16 (4), pp. 415–65. Masch, V. (2015), Shifting ‘the dodo paradigm’: to be or not to be. World Journal of Social Sciences, 5 (3), September, pp. 123–42. Masch, V. A. (2017), Balancing global trade: ‘compensated free trade’.