constrained optimization Concepts

Cogs and Monsters: What Economics Is, and What It Should Be

by Diane Coyle · 11 Oct 2021 · 305pp · 75,697 words

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

Mathematics for Economics and Finance

by Michael Harrison and Patrick Waldron · 19 Apr 2011 · 153pp · 12,501 words

functions . 3.2.3 Convexity and differentiability . . 3.2.4 Variations on the convexity theme 3.3 Unconstrained Optimisation . . . . . . . 3.4 Equality Constrained Optimisation: The Lagrange Multiplier Theorems . . . . 3.5 Inequality Constrained Optimisation: The Kuhn-Tucker Theorems . . . . . . . 3.6 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 21 23 24 25 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 27 27 27 29 30 34 39

The End of Alchemy: Money, Banking and the Future of the Global Economy

by Mervyn King · 3 Mar 2016 · 464pp · 139,088 words

right answer, only a problem of coping with the unknown. A different way of thinking about behaviour as neither irrational nor the product of a constrained optimisation problem is, I believe, helpful in understanding what happened both before and after the crisis. In other words, we need an alternative to both optimising

Artificial Intelligence: A Modern Approach

by Stuart Russell and Peter Norvig · 14 Jul 2019 · 2,466pp · 668,761 words

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

…

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

…

,” 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

…

never do. 16.2.3Linear programming Linear programming or LP, which was mentioned briefly in Chapter 4 (page 139), is a general approach for formulating constrained optimization problems, and there are many industrial-strength LP solvers available. Given that the Bellman equations involve a lot of sums and maxes, it is perhaps

…

of propositional STRIPS planning. AIJ, 69, 165–204. Byrd, R. H., Lu, P., Nocedal, J., and Zhu, C. (1995). A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific and Statistical Computing, 16, 1190–1208. Cabeza, R. and Nyberg, L. (2001). Imaging cognition II: An empirical review of 275 PET

…

of a CSP assignment, 165 of a heuristic, 106 path, 172, 188 consistent estimation, 455 consistent hypothesis, 671 conspiracy number, 221 constant symbol, 275, 277 constrained optimization problem, 139, 169 constraint binary, 168 global, 168, 172 nonlinear, 167 preference constraint, 169 propagation, 169, 169–175, 178–179 resource constraint, 173 symmetry-breaking

A Primer for the Mathematics of Financial Engineering

by Dan Stefanica · 4 Apr 2008

(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

…

}. 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

…

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

…

(V"d) is qred(Vred) = q( v) = v~ = t qred from (8.15) is positive definite since Answer: We first reformulate the problem as a constrained optimization problem. Let U = I1i=I:3(0,oo) C ffi.3 and let x = (XI,X2,X3) E U. The functions f : U ----7 ffi

…

, i = 1 : n. = 1. (8.20) 244 CHAPTER 8. LAGRANGE MULTIPLIERS. NEWTON'S METHOD. 8.1. LAGRANGE MULTIPLIERS We formulate problem (8.20) as a constrained optimization problem. Let U = I17=1 (0, oo)n and let x = (Xl, X2, . .. , Xn) E U. The functions f : U -+ ~ and g : U -+ ~ are defined

…

eJp, find Wi, i jLp, such that var(R) is minimal. = 1 : n, with var(R) = eJ~, such that E[R] is maximal. These are constrained optimization problems and can be solved using Lagrange multipliers. Rather than discuss the general case of n assets, we provide more details for a particular example

…

the following (row) vector: 2W10"r + 2W20"10"2P1,2 From (8.52-8.54), it follows that this problem can be written as a constrained optimization problem as follows: find Wo such that min f(w) 263 2W20"~ + 2W10"10"2P1,2 + 2W30"20"3P2,3 + Al + A2JL2 + W2 W30"20

…

solution of (8.59). Since condition (8.9) IS satIsfied, we know from Theorem 8.1 that Wo is the only possible solution ~or the constrained optimization problem (8.55). To identify whether w is Indeed, a constrained minimum, we need to construct the reduced quadr~ti~ form qred(Vred) given by

…

the first four entries of the solution to the linear system (8.59) that identifies the critical points of the Lagrangian function of the corresponding constrained optimization problem. By direct computation, we find that the system (8.59) can be written as 1 0 0.08 -0.06 0.049 1 0

Why Machines Learn: The Elegant Math Behind Modern AI

by Anil Ananthaswamy · 15 Jul 2024 · 416pp · 118,522 words

find the minimum. But minimizing while accounting for the second set of equations yi(w.xi + b) ≥ 1, complicates things somewhat. We now have a constrained optimization problem. We must descend the bowl to a location that simultaneously satisfies the constraint and is a minimum. One solution for such a problem was

…

least altitude, so that when you do drill down, it’ll require the minimum amount of digging. What we have just done is pose a constrained optimization problem. If you had simply been told to find the place with the least altitude in the valley (the minimum), well, that would have been

…

sides. Such a surface has a saddle point, the flat bit in the middle, but it has no maximum or minimum. Now think of our constrained optimization problem. Let’s add the constraint that the (x, y) coordinates must lie on a circle of radius 2. So, the (x, y) coordinates are

…

the constraining curve is a circle. Here’s what the points look like in the 2D and 3D contour plots: More generally, the problem of constrained optimization can be thought of as finding the extrema of the so-called Lagrange function, given by: L(x,λ) = f(x) -λg(x) The logic

…

these multipliers as there are constraining equations, and we have one such equation for each data point.) We’ll focus on the results of the constrained optimization. The first result is that the weight vector turns out to be given by this formula: Each αi (alpha sub-i) is a scalar and

…

able to, so I need another six months,’ ” Hinton said. “It kept going like that.” Hinton did finish his Ph.D. His work involved solving constrained optimization problems using neural networks. “But they weren’t learning,” he said of his neural networks. He was convinced, however, that multi-layer neural networks could

…

theory, 377 Computational Learning Theory (COLT) conference, 238 computational neuroscience, 243–44 Computer Vision and Pattern Recognition (CVPR) conference, 377–78 conditional probability distribution, 417 constrained optimization problem constraining function, 212–16 contour lines, 214–15 equation for, 211–13 finding the exterma, 214–15 Hinton’s work on, 304 kernel operation

…

on LeNet, 374 Microsoft and, 376–77 on Minsky-Papert proof, 302 Ph.D. work, 304 Rosenblatt and, 307 Rumelhart and, 308, 338, 341 solving constrained optimization problems, 304 Sutherland and, 340 Sutskever and, 5, 379 on symbolic AI (artificial intelligence), 303–4 at White Lion Street Free School, 308 Hochreiter, Sepp

Market Risk Analysis, Quantitative Methods in Finance

by Carol Alexander · 2 Jan 2007 · 320pp · 33,385 words

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

…

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

…

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

…

the objective function and thence maximize a function of a single variable. However, we want to illustrate the use of the Lagrangian function to solve constrained optimization problems, so we shall find the solution the long way in this example. 26 See Section I.2.4. Basic Calculus for Finance 31 is

…

surfaces, depending on the method used. I.5.4 OPTIMIZATION Optimization is the process of finding a maximum or minimum value of a function. In constrained optimization problems the possible optima are constrained to lie within a certain feasible set. One of the most famous optimization problems in finance has an analytic

…

Finance 201 method, often a gradient algorithm, to find the maximum or minimum value of the function in the feasible domain. Other financial applications of constrained optimization include the calibration of stochastic volatility option pricing models using a least squares algorithm and the estimation of the parameters of a statistical distribution. In

…

rates and implied volatility surfaces. Algorithms for finding the maximum or minimum value of a multivariate function, subject to certain constraints on the parameters. These constrained optimization techniques have many applications to portfolio optimization. Techniques for approximating a function’s derivatives, based on finite differences. We have outlined how to use these

…

193 Consistent OLS estimators 156–8 Constant absolute risk aversion (CARA) 233–4 Constant relative risk aversion (CRRA) 232–4 Constant term, regression 143–4 Constrained optimization 29–31 Constraint, minimum variance portfolio 245–6 Continuous compounding, return 22–3 Continuous distribution 114 Continuous function 5–6, 35 Continuous time 134–9

Hands-On Machine Learning With Scikit-Learn and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems

by Aurélien Géron · 13 Mar 2017 · 1,331pp · 163,200 words

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

…

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

…

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

…

wi,j = 0 if x(j) is not one of the k closest neighbors of x(i). Thus the first step of LLE is the constrained optimization problem described in Equation 8-4, where W is the weight matrix containing all the weights wi,j. The second constraint simply normalizes the weights

…

(2012). Appendix C. SVM Dual Problem To understand duality, you first need to understand the Lagrange multipliers method. The general idea is to transform a constrained optimization objective into an unconstrained one, by moving the constraints into the objective function. Let’s look at a simple example. Suppose you want to find

…

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

…

) 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

…

we can easily find that , , and . This is the only stationary point, and as it respects the constraint, it must be the solution to the constrained optimization problem. However, this method applies only to equality constraints. Fortunately, under some regularity conditions (which are respected by the SVM objectives), this method can be

…

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

…

C-4. Dual form of the SVM problem The goal is now to find the vector that minimizes this function, with for all instances. This constrained optimization problem is the dual problem we were looking for. Once you find the optimal , you can compute using the first line of Equation C-3

…

config.gpu_options, Managing the GPU RAM ConfigProto, Managing the GPU RAM confusion matrix, Confusion Matrix-Confusion Matrix, Error Analysis-Error Analysis connectionism, The Perceptron constrained optimization, Training Objective, SVM Dual Problem Contrastive Divergence, Restricted Boltzmann Machines control dependencies, Control Dependencies conv1d(), ResNet conv2d_transpose(), ResNet conv3d(), ResNet convergence rate, Batch Gradient

Hands-On Machine Learning With Scikit-Learn, Keras, and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems

by Aurelien Geron · 14 Aug 2019

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

…

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

…

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

…

wi,j = 0 if x(j) is not one of the k closest neighbors of x(i). Thus the first step of LLE is the constrained optimization problem described in Equation 8-4, where W is the weight matrix containing all the weights wi,j. The second constraint simply normalizes the weights

Elements of Mathematics for Economics and Finance

by Vassilis C. Mavron and Timothy N. Phillips · 30 Sep 2006 · 320pp · 24,110 words

8.7.4 Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 8.7.5 Graphical Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 9. 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

…

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

…

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 constraint function, k the constraint constant, and g(x, y) = k the constraint equation (or simply the constraint). There are various methods used for constrained optimization. We will consider two important techniques: the substitution method and the Lagrange Multiplier method. 9.3.1 Substitution Method If the constraint equation allows one

…

total input costs when production is constant at 1,200 units. 9. Optimization 197 9.3.2 Lagrange Multipliers The method of Lagrange multipliers for constrained optimization can be applied generally; unlike the substitution method. The latter requires that one variable can be expressed explicitly in terms of the others, using the

…

may appear that by turning a two variable problem into a three variable one makes the problem harder. However, the method transforms a problem of constrained optimization to one of unconstrained optimization. We state this as follows: The pairs of values of x, y that optimize the function f (x, y), subject

…

but is nevertheless useful to know, as we shall see. 9.3.3 The Lagrange Multiplier λ: An Interpretation The Lagrange multiplier λ used in constrained optimization appears at first glance to have no use as it is eliminated from the equations determining a stationary point and does not appear in the

…

a production, profit, or cost function. Then the iso curves are known respectively, as isoquants, isoprofit, or isocost curves. Using iso curves we can visualize constrained optimization. We illustrate this using the utility function U = 30x2/5 y 1/3 of Example 9.12. The constraint is the budget of e1,100