linear programming

Artificial Intelligence: A Modern Approach

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

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. The time complexity of

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linear programming is polynomial in the number of variables. Linear programming is probably the most widely studied and broadly useful method for optimization. It is a special case of the more general problem

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developed, including simulated annealing, which returns optimal solutions when given an appropriate cooling schedule. •Many local search methods apply also to problems in continuous spaces. Linear programming and convex optimization problems obey certain restrictions on the shape of the state space and the nature of the objective function, and admit polynomial-time

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)); entomology (ant colony (Dorigo et al., 2008), bee colony (Karaboga and Basturk, 2007), firefly (Yang, 2009) and glowworm (Krishnanand and Ghose, 2009) optimization); and others. Linear programming (LP) was first studied systematically by the mathematician Leonid Kantorovich (1939). It was one of the first applications of computers; the simplex algorithm (Dantzig, 1949

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obey a variety of astronomical, precedence, and power constraints. The best-known category of continuous-domain CSPs is that of linear programming problems, where constraints must be linear equalities or inequalities. Linear programming problems can be solved in time polynomial in the number of variables. Problems with different types of constraints and objective functions

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on one of four different colors.) Infinite-domain CSPs—for example, with integer- or realvalued variables—require quite different algorithms, such as bounds propagation or linear programming. Consider the following example. We define triangle(X,Y,Z) as a predicate that holds if the three arguments are numbers that satisfy the triangle

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constraint-solving algorithms for the constraints allowed in the language. For example, a system that allows linear inequalities on real-valued variables might include a linear programming algorithm for solving those constraints. CLP systems also adopt a much more flexible approach to solving standard logic programming queries. For example, instead of depth

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Tetris MDP. 16.2Algorithms for MDPs In this section, we present four different algorithms for solving MDPs. The first three, value iteration, policy iteration, and linear programming, generate exact solutions offline. The fourth is a family of online approximate algorithms that includes Monte Carlo planning. 16.2.1Value Iteration The Bellman equation

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be reached by a good policy. There’s no sense planning for the results of an action you will 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

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for every state s and every action a. This creates a connection from dynamic programming to linear programming, for which algorithms and complexity issues have been studied in great depth. For example, from the fact that linear programming is solvable in polynomial time, one can show that MDPs can be solved in time polynomial

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and Atkeson, 1993; Andre et al., 1998; Wingate and Seppi, 2005). The formulation of MDP-solving as a linear program is due to de Ghellinck (1960), Manne (1960), and D’Épenoux (1963). Although linear programming has traditionally been considered inferior to dynamic programming as an exact solution method for MDPs, de Farias and Roy

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(2003) show that it is possible to use linear programming and a linear representation of the utility function to obtain provably good approximate solutions to very large MDPs. Papadimitriou and Tsitsiklis (1987) and Littman et

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al. (1995) provide general results on the computational complexity of MDPs. Yinyu Ye (2011) analyzes the relationship between policy iteration and the simplex method for linear programming and proves that for fixed γ, the runtime of policy iteration is polynomial in the number of states and actions. Seminal work by Sutton (1988

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), the optimal choice at the root is the highest (or lowest) intersection point of the remaining hyperplanes. Finding this choice is an example of a linear programming problem: maximizing an objective function subject to linear constraints. Such problems can be solved by standard techniques in time polynomial in the number of actions

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that are similar to the ones used in the zero–sum case. For two players these equations are linear and can be solved with basic linear programming techniques, but for three or more players they are nonlinear and may be very difficult to solve. 17.2.3Repeated games So far, we have

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1. But in general we can solve extensive games by converting to normal form and then finding a solution (usually a mixed strategy) using standard linear programming methods. That works in theory. But if a player has I information sets and a actions per set, then that player will have aI pure

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than exponential. Rather than represent strategies, it represents paths through the tree; the number of paths is equal to the number of terminal nodes. Standard linear programming methods can again be applied to this representation. The resulting system can solve poker variants with 25,000 states in a minute or two. This

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second. Within such a logic, one can prove, for example, that Satisfiability of sets of probability assertions can be determined in the propositional case by linear programming (Hailperin, 1984; Nilsson, 1986). Thus, we have a “probability logic” in the same sense as “temporal logic”—a logical system specialized for probabilistic reasoning. To

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Clamp, S. E. (1981). Geographical variation in disease presentation. Medical Decision Making, 1, 59–69. de Farias, D. P and Roy, B. V. (2003). The linear programming approach to approximate dynamic programming. Operations Research, 51, 839–1016. de Finetti, B. (1937). Le prévision: ses lois logiques, ses sources subjectives. Ann. Inst. Poincaré

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, C. A. (1989). Adaptive importance sampling. In Proc. Fifth International Conference on Structural Safety and Reliability. Karmarkar, N. (1984). A new polynomial–time algorithm for linear programming. Combinatorica, 4, 373–395. Karp, R. M. (1972). Reducibility among combinatorial problems. In Miller, R. E. and Thatcher, J. W. (Eds.), Complexity of Computer Computations

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, 14, 151–165. Manna, Z. and Waldinger, R. (1985). The Logical Basis for Computer Programming: Volume 1: Deductive Reasoning. Addison-Wesley. Manne, A. S. (1960). Linear programming and sequential decisions. Management Science, 6, 259–267. Manning, C. and Schutze, H. (1999). Foundations of Statistical Natural Language Processing. MIT Press. Manning, C., Raghavan

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., 356, 1104 Lindsten, F., 517, 1104 linear-Gaussian, 440, 473, 497, 656, 779 linear algebra, 1076–1077 linear constraint, 167 linear function, 694 linearization, 942 linear programming, 139, 159, 161, 167, 562, 603 linear quadratic regulator (LQR), 962, 982 linear regression, see regression (in machine learning) linear resolution, 326 linear separability, 700

The Art of Computer Programming: Fundamental Algorithms

by Donald E. Knuth  · 1 Jan 1974

. Representation of matrix A2), with nodes in the format List heads appear at the left and at the top. LEFT UP ROW COL VAL solving linear programming problems by the simplex method. A pivot step is the following matrix transformation: / Pivot row Any other row • • Before pivot step After Any Pivot other

Optimization Methods in Finance

by Gerard Cornuejols and Reha Tutuncu  · 2 Jan 2006  · 130pp  · 11,880 words

Asset Allocation 1.3.2 Pricing and Hedging of Options . . . . . 1.3.3 Risk Management . . . . . . . . . . . . . 1.3.4 Asset Liability Management . . . . . . . . . . . . . . . . . . . 2 Linear Programming: Theory and Algorithms 2.1 The Linear Programming Problem . . . . . . . . 2.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Optimality Conditions . . . . . . . . . . . . . . . 2.4 The Simplex Method . . . . . . . . . . . . . . . . 2.4.1 Basic Solutions . . . . . . . . . . . . . . . 2.4.2

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Finance 3.1 Derivative Securities and The Fundamental Theorem of 3.1.1 Replication . . . . . . . . . . . . . . . . . . . . 3.1.2 Risk-Neutral Probabilities . . . . . . . . . . . . 3.2 Arbitrage Detection Using Linear Programming . . . . 3.3 Risk Measures: Conditional Value-at-Risk . . . . . . . iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asset Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 3 3 4 4 5 6 7 8 9 11 11 . . . . . . . .

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gov/otc/Guide/OptWeb/. 1.1.1 Linear Optimization One of the most common and easiest optimization problems is the linear optimization (LO) or the linear programming (LP) problem: the problem of optimizing a linear objective function subject to linear equality and inequality constraints. This corresponds to the case where f

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as equalities after the introduction of a so-called slack or surplus variable that is restricted to be nonnegative. For example, 13 14 CHAPTER 2. LINEAR PROGRAMMING: THEORY AND ALGORITHMS −x1 − x2 2x1 + x2 ≤ 12 x1 + 2x2 ≤ 9 x1 , x2 ≥ 0 (2.3) −x1 − x2 2x1 + x2 + x3 = 12 x1 + 2x2 +

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find y1 and y2 that achieve the maximum combination of the right-hand-side values: max 12y1 + 9y2 . 16 CHAPTER 2. LINEAR PROGRAMMING: THEORY AND ALGORITHMS This process results in a linear programming problem that is strongly related to the LP we are solving. We want to max 12y1 + 9y2 2y1 + y2 y1 +

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omit the (elementary) proof of this theorem since it requires some additional tools. The reader can find a proof of this result on most standard linear programming textbooks. The strong duality theorem provides us with conditions to identify optimal solutions (called optimality conditions): x ∈ <n is an optimal solution of (2

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the simplex method. We discuss the latter here and postpone our discussion of interior-point methods till we study quadratic programming problems. 18 CHAPTER 2. LINEAR PROGRAMMING: THEORY AND ALGORITHMS 2.4 The Simplex Method To motivate our discussion of the simplex method, we consider the following example from the book Introduction

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= BxB + NxN = b, or by multiplying both sides by B −1 from left, xB + B−1 NxN = B−1 b. 0 0 .. .    ,   20 CHAPTER 2. LINEAR PROGRAMMING: THEORY AND ALGORITHMS So the following systems of equations are equivalent; any solution to the first will be a solution for the next two, and

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to be able to figure out the net effect of changing a nonbasic variable on the objective function. A key observation is that when a linear programming problem has an optimal solution, it must have an optimal basic feasible solution. The significance of this result lies in the fact that when

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we are looking for a solution of a linear programming problem what we really need to check is the objective value of each basic solution. There are only finitely many of them, so this

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space from an infinite space to a finite one. 2.4. THE SIMPLEX METHOD 2.4.2 21 Simplex Iterations The simplex method solves a linear programming problem by moving from one basic feasible solution to another. Since one of these solutions is optimal, presumably, the method will eventually get there.

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to be increased to maintain the equality in that row; but then we would not worry about that basic variable becoming negative. 22 CHAPTER 2. LINEAR PROGRAMMING: THEORY AND ALGORITHMS so we should not increase x1 more than min{100, 75, 800} = 75, if we want to maintain the non-negativity

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understand why this last basic solution is optimal. From our discussion at the beginning of this section, recall that any feasible solution to a linear programming 24 CHAPTER 2. LINEAR PROGRAMMING: THEORY AND ALGORITHMS problem in the standard form can be expressed relative to a basis matrix B by xB = B −1 b −

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solution, which means that the current solution is the best, the optimal. 2.4.3 The Tableau Form of the Simplex Method In most linear programming textbooks, the simplex method is described using tableaus that summarize the information in the different representations of the problem we saw above. Since the reader

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) indefinitely, the equalities can be balanced by increasing the basic variables appropriately, and none of the nonnegativity constraints will be violated. 26 CHAPTER 2. LINEAR PROGRAMMING: THEORY AND ALGORITHMS Step 2. Consider the column picked in Step 1. For each positive entry in this column, calculate the ratio of the right

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-15 5 1 50 This solution is optimal since all the coefficients in the objective row are nonnegative. 2.5 Exercises 1. Consider the following linear programming problem: max 4x1 + 3x2 3x1 + x2 ≤ 9 3x1 + 2x2 ≤ 10 x1 + x2 ≤ 4 x1 , x2 ≥ 0. First, transform this problem into the “standard form”.

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many basic feasible solutions? What are the basic feasible solutions and what are the extreme points of the feasible region? 2. We say that two linear programming problems are equivalent if one can be obtained from the other by (i) multiplying the objective function by -1 and changing it from min

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We complete this section by stating and proving the first fundamental theorem of asset pricing. For finite Ω this proof is a simple exercise in linear programming duality that also utilizes the following well-known result of Goldman and Tucker on the existence of strictly complementary optimal solutions of LPs: 3.1

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. DERIVATIVE SECURITIES AND THE FUNDAMENTAL THEOREM OF ASSET PRICING33 Theorem 3.1 When both the primal and dual linear programming problems minx cT x Ax = b x ≥ 0. (LP) and (LD) maxy bT y AT y ≤ c, (3.2) (3.3) have feasible solutions,

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dual solution, any optimal solution of the primal must have tight constraints, indicating that there is no type-B arbitrage. 3.2 Arbitrage Detection Using Linear Programming The linear programming problems (3.4) and (3.4) we formulated for the proof of Theorem 3.2 can naturally be used for detection of arbitrage opportunities

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) i=1 The cost of the portfolio x is given by n X i=1 S0i xi . (3.7) 3.2. ARBITRAGE DETECTION USING LINEAR PROGRAMMING 35 To determine whether there exists a static arbitrage opportunity in the current prices S0i , we consider the following problem: What is the smallest cost

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Mean Expected Loss, Mean Shortfall, and Tail VaR. We now describe this risk measure in more detail and discuss how it can be optimized using linear programming techniques when the loss function is linear in the portfolio positions. Our discussion follows parts of articles by Rockafellar and Uryasev [12, 17]. We

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x, y) is linear in x, all the expressions zs ≥ f (x, ys ) − γ represent linear constraints and therefore the problem (3.21) is a linear programming problem that can be solved using the simplex method or alternative LP algorithms. Alternative optimization problems are often formulated within the context of risk management

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the given period. Let R = 1 + r. Show that if u > R > d is not satisfied there is an arbitrage opportunity. 2. Recall the linear programming problem (3.9) that we developed to detect arbitrage opportunities in the prices European call options with a common underlying security and common maturity (but

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other market quote websites. Delayed data is available for free, so do not use the sites that charge fees for real-time data. Formulate the linear programming problem (33) (or, rather the 42 CHAPTER 3. LP MODELS AND TOOLS IN FINANCE version you developed for Problem 4 since market quotes will

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-neos.mcs.anl.gov/neos/ to access the Network Enable Optimization Server. Using this site, and their JAVA submission tool, you can submit a linear programming problem (in some standard format) and have a remote computer solve your problem using one of the several solver options. You will then receive the

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is positive semidefinite, the QP (4.1) is a convex optimization problem. As such, its local optimal solutions are also global optimal solutions. As in linear programming, we can develop a dual of quadratic programming problems. The dual of the problem (4.1) is given below: (QD) maxx,y,s bT

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x Qx 2 Qx + s = c x , s ≥ 0. (4.2) 44CHAPTER 4. QUADRATIC PROGRAMMING: THEORY AND ALGORITHMS Note that, unlike the case of linear programming, the variables of the primal quadratic programming problem also appear in the dual QP. 4.2 Optimality Conditions One of the fundamental tools in the

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) are both necessary and sufficient for x, y, and s to describe a global optimal solution of the QP problem. In a manner similar to linear programming, optimality conditions (4.3)-(4.7) can be seen as a collection of conditions for 1. primal feasibility: Ax = b, x ≥ 0, 2. dual

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, e is an n-dimensional vector of ones. 4.3 Interior-Point Methods We have seen that simplex method can be used to solve linear programming problems efficiently and its variants can be used for quadratic programming problems. Although these methods demonstrate satisfactory performance for the solution of most practical problems

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simplex method is exponential (and, therefore, that simplex method is not a polynomial-time algorithm) there was an effort to identify alternative methods for linear programming with polynomial time complexity. The first such method, called the ellipsoid method was developed by Yudin and Nemirovski in 1979. But the more exciting and

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that work with infeasible iterates. In these versions of the algorithms, search for feasibility and optimality are performed simultaneously. 4.6 QP software As for linear programs, there are several software options for solving practical quadratic programming problems. Many of the commercial software options are very efficient and solve very large QPs

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weight on violations or margin, depending on one’s preference. 4. The classification problems we discussed in the previous exercise can also be formulated as linear programming problems, if one agrees to use 1-norm rather P than 2-norm of w in the objective function. Recall that kwk1 = i |wi |.

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A2K2 x2K2 BK 2 1 2 x, x1 , ... x2K2 = b1 = b21 .. . = b2K2 ≥ 0. (6.6) 74 CHAPTER 6. STOCHASTIC PROGRAMMING MODELS This is a deterministic linear programming problem called the deterministic equivalent of the original uncertain problem. This problem has k copies of the second-stage decision variables and therefore, can be

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movements can be modeled by random variables this problem presents an attractive setting for the use of stochastic programming techniques. Below, we discuss a deterministic linear programming equivalent of stochastic LP model for the debt management problem. We consider a multi-period framework with T time periods. We will use the

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, i = 0, . . . , n, l = 1, . . . , L. (7.5) P P 86CHAPTER 7. ROBUST OPTIMIZATION MODELS AND TOOLS IN FINANCE This is, in fact, a linear programming problem that can be solved easily using the simplex method or interior-point methods. The nonnegativity constraints on xli ’s disallow short sales and borrowing

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closed convex cone. When X = <n and C = <n+ , this problem is the standard form LP. However, this setting is much more general than linear programming since we can use non-polyhedral cones C in the description of these problems. Conic optimization offers a convenient setting where the sophisticated interior-point

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algorithms for linear programming problems can be generalized and used very efficiently to solve a large class of convex optimization problems. An advanced discussion on this subject can be

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a class of saddle point problems. Journal of Optimization Theory and Applications, 116(3):559– 590, 2003. [7] S. Herzel. Arbitrage opportunities on derivatives: A linear programming approach. Technical report, Department of Economics, University of Perugia, 2000. [8] R. O. Michaud. The Markowitz optimization enigma: Is optimized optimal? Financial Analysts Journal,

Algorithms in C++ Part 5: Graph Algorithms

by Robert Sedgewick  · 2 Jan 1992

-scheduling problem instance illustrated in Figure 21.22. For example, the equation x8x7 +. 32 means that job 8 cannot start until job 7 is completed. Linear programming Assign nonnegative values to a set of variables x 0 through x n that minimize the value of a specified linear combination of the variables

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on the variables, each of which specifies that a given linear combination of the variables must be greater than or equal to a given constant. Linear programming is a widely used general approach to solving a broad class of optimization problems that we will not consider it in detail until Part 8

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. Clearly, the difference-constraints problem reduces to linear programming, as do many other problems. For the moment, our interest is in the relationships among the difference-constraints, job-scheduling, and shortest-paths problems. Property

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them, by reducing other problems to them. It is also important to be aware of a hierarchy among increasingly general problem-formulation models. For example, linear programming is a general formulation that is important not just because many problems reduce to it but also because it is not known to be NP

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-hard. In other words, there is no known way to reduce the general shortest-paths problem (or any other NP-hard problem) to linear programming. We discuss such issues in Part 8. Not all problems can be solved, but good general models have been devised that are suitable for broad

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the difference-constraints problem by the construction used in the proof of Property 21.15, and the difference-constraints problem reduces trivially to linear programming, so, by Property 21.17, linear programming is NP-hard. 21.118 Does the shortest-paths problem in networks with no negative cycles reduce to the job-scheduling problem

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we cannot solve. They are the first of several other classes of problems with a similar character that we consider, including network-flow problems and linear programming. As in shortest paths, there is a fine line between easy and intractable problems in those areas. Not only are numerous efficient algorithms available when

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: the maxflow problem and the mincost-flow problem. We see specific relationships among these problem-solving models, the shortest-path model of Chapter 21, the linear-programming (LP) model of Part 8, and numerous specific problem models including some of those just discussed. Figure 22.4 Cutting supply lines This diagram represents

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—significant first steps in coping with a new combinatorial problem. We conclude this section by considering a strictly mathematical formulation of the maxflow problem, using linear programming (LP) (see Section 21.6). This exercise is useful because it helps us to see relationships to other problems that can be so formulated. The

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introduced many of the fundamental concepts. We have briefly introduced numerous advanced topics from (the yet to be published) Part 8, including reducibility, intractability, and linear programming, among several others. This reference list is focused on the material that we cover in detail and cannot do justice to these advanced topics. The

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, 258–263, 268 Kuratowski’s theorem, 73 Least common ancestor (LCA), 459–460 Length, path, 11 Library programming, 336, 485 LIFO stack, 132–134, 138 Linear programming (LP), 333, 342–343, 365 maxflow, 439–440 mincost flow, 425, 479 multicommodity flow, 483 network flow algorithms, 371–372 Linear quantity, 59 Link, 4

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, 427–429 augmenting-path method, 382–407 bipartite matching, 433–436 capacity constraints, 426 connectivity, 437–438 feasible flow, 430–432 general networks, 425–426 linear programming, 439–440 and mincost flow problem, 425, 472–473 mincost maxflow, 443–447 preflow-push method, 410–423 reductions, 370, 406, 425–440 running time

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Reduced cost, 454–457 Reduction, 177, 283, 484 difference constraints, 333–336, 338 equivalent problems, 328 flow networks, 367 implications, 341 job scheduling, 332–338 linear programming, 333, 342–343 maxflow, 370, 406, 425–440 mincost flow, 472–479 polynomial-time, 439 shortest paths, 328–343 transitive, 223 upper/lower bounds, 342

Programming Python

by Mark Lutz  · 5 Jan 2011

, any information needed across events must be stored in long-lived references such as global variables and class instance attributes. The notion of a traditional linear program control flow doesn’t really apply in the GUI domain; you need to think in terms of smaller chunks. In Python, Tk also becomes object

Television disrupted: the transition from network to networked TV

by Shelly Palmer  · 14 Apr 2006  · 406pp  · 88,820 words

(early morning, daytime, family time, prime time access, etc.) Pay Per View • On-demand • Subscription • Ad-supported Distribution Traditional network television was developed to deliver linear programming using a fairly efficient one-to-many system. Alternatively, networked television systems offer one-to-one and optionally, non-linear distribution. Linear Distribution Methodology • Broadcast

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, Near-VOD) • Consumer-based time-shifted (personal video recorder, TiVo®, Media Center) Technically speaking, all of the linear methodologies could be used to distribute non-linear programming as well. But in common practice, only cable, IPTV and the Internet are used to do so. Copyright © 2006, Shelly Palmer. All rights reserved. 1

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IPTV systems. The unique attribute of linear television is that it is scheduled for you by the network or station programmers. Marketers sometimes refer to linear programming as “destination television” because you are supposed to watch the program when it is presented. (Of course, consumers don’t always do what they’re

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subscription, pure subscription, pay-per-view, rental and purchase. • The major components of the television business are form factors, packaging and distribution. • Linear programming is scheduled for consumers, Non-linear programming is on-demand. • There are highly evolved business rules and negotiable currencies used in the traditional television business, and these must evolve for

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network advertising executives. Of course, you can use the enhancements to help call attention to the advertisements just as easily as you can with the linear programming content. This actually works and, for the limited number of people who enjoy playing enhanced television games, is more effective than its pure linear counterpart

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7:21 AM Page 58 58 C H A P T E R 4 Existing Wireless Networks consumers want or need a way to watch linear programming on a mobile phone. But, this won’t stop content owners from making every possible piece of branded content available in the format. The “cool

Masterminds of Programming: Conversations With the Creators of Major Programming Languages

by Federico Biancuzzi and Shane Warden  · 21 Mar 2009  · 496pp  · 174,084 words

just no substitute for years of hard work. So here you are in the middle of some project and you decide you need to understand linear programming to solve your problem; probably what you’ll get from the Internet will not be helpful, and if you have to solve this problem within

How I Became a Quant: Insights From 25 of Wall Street's Elite

by Richard R. Lindsey and Barry Schachter  · 30 Jun 2007

, if not all, such restrictions. Could a computer allocate the funds in a more efficient manner? This seemed to be an ideal opportunity for a linear programming application: Maximize expected return while meeting all regulatory restrictions. The results demonstrated that manual adherence to each restriction with cushions was expensive in terms of

Capital Ideas: The Improbable Origins of Modern Wall Street

by Peter L. Bernstein  · 19 Jun 2005  · 425pp  · 122,223 words

Prize in economic sciences in 1975 for his work in this area. Koopmans developed an analytic method known as linear programming or activity analysis that falls under the general heading of operations research. Linear programming solves problems that involve combinations of inputs and outputs. Assume, for example, that an airline has a limited number

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economizing on crew time its most important objective? Or if it wanted to make as many landings as possible in the New York City area? Linear programming identifies the combinations of inputs and outputs that are achievable, defines the combinations that minimize the inputs and maximize the outputs, and then identifies the

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and developed a systematic means to avoid it. Markowitz’s reflections on diversification and risk led him to explore the subject more thoroughly in a linear programming course he was taking under Koopmans. Koopmans had asked the class to describe a resource allocation problem and state whether or not it was a

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linear programming problem. Markowitz took the occasion to analyze the choices facing an investor who must decide between seeking high returns and attempting to hold down risk

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at the same time. He concluded that the solution to this problem was even more complex than linear programming. Koopmans gave him an A on his paper and noted, “The problem does not seem that hard. Why don’t you solve it?”10 That

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between risk and reward in selecting a portfolio; this is the subject matter of his 1952 article and is closely related to the techniques of linear programming that Markowitz learned from Koopmans. The second track tells how each investor should go about selecting the single portfolio that most closely conforms to the

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an envelope. The task is simpler if you have access to a computer. Markowitz combined his skill with the computer with Koopmans’s achievements in linear programming to make the task at least manageable. Still, the audience for such technical matters was still limited in the 1950s. The Naval Research Logistics Quarterly

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up residence in the Los Angeles area at a think tank called RAND—an acronym for R&D, or research-and-development—to work on linear programming applications for industrial firms. RAND had been established during World War II, primarily to do research for the armed forces. Its sphere of interest has

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the Dutch economist Tjalling Koopmans. Koopmans was then working in New York for a Dutch shipping company, where he was applying his new technique of linear programming to the company’s scheduling problems. Modigliani then launched forth on a varied teaching career. He began at the New Jersey College for Women. In

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breakdowns. Analysts were developing long-run forecasts to implement the dividend discount model. Computer-based portfolios were up and running, with risk-controlled procedures and linear programming to combine the holdings into the optimal portfolio. In 1970 further impetus came unexpectedly from outside. Keith Schwayder, a young man who had just completed

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’Brien Rubinstein Associates, Inc. (LOR) Leland-Rubinstein Associates Leverage Leveraged buyouts Liquidity management market money Preference theory stock “Liquidity Preference as Behavior Toward Risk” (Tobin) Linear programming Loading charges: see Brokerage commissions London School of Economics (LSE) London Stock Exchange Macroeconomics Management Science Marginal utility concept “Market and Industry Factors in Stock

Commodity Trading Advisors: Risk, Performance Analysis, and Selection

by Greg N. Gregoriou, Vassilios Karavas, François-Serge Lhabitant and Fabrice Douglas Rouah  · 23 Sep 2004

outputs of the DMU. Following Banker, Charnes, and Cooper (BCC) (1984) we can measure the output-oriented efficiency of the ith DMU by solving this linear programming problem:17 Max fi Subject to N ∑ λ j yrj j =1 N ∑ λ j xsj j =1 N ∑ λj ≥ φi yri r = 1, 2

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an inefficient DMU fi > 1. On the other hand, an input-oriented measure of efficiency can be obtained for the ith DMU by solving the linear programming problem: 16The concept of efficiency used here is that of technical efficiency. It is used in the context of an expanded efficient frontier with n

Site Reliability Engineering: How Google Runs Production Systems

by Betsy Beyer, Chris Jones, Jennifer Petoff and Niall Richard Murphy  · 15 Apr 2016  · 719pp  · 181,090 words

In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation

by William J. Cook  · 1 Jan 2011  · 245pp  · 12,162 words

The Mathematics of Banking and Finance

by Dennis W. Cox and Michael A. A. Cox  · 30 Apr 2006  · 312pp  · 35,664 words

Trend Following: How Great Traders Make Millions in Up or Down Markets

by Michael W. Covel  · 19 Mar 2007  · 467pp  · 154,960 words

Statistics in a Nutshell

by Sarah Boslaugh  · 10 Nov 2012

Tools for Computational Finance

by Rüdiger Seydel  · 2 Jan 2002  · 313pp  · 34,042 words

How Not to Network a Nation: The Uneasy History of the Soviet Internet (Information Policy)

by Benjamin Peters  · 2 Jun 2016  · 518pp  · 107,836 words

Algorithms to Live By: The Computer Science of Human Decisions

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