1. Introduction
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4. Convex Optimization Problems
Convex Optimization — Boyd & Vandenberghe 4. Convex optimization problems optimization problem in standard form • convex optimization problems • quasiconvex optimization • linear optimization • quadratic optimization • geometric programming • generalized inequality constraints • semidefinite programming • vector optimization • 4–1 Optimization problem in standard form minimize f0(x) subject to f (x) 0, i =1,...,m i ≤ hi(x)=0, i =1,...,p x Rn is the optimization variable • ∈ f : Rn R is the objective or cost function • 0 → f : Rn R, i =1,...,m, are the inequality constraint functions • i → h : Rn R are the equality constraint functions • i → optimal value: p⋆ = inf f (x) f (x) 0, i =1,...,m, h (x)=0, i =1,...,p { 0 | i ≤ i } p⋆ = if problem is infeasible (no x satisfies the constraints) • ∞ p⋆ = if problem is unbounded below • −∞ Convex optimization problems 4–2 Optimal and locally optimal points x is feasible if x dom f and it satisfies the constraints ∈ 0 ⋆ a feasible x is optimal if f0(x)= p ; Xopt is the set of optimal points x is locally optimal if there is an R> 0 such that x is optimal for minimize (over z) f0(z) subject to fi(z) 0, i =1,...,m, hi(z)=0, i =1,...,p z x≤ R k − k2 ≤ examples (with n =1, m = p =0) f (x)=1/x, dom f = R : p⋆ =0, no optimal point • 0 0 ++ f (x)= log x, dom f = R : p⋆ = • 0 − 0 ++ −∞ f (x)= x log x, dom f = R : p⋆ = 1/e, x =1/e is optimal • 0 0 ++ − f (x)= x3 3x, p⋆ = , local optimum at x =1 • 0 − −∞ Convex optimization problems 4–3 Implicit constraints the standard form optimization problem has an implicit -
Metaheuristics1
METAHEURISTICS1 Kenneth Sörensen University of Antwerp, Belgium Fred Glover University of Colorado and OptTek Systems, Inc., USA 1 Definition A metaheuristic is a high-level problem-independent algorithmic framework that provides a set of guidelines or strategies to develop heuristic optimization algorithms (Sörensen and Glover, To appear). Notable examples of metaheuristics include genetic/evolutionary algorithms, tabu search, simulated annealing, and ant colony optimization, although many more exist. A problem-specific implementation of a heuristic optimization algorithm according to the guidelines expressed in a metaheuristic framework is also referred to as a metaheuristic. The term was coined by Glover (1986) and combines the Greek prefix meta- (metá, beyond in the sense of high-level) with heuristic (from the Greek heuriskein or euriskein, to search). Metaheuristic algorithms, i.e., optimization methods designed according to the strategies laid out in a metaheuristic framework, are — as the name suggests — always heuristic in nature. This fact distinguishes them from exact methods, that do come with a proof that the optimal solution will be found in a finite (although often prohibitively large) amount of time. Metaheuristics are therefore developed specifically to find a solution that is “good enough” in a computing time that is “small enough”. As a result, they are not subject to combinatorial explosion – the phenomenon where the computing time required to find the optimal solution of NP- hard problems increases as an exponential function of the problem size. Metaheuristics have been demonstrated by the scientific community to be a viable, and often superior, alternative to more traditional (exact) methods of mixed- integer optimization such as branch and bound and dynamic programming. -
Convex Optimisation Matlab Toolbox
UNLOCBOX: USER’S GUIDE MATLAB CONVEX OPTIMIZATION TOOLBOX Lausanne - February 2014 PERRAUDIN Nathanaël, KALOFOLIAS Vassilis LTS2 - EPFL Abstract Nowadays the trend to solve optimization problems is to use specific algorithms rather than very gen- eral ones. The UNLocBoX provides a general framework allowing the user to design his own algorithms. To do so, the framework try to stay as close from the mathematical problem as possible. More precisely, the UNLocBoX is a Matlab toolbox designed to solve convex optimization problem of the form K min ∑ fn(x); x2C n=1 using proximal splitting techniques. It is mainly composed of solvers, proximal operators and demonstra- tion files allowing the user to quickly implement a problem. Contents 1 Introduction 1 2 Literature 2 3 Installation and initialization2 3.1 Dependencies.........................................2 3.2 GPU computation.......................................2 4 Structure of the UNLocBoX2 5 Problems of interest4 6 Solvers 4 6.1 Defining functions......................................5 6.2 Selecting a solver.......................................5 6.3 Optional parameters......................................6 6.4 Plug-ins: how to tune your algorithm.............................6 6.5 Returned arguments......................................8 7 Proximal operators9 7.1 Constraints..........................................9 8 Example 10 References 16 LTS2 - EPFL 1 Introduction During your research, you have most likely faced the situation where you need to solve a convex optimiza- tion problem. Maybe you were trying to find an optimal combination of parameters, you were transforming some process into something automatic or, even more simply, your algorithm was equivalent to solving convex problem. In this situation, you usually face a dilemma: you need a specific algorithm to solve your problem, but you do not want to spend hours writing it. -
A Lifted Linear Programming Branch-And-Bound Algorithm for Mixed Integer Conic Quadratic Programs Juan Pablo Vielma, Shabbir Ahmed, George L
A Lifted Linear Programming Branch-and-Bound Algorithm for Mixed Integer Conic Quadratic Programs Juan Pablo Vielma, Shabbir Ahmed, George L. Nemhauser, H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 765 Ferst Drive NW, Atlanta, GA 30332-0205, USA, {[email protected], [email protected], [email protected]} This paper develops a linear programming based branch-and-bound algorithm for mixed in- teger conic quadratic programs. The algorithm is based on a higher dimensional or lifted polyhedral relaxation of conic quadratic constraints introduced by Ben-Tal and Nemirovski. The algorithm is different from other linear programming based branch-and-bound algo- rithms for mixed integer nonlinear programs in that, it is not based on cuts from gradient inequalities and it sometimes branches on integer feasible solutions. The algorithm is tested on a series of portfolio optimization problems. It is shown that it significantly outperforms commercial and open source solvers based on both linear and nonlinear relaxations. Key words: nonlinear integer programming; branch and bound; portfolio optimization History: February 2007. 1. Introduction This paper deals with the development of an algorithm for the class of mixed integer non- linear programming (MINLP) problems known as mixed integer conic quadratic program- ming problems. This class of problems arises from adding integrality requirements to conic quadratic programming problems (Lobo et al., 1998), and is used to model several applica- tions from engineering and finance. Conic quadratic programming problems are also known as second order cone programming problems, and together with semidefinite and linear pro- gramming (LP) problems are special cases of the more general conic programming problems (Ben-Tal and Nemirovski, 2001a). -
Linear Programming
Lecture 18 Linear Programming 18.1 Overview In this lecture we describe a very general problem called linear programming that can be used to express a wide variety of different kinds of problems. We can use algorithms for linear program- ming to solve the max-flow problem, solve the min-cost max-flow problem, find minimax-optimal strategies in games, and many other things. We will primarily discuss the setting and how to code up various problems as linear programs At the end, we will briefly describe some of the algorithms for solving linear programming problems. Specific topics include: • The definition of linear programming and simple examples. • Using linear programming to solve max flow and min-cost max flow. • Using linear programming to solve for minimax-optimal strategies in games. • Algorithms for linear programming. 18.2 Introduction In the last two lectures we looked at: — Bipartite matching: given a bipartite graph, find the largest set of edges with no endpoints in common. — Network flow (more general than bipartite matching). — Min-Cost Max-flow (even more general than plain max flow). Today, we’ll look at something even more general that we can solve algorithmically: linear pro- gramming. (Except we won’t necessarily be able to get integer solutions, even when the specifi- cation of the problem is integral). Linear Programming is important because it is so expressive: many, many problems can be coded up as linear programs (LPs). This especially includes problems of allocating resources and business 95 18.3. DEFINITION OF LINEAR PROGRAMMING 96 supply-chain applications. In business schools and Operations Research departments there are entire courses devoted to linear programming. -
Conditional Gradient Sliding for Convex Optimization ∗
CONDITIONAL GRADIENT SLIDING FOR CONVEX OPTIMIZATION ∗ GUANGHUI LAN y AND YI ZHOU z Abstract. In this paper, we present a new conditional gradient type method for convex optimization by calling a linear optimization (LO) oracle to minimize a series of linear functions over the feasible set. Different from the classic conditional gradient method, the conditional gradient sliding (CGS) algorithm developed herein can skip the computation of gradients from time to time, and as a result, can achieve the optimal complexity bounds in terms of not only the number of callsp to the LO oracle, but also the number of gradient evaluations. More specifically, we show that the CGS method requires O(1= ) and O(log(1/)) gradient evaluations, respectively, for solving smooth and strongly convex problems, while still maintaining the optimal O(1/) bound on the number of calls to LO oracle. We also develop variants of the CGS method which can achieve the optimal complexity bounds for solving stochastic optimization problems and an important class of saddle point optimization problems. To the best of our knowledge, this is the first time that these types of projection-free optimal first-order methods have been developed in the literature. Some preliminary numerical results have also been provided to demonstrate the advantages of the CGS method. Keywords: convex programming, complexity, conditional gradient method, Frank-Wolfe method, Nesterov's method AMS 2000 subject classification: 90C25, 90C06, 90C22, 49M37 1. Introduction. The conditional gradient (CndG) method, which was initially developed by Frank and Wolfe in 1956 [13] (see also [11, 12]), has been considered one of the earliest first-order methods for solving general convex programming (CP) problems. -
Integer Linear Programs
20 ________________________________________________________________________________________________ Integer Linear Programs Many linear programming problems require certain variables to have whole number, or integer, values. Such a requirement arises naturally when the variables represent enti- ties like packages or people that can not be fractionally divided — at least, not in a mean- ingful way for the situation being modeled. Integer variables also play a role in formulat- ing equation systems that model logical conditions, as we will show later in this chapter. In some situations, the optimization techniques described in previous chapters are suf- ficient to find an integer solution. An integer optimal solution is guaranteed for certain network linear programs, as explained in Section 15.5. Even where there is no guarantee, a linear programming solver may happen to find an integer optimal solution for the par- ticular instances of a model in which you are interested. This happened in the solution of the multicommodity transportation model (Figure 4-1) for the particular data that we specified (Figure 4-2). Even if you do not obtain an integer solution from the solver, chances are good that you’ll get a solution in which most of the variables lie at integer values. Specifically, many solvers are able to return an ‘‘extreme’’ solution in which the number of variables not lying at their bounds is at most the number of constraints. If the bounds are integral, all of the variables at their bounds will have integer values; and if the rest of the data is integral, many of the remaining variables may turn out to be integers, too. -
Chapter 11: Basic Linear Programming Concepts
CHAPTER 11: BASIC LINEAR PROGRAMMING CONCEPTS Linear programming is a mathematical technique for finding optimal solutions to problems that can be expressed using linear equations and inequalities. If a real-world problem can be represented accurately by the mathematical equations of a linear program, the method will find the best solution to the problem. Of course, few complex real-world problems can be expressed perfectly in terms of a set of linear functions. Nevertheless, linear programs can provide reasonably realistic representations of many real-world problems — especially if a little creativity is applied in the mathematical formulation of the problem. The subject of modeling was briefly discussed in the context of regulation. The regulation problems you learned to solve were very simple mathematical representations of reality. This chapter continues this trek down the modeling path. As we progress, the models will become more mathematical — and more complex. The real world is always more complex than a model. Thus, as we try to represent the real world more accurately, the models we build will inevitably become more complex. You should remember the maxim discussed earlier that a model should only be as complex as is necessary in order to represent the real world problem reasonably well. Therefore, the added complexity introduced by using linear programming should be accompanied by some significant gains in our ability to represent the problem and, hence, in the quality of the solutions that can be obtained. You should ask yourself, as you learn more about linear programming, what the benefits of the technique are and whether they outweigh the additional costs. -
Chapter 7. Linear Programming and Reductions
Chapter 7 Linear programming and reductions Many of the problems for which we want algorithms are optimization tasks: the shortest path, the cheapest spanning tree, the longest increasing subsequence, and so on. In such cases, we seek a solution that (1) satisfies certain constraints (for instance, the path must use edges of the graph and lead from s to t, the tree must touch all nodes, the subsequence must be increasing); and (2) is the best possible, with respect to some well-defined criterion, among all solutions that satisfy these constraints. Linear programming describes a broad class of optimization tasks in which both the con- straints and the optimization criterion are linear functions. It turns out an enormous number of problems can be expressed in this way. Given the vastness of its topic, this chapter is divided into several parts, which can be read separately subject to the following dependencies. Flows and matchings Introduction to linear programming Duality Games and reductions Simplex 7.1 An introduction to linear programming In a linear programming problem we are given a set of variables, and we want to assign real values to them so as to (1) satisfy a set of linear equations and/or linear inequalities involving these variables and (2) maximize or minimize a given linear objective function. 201 202 Algorithms Figure 7.1 (a) The feasible region for a linear program. (b) Contour lines of the objective function: x1 + 6x2 = c for different values of the profit c. x x (a) 2 (b) 2 400 400 Optimum point Profit = $1900 ¡ ¡ ¡ ¡ ¡ 300 ¢¡¢¡¢¡¢¡¢ 300 ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ 200 200 c = 1500 ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ c = 1200 ¡ ¡ ¡ ¡ ¡ 100 ¢¡¢¡¢¡¢¡¢ 100 ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ c = 600 ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ x1 x1 0 100 200 300 400 0 100 200 300 400 7.1.1 Example: profit maximization A boutique chocolatier has two products: its flagship assortment of triangular chocolates, called Pyramide, and the more decadent and deluxe Pyramide Nuit. -
Linear Programming
Stanford University | CS261: Optimization Handout 5 Luca Trevisan January 18, 2011 Lecture 5 In which we introduce linear programming. 1 Linear Programming A linear program is an optimization problem in which we have a collection of variables, which can take real values, and we want to find an assignment of values to the variables that satisfies a given collection of linear inequalities and that maximizes or minimizes a given linear function. (The term programming in linear programming, is not used as in computer program- ming, but as in, e.g., tv programming, to mean planning.) For example, the following is a linear program. maximize x1 + x2 subject to x + 2x ≤ 1 1 2 (1) 2x1 + x2 ≤ 1 x1 ≥ 0 x2 ≥ 0 The linear function that we want to optimize (x1 + x2 in the above example) is called the objective function.A feasible solution is an assignment of values to the variables that satisfies the inequalities. The value that the objective function gives 1 to an assignment is called the cost of the assignment. For example, x1 := 3 and 1 2 x2 := 3 is a feasible solution, of cost 3 . Note that if x1; x2 are values that satisfy the inequalities, then, by summing the first two inequalities, we see that 3x1 + 3x2 ≤ 2 that is, 1 2 x + x ≤ 1 2 3 2 1 1 and so no feasible solution has cost higher than 3 , so the solution x1 := 3 , x2 := 3 is optimal. As we will see in the next lecture, this trick of summing inequalities to verify the optimality of a solution is part of the very general theory of duality of linear programming. -
Implementing Customized Pivot Rules in COIN-OR's CLP with Python
Cahier du GERAD G-2012-07 Customizing the Solution Process of COIN-OR's Linear Solvers with Python Mehdi Towhidi1;2 and Dominique Orban1;2 ? 1 Department of Mathematics and Industrial Engineering, Ecole´ Polytechnique, Montr´eal,QC, Canada. 2 GERAD, Montr´eal,QC, Canada. [email protected], [email protected] Abstract. Implementations of the Simplex method differ only in very specific aspects such as the pivot rule. Similarly, most relaxation methods for mixed-integer programming differ only in the type of cuts and the exploration of the search tree. Implementing instances of those frame- works would therefore be more efficient if linear and mixed-integer pro- gramming solvers let users customize such aspects easily. We provide a scripting mechanism to easily implement and experiment with pivot rules for the Simplex method by building upon COIN-OR's open-source linear programming package CLP. Our mechanism enables users to implement pivot rules in the Python scripting language without explicitly interact- ing with the underlying C++ layers of CLP. In the same manner, it allows users to customize the solution process while solving mixed-integer linear programs using the CBC and CGL COIN-OR packages. The Cython pro- gramming language ensures communication between Python and COIN- OR libraries and activates user-defined customizations as callbacks. For illustration, we provide an implementation of a well-known pivot rule as well as the positive edge rule|a new rule that is particularly efficient on degenerate problems, and demonstrate how to customize branch-and-cut node selection in the solution of a mixed-integer program. -
IEOR 269, Spring 2010 Integer Programming and Combinatorial Optimization
IEOR 269, Spring 2010 Integer Programming and Combinatorial Optimization Professor Dorit S. Hochbaum Contents 1 Introduction 1 2 Formulation of some ILP 2 2.1 0-1 knapsack problem . 2 2.2 Assignment problem . 2 3 Non-linear Objective functions 4 3.1 Production problem with set-up costs . 4 3.2 Piecewise linear cost function . 5 3.3 Piecewise linear convex cost function . 6 3.4 Disjunctive constraints . 7 4 Some famous combinatorial problems 7 4.1 Max clique problem . 7 4.2 SAT (satisfiability) . 7 4.3 Vertex cover problem . 7 5 General optimization 8 6 Neighborhood 8 6.1 Exact neighborhood . 8 7 Complexity of algorithms 9 7.1 Finding the maximum element . 9 7.2 0-1 knapsack . 9 7.3 Linear systems . 10 7.4 Linear Programming . 11 8 Some interesting IP formulations 12 8.1 The fixed cost plant location problem . 12 8.2 Minimum/maximum spanning tree (MST) . 12 9 The Minimum Spanning Tree (MST) Problem 13 i IEOR269 notes, Prof. Hochbaum, 2010 ii 10 General Matching Problem 14 10.1 Maximum Matching Problem in Bipartite Graphs . 14 10.2 Maximum Matching Problem in Non-Bipartite Graphs . 15 10.3 Constraint Matrix Analysis for Matching Problems . 16 11 Traveling Salesperson Problem (TSP) 17 11.1 IP Formulation for TSP . 17 12 Discussion of LP-Formulation for MST 18 13 Branch-and-Bound 20 13.1 The Branch-and-Bound technique . 20 13.2 Other Branch-and-Bound techniques . 22 14 Basic graph definitions 23 15 Complexity analysis 24 15.1 Measuring quality of an algorithm .