First-Order Methods for Semidefinite Programming
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An Algorithm Based on Semidefinite Programming for Finding Minimax
Takustr. 7 Zuse Institute Berlin 14195 Berlin Germany BELMIRO P.M. DUARTE,GUILLAUME SAGNOL,WENG KEE WONG An algorithm based on Semidefinite Programming for finding minimax optimal designs ZIB Report 18-01 (December 2017) Zuse Institute Berlin Takustr. 7 14195 Berlin Germany Telephone: +49 30-84185-0 Telefax: +49 30-84185-125 E-mail: [email protected] URL: http://www.zib.de ZIB-Report (Print) ISSN 1438-0064 ZIB-Report (Internet) ISSN 2192-7782 An algorithm based on Semidefinite Programming for finding minimax optimal designs Belmiro P.M. Duarte a,b, Guillaume Sagnolc, Weng Kee Wongd aPolytechnic Institute of Coimbra, ISEC, Department of Chemical and Biological Engineering, Portugal. bCIEPQPF, Department of Chemical Engineering, University of Coimbra, Portugal. cTechnische Universität Berlin, Institut für Mathematik, Germany. dDepartment of Biostatistics, Fielding School of Public Health, UCLA, U.S.A. Abstract An algorithm based on a delayed constraint generation method for solving semi- infinite programs for constructing minimax optimal designs for nonlinear models is proposed. The outer optimization level of the minimax optimization problem is solved using a semidefinite programming based approach that requires the de- sign space be discretized. A nonlinear programming solver is then used to solve the inner program to determine the combination of the parameters that yields the worst-case value of the design criterion. The proposed algorithm is applied to find minimax optimal designs for the logistic model, the flexible 4-parameter Hill homoscedastic model and the general nth order consecutive reaction model, and shows that it (i) produces designs that compare well with minimax D−optimal de- signs obtained from semi-infinite programming method in the literature; (ii) can be applied to semidefinite representable optimality criteria, that include the com- mon A−; E−; G−; I− and D-optimality criteria; (iii) can tackle design problems with arbitrary linear constraints on the weights; and (iv) is fast and relatively easy to use. -
A Framework for Solving Mixed-Integer Semidefinite Programs
A FRAMEWORK FOR SOLVING MIXED-INTEGER SEMIDEFINITE PROGRAMS TRISTAN GALLY, MARC E. PFETSCH, AND STEFAN ULBRICH ABSTRACT. Mixed-integer semidefinite programs arise in many applications and several problem-specific solution approaches have been studied recently. In this paper, we investigate a generic branch-and-bound framework for solving such problems. We first show that strict duality of the semidefinite relaxations is inherited to the subproblems. Then solver components like dual fixing, branch- ing rules, and primal heuristics are presented. We show the applicability of an implementation of the proposed methods on three kinds of problems. The re- sults show the positive computational impact of the different solver components, depending on the semidefinite programming solver used. This demonstrates that practically relevant mixed-integer semidefinite programs can successfully be solved using a general purpose solver. 1. INTRODUCTION The solution of mixed-integer semidefinite programs (MISDPs), i.e., semidefinite programs (SDPs) with integrality constraints on some of the variables, received increasing attention in recent years. There are two main application areas of such problems: In the first area, they arise because they capture an essential structure of the problem. For instance, SDPs are important for expressing ellipsoidal uncer- tainty sets in robust optimization, see, e.g., Ben-Tal et al. [10]. In particular, SDPs appear in the context of optimizing robust truss structures, see, e.g., Ben-Tal and Nemirovski [11] as well as Bendsøe and Sigmund [12]. Allowing to change the topological structure then yields MISDPs; Section 9.1 will provide more details. Another application arises in the design of downlink beamforming, see, e.g., [56]. -
Abstracts Exact Separation of K-Projection Polytope Constraints Miguel F
13th EUROPT Workshop on Advances in Continuous Optimization 8-10 July [email protected] http://www.maths.ed.ac.uk/hall/EUROPT15 Abstracts Exact Separation of k-Projection Polytope Constraints Miguel F. Anjos Cutting planes are often used as an efficient means to tighten continuous relaxations of mixed-integer optimization problems and are a vital component of branch-and-cut algorithms. A critical step of any cutting plane algorithm is to find valid inequalities, or cuts, that improve the current relaxation of the integer-constrained problem. The maximally violated valid inequality problem aims to find the most violated inequality from a given family of cuts. $k$-projection polytope constraints are a family of cuts that are based on an inner description of the cut polytope of size $k$ and are applied to $k\times k$ principal minors of a semidefinite optimization relaxation. We propose a bilevel second order cone optimization approach to solve the maximally violated $k$ projection polytope constraint problem. We reformulate the bilevel problem as a single-level mixed binary second order cone optimization problem that can be solved using off-the-shelf conic optimization software. Additionally we present two methods for improving the computational performance of our approach, namely lexicographical ordering and a reformulation with fewer binary variables. All of the proposed formulations are exact. Preliminary results show the computational time and number of branch-and-bound nodes required to solve small instances in the context of the max-cut problem. Global convergence of the Douglas{Rachford method for some nonconvex feasibility problems Francisco J. -
PENALTY METHODS for the SOLUTION of GENERALIZED NASH EQUILIBRIUM PROBLEMS (WITH COMPLETE TEST PROBLEMS) Francisco Facchinei1
PENALTY METHODS FOR THE SOLUTION OF GENERALIZED NASH EQUILIBRIUM PROBLEMS (WITH COMPLETE TEST PROBLEMS) Francisco Facchinei1 and Christian Kanzow2 1 Sapienza University of Rome Department of Computer and System Sciences “A. Ruberti” Via Ariosto 25, 00185 Roma, Italy e-mail: [email protected] 2 University of W¨urzburg Institute of Mathematics Am Hubland, 97074 W¨urzburg, Germany e-mail: [email protected] February 11, 2009 Abstract The generalized Nash equilibrium problem (GNEP) is an extension of the classical Nash equilibrium problem where both the objective functions and the constraints of each player may depend on the rivals’ strategies. This class of problems has a multitude of important engineering applications and yet solution algorithms are extremely scarce. In this paper, we analyze in detail a globally convergent penalty method that has favorable theoretical properties. We also consider strengthened results for a particular subclass of problems very often considered in the literature. Basically our method reduces the GNEP to a single penalized (and nonsmooth) Nash equilibrium problem. We suggest a suitable method for the solution of the latter penalized problem and present extensive numerical results. Key Words: Nash equilibrium problem, Generalized Nash equilibrium problem, Jointly convex problem, Exact penalty function, Global convergence. 1 Introduction In this paper, we consider the Generalized Nash Equilibrium Problem (GNEP) and describe new globally convergent methods for its solution. The GNEP is an extension of the classical Nash Equilibrium Problem (NEP) where the feasible sets of each player may depend on ν n the rivals’ strategies. We assume there are N players and denote by x ∈ R ν the vector representing the ν-th player’s strategy. -
A Survey of Numerical Methods for Nonlinear Semidefinite Programming
Journal of the Operations Research Society of Japan ⃝c The Operations Research Society of Japan Vol. 58, No. 1, January 2015, pp. 24–60 A SURVEY OF NUMERICAL METHODS FOR NONLINEAR SEMIDEFINITE PROGRAMMING Hiroshi Yamashita Hiroshi Yabe NTT DATA Mathematical Systems Inc. Tokyo University of Science (Received September 16, 2014; Revised December 22, 2014) Abstract Nonlinear semidefinite programming (SDP) problems have received a lot of attentions because of large variety of applications. In this paper, we survey numerical methods for solving nonlinear SDP problems. Three kinds of typical numerical methods are described; augmented Lagrangian methods, sequential SDP methods and primal-dual interior point methods. We describe their typical algorithmic forms and discuss their global and local convergence properties which include rate of convergence. Keywords: Optimization, nonlinear semidefinite programming, augmented Lagrangian method, sequential SDP method, primal-dual interior point method 1. Introduction This paper is concerned with the nonlinear SDP problem: minimize f(x); x 2 Rn; (1.1) subject to g(x) = 0;X(x) º 0 where the functions f : Rn ! R, g : Rn ! Rm and X : Rn ! Sp are sufficiently smooth, p p and S denotes the set of pth-order real symmetric matrices. We also define S+ to denote the set of pth-order symmetric positive semidefinite matrices. By X(x) º 0 and X(x) Â 0, we mean that the matrix X(x) is positive semidefinite and positive definite, respectively. When f is a convex function, X is a concave function and g is affine, this problem is a convex optimization problem. Here the concavity of X(x) means that X(¸u + (1 ¡ ¸)v) ¡ ¸X(u) ¡ (1 ¡ ¸)X(v) º 0 holds for any u; v 2 Rn and any ¸ satisfying 0 · ¸ · 1. -
Introduction to Semidefinite Programming (SDP)
Introduction to Semidefinite Programming (SDP) Robert M. Freund March, 2004 2004 Massachusetts Institute of Technology. 1 1 Introduction Semidefinite programming (SDP ) is the most exciting development in math- ematical programming in the 1990’s. SDP has applications in such diverse fields as traditional convex constrained optimization, control theory, and combinatorial optimization. Because SDP is solvable via interior point methods, most of these applications can usually be solved very efficiently in practice as well as in theory. 2 Review of Linear Programming Consider the linear programming problem in standard form: LP : minimize c · x s.t. ai · x = bi,i=1 ,...,m n x ∈+. x n c · x Here is a vector of variables, and we write “ ” for the inner-product n “ j=1 cj xj ”, etc. n n n Also, + := {x ∈ | x ≥ 0},and we call + the nonnegative orthant. n In fact, + is a closed convex cone, where K is called a closed a convex cone if K satisfies the following two conditions: • If x, w ∈ K, then αx + βw ∈ K for all nonnegative scalars α and β. • K is a closed set. 2 In words, LP is the following problem: “Minimize the linear function c·x, subject to the condition that x must solve m given equations ai · x = bi,i =1,...,m, and that x must lie in the closed n convex cone K = +.” We will write the standard linear programming dual problem as: m LD : maximize yibi i=1 m s.t. yiai + s = c i=1 n s ∈+. Given a feasible solution x of LP and a feasible solution (y, s)ofLD , the m m duality gap is simply c·x− i=1 yibi =(c−i=1 yiai) ·x = s·x ≥ 0, because x ≥ 0ands ≥ 0. -
Semidefinite Optimization ∗
Semidefinite Optimization ∗ M. J. Todd † August 22, 2001 Abstract Optimization problems in which the variable is not a vector but a symmetric matrix which is required to be positive semidefinite have been intensely studied in the last ten years. Part of the reason for the interest stems from the applicability of such problems to such diverse areas as designing the strongest column, checking the stability of a differential inclusion, and obtaining tight bounds for hard combinatorial optimization problems. Part also derives from great advances in our ability to solve such problems efficiently in theory and in practice (perhaps “or” would be more appropriate: the most effective computational methods are not always provably efficient in theory, and vice versa). Here we describe this class of optimization problems, give a number of examples demonstrating its significance, outline its duality theory, and discuss algorithms for solving such problems. ∗Copyright (C) by Cambridge University Press, Acta Numerica 10 (2001) 515–560. †School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York 14853, USA ([email protected]). Research supported in part by NSF through grant DMS-9805602 and ONR through grant N00014-96-1-0050. 1 1 Introduction Semidefinite optimization is concerned with choosing a symmetric matrix to optimize a linear function subject to linear constraints and a further crucial constraint that the matrix be positive semidefinite. It thus arises from the well-known linear programming problem by replacing the -
Constraint-Handling Techniques for Highly Constrained Optimization
Andreas Petrow Constraint-Handling Techniques for highly Constrained Optimization Intelligent Cooperative Systems Computational Intelligence Constraint-Handling Techniques for highly Constrained Optimization Master Thesis Andreas Petrow March 22, 2019 Supervisor: Prof. Dr.-Ing. habil. Sanaz Mostaghim Advisor: M.Sc. Heiner Zille Advisor: Dr.-Ing. Markus Schori Andreas Petrow: Constraint-Handling Techniques for highly Con- strained Optimization Otto-von-Guericke Universität Intelligent Cooperative Systems Computational Intelligence Magdeburg, 2019. Abstract This thesis relates to constraint handling techniques for highly constrained opti- mization. Many real world optimization problems are non-linear and restricted to constraints. To solve non-linear optimization problems population-based meta- heuristics are used commonly. To enable such optimization methods to fulfill the optimization with respect to the constraints modifications are required. Therefore a research field considering such constraint handling techniques establishes. How- ever the combination of large-scale optimization problems with constraint handling techniques is rare. Therefore this thesis involves with common constraint handling techniques and evaluates these on two benchmark problems. I Contents List of Figures VII List of TablesXI List of Acronyms XIII 1. Introduction and Motivation1 1.1. Aim of this thesis...........................2 1.2. Structure of this thesis........................3 2. Basics5 2.1. Constrained Optimization Problem..................5 2.2. Optimization Method.........................6 2.2.1. Classical optimization....................7 2.2.2. Population-based Optimization Methods...........7 3. Related Work 11 3.1. Penalty Methods........................... 12 3.1.1. Static Penalty Functions................... 12 3.1.2. Co-Evolution Penalty Methods................ 13 3.1.3. Dynamic Penalty Functions................. 13 3.1.4. Adaptive Penalty Functions................. 13 3.2. -
Linear Programming Notes
Linear Programming Notes Lecturer: David Williamson, Cornell ORIE Scribe: Kevin Kircher, Cornell MAE These notes summarize the central definitions and results of the theory of linear program- ming, as taught by David Williamson in ORIE 6300 at Cornell University in the fall of 2014. Proofs and discussion are mostly omitted. These notes also draw on Convex Optimization by Stephen Boyd and Lieven Vandenberghe, and on Stephen Boyd's notes on ellipsoid methods. Prof. Williamson's full lecture notes can be found here. Contents 1 The linear programming problem3 2 Duailty 5 3 Geometry6 4 Optimality conditions9 4.1 Answer 1: cT x = bT y for a y 2 F(D)......................9 4.2 Answer 2: complementary slackness holds for a y 2 F(D)........... 10 4.3 Answer 3: the verifying y is in F(D)...................... 10 5 The simplex method 12 5.1 Reformulation . 12 5.2 Algorithm . 12 5.3 Convergence . 14 5.4 Finding an initial basic feasible solution . 15 5.5 Complexity of a pivot . 16 5.6 Pivot rules . 16 5.7 Degeneracy and cycling . 17 5.8 Number of pivots . 17 5.9 Varieties of the simplex method . 18 5.10 Sensitivity analysis . 18 5.11 Solving large-scale linear programs . 20 1 6 Good algorithms 23 7 Ellipsoid methods 24 7.1 Ellipsoid geometry . 24 7.2 The basic ellipsoid method . 25 7.3 The ellipsoid method with objective function cuts . 28 7.4 Separation oracles . 29 8 Interior-point methods 30 8.1 Finding a descent direction that preserves feasibility . 30 8.2 The affine-scaling direction . -
Semidefinite Programming
Semidefinite Programming Yinyu Ye December 2002 i ii Preface This is a monograph for MS&E 314, “Semidefinite Programming”, which I am teaching at Stanford. Information, supporting materials, and computer programs related to this book may be found at the following address on the World-Wide Web: http://www.stanford.edu/class/msande314 Please report any question, comment and error to the address: [email protected] YINYU YE Stanford, 2002 iii iv PREFACE Chapter 1 Introduction and Preliminaries 1.1 Introduction Semidefinite Programming, hereafter SDP, is a natural extension of Linear programming (LP) that is a central decision model in Management Science and Operations Research. LP plays an extremely important role in the theory and application of Optimization. In one sense it is a continuous optimization problem in minimizing a linear objective function over a con- vex polyhedron; but it is also a combinatorial problem involving selecting an extreme point among a finite set of possible vertices. Businesses, large and small, use linear programming models to optimize communication sys- tems, to schedule transportation networks, to control inventories, to adjust investments, and to maximize productivity. In LP, the variables form a vector which is regiured to be nonnegative, where in SDP they are components of a matrix and it is constrained to be positive semidefinite. Both of them may have linear equality constraints as well. One thing in common is that interior-point algorithms developed in past two decades for LP are naturally applied to solving SDP. Interior-point algorithms are continuous iterative algorithms. Compu- tation experience with sophisticated procedures suggests that the num- ber of iterations necessarily grows much more slowly than the dimension grows. -
Numerical Solution of Saddle Point Problems
Acta Numerica (2005), pp. 1–137 c Cambridge University Press, 2005 DOI: 10.1017/S0962492904000212 Printed in the United Kingdom Numerical solution of saddle point problems Michele Benzi∗ Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322, USA E-mail: [email protected] Gene H. Golub† Scientific Computing and Computational Mathematics Program, Stanford University, Stanford, California 94305-9025, USA E-mail: [email protected] J¨org Liesen‡ Institut f¨ur Mathematik, Technische Universit¨at Berlin, D-10623 Berlin, Germany E-mail: [email protected] We dedicate this paper to Gil Strang on the occasion of his 70th birthday Large linear systems of saddle point type arise in a wide variety of applica- tions throughout computational science and engineering. Due to their indef- initeness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems. ∗ Supported in part by the National Science Foundation grant DMS-0207599. † Supported in part by the Department of Energy of the United States Government. ‡ Supported in part by the Emmy Noether Programm of the Deutsche Forschungs- gemeinschaft. 2 M. Benzi, G. H. Golub and J. Liesen CONTENTS 1 Introduction 2 2 Applications leading to saddle point problems 5 3 Properties of saddle point matrices 14 4 Overview of solution algorithms 29 5 Schur complement reduction 30 6 Null space methods 32 7 Coupled direct solvers 40 8 Stationary iterations 43 9 Krylov subspace methods 49 10 Preconditioners 59 11 Multilevel methods 96 12 Available software 105 13 Concluding remarks 107 References 109 1. -
Open Source Tools for Optimization in Python
Open Source Tools for Optimization in Python Ted Ralphs Sage Days Workshop IMA, Minneapolis, MN, 21 August 2017 T.K. Ralphs (Lehigh University) Open Source Optimization August 21, 2017 Outline 1 Introduction 2 COIN-OR 3 Modeling Software 4 Python-based Modeling Tools PuLP/DipPy CyLP yaposib Pyomo T.K. Ralphs (Lehigh University) Open Source Optimization August 21, 2017 Outline 1 Introduction 2 COIN-OR 3 Modeling Software 4 Python-based Modeling Tools PuLP/DipPy CyLP yaposib Pyomo T.K. Ralphs (Lehigh University) Open Source Optimization August 21, 2017 Caveats and Motivation Caveats I have no idea about the background of the audience. The talk may be either too basic or too advanced. Why am I here? I’m not a Sage developer or user (yet!). I’m hoping this will be a chance to get more involved in Sage development. Please ask lots of questions so as to guide me in what to dive into! T.K. Ralphs (Lehigh University) Open Source Optimization August 21, 2017 Mathematical Optimization Mathematical optimization provides a formal language for describing and analyzing optimization problems. Elements of the model: Decision variables Constraints Objective Function Parameters and Data The general form of a mathematical optimization problem is: min or max f (x) (1) 8 9 < ≤ = s.t. gi(x) = bi (2) : ≥ ; x 2 X (3) where X ⊆ Rn might be a discrete set. T.K. Ralphs (Lehigh University) Open Source Optimization August 21, 2017 Types of Mathematical Optimization Problems The type of a mathematical optimization problem is determined primarily by The form of the objective and the constraints.