Hoàng Tuỵ được coi là “cha đẻ của tối ưu toàn cục” (the father of Global Optimization). Ngày nay, bất cứ ai trên thế giới muốn đi vào chuyên ngành này, đều phải học những điều đã trở thành kinh điển như Tuy’s cut (lát cắt Tuỵ), Tuytype algorithm (thuật toán kiểu Tuỵ), Tuy’s inconsistency condition (điều kiện không tương thích Tuỵ. Cuốn sách toán tiếng Anh do GS Hoàng Tuỵ viết chung với GS Reiner Horst (CHLB Đức) nhan đề Global Optimization Deterministic Approches (Tối ưu toàn cục tiếp cận tất định) dày 694 trang, được nhà xuất bản Springer Verlag in lần đầu năm 1990, lần thứ hai năm 1993, lần thứ ba (có sửa chữa) năm 1996.
Reiner Horst· Hoang Tuy Global Optimization Deterministic Approaches With 55 Figures Springer-Verlag Berlin Heidelberg GmbH Professor Or Reiner Horst University of Trier Oepartment of Mathematics P.O.Box 3825 0-5500 Trier, FRG Professor Or Hoang Tuy Vien Toan Hoc Institute of Mathematics P.O.Box 631, BO-HO 10000 Hanoi, Vietnam ISBN 978-3-662-02600-7 ISBN 978-3-662-02598-7 (eBook) DOI 10.1007/978-3-662-02598-7 This work is subject to copyright Ali rights are reserved, whether the whole or part ofthe material is concemed, specifically the rights oftranslation, reprinting, reuse ofillustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks Duplication ofthis publication or parts thereof is ooly permitted under the provisions ofthe German Copyright Law ofSeptember 9,1965, in its version ofJune 24, 1985, and a copyright fee must always be paid Violations fali under the prosecution act ofthe German Copyright Law © Springer-Verlag Berlin Heidelberg 1990 OriginaIly published by Springer-Verlag Berlin Heidelberg New York Tokyo in 1990 Softcover reprint of the hardcover 1st edition 1990 The use ofregistered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use 2142/7130-543210 PREFACE The enormous practical need for solving global optimization problems coupled with a rapidly advancing computer technology has allowed one to consider problems which a few years ago would have been considered computationally intractable As a consequence, we are seeing the creation of a large and increasing number of diverse algorithms for solving a wide variety of multiextremal global optimization problems The goal of this book is to systematically clarify and unify these diverse approaches in order to provide insight into the underlying concepts and their properties Aside from a coherent view of the field much new material is presented By definition, a multiextremal global optimization problem seeks at least one global minimizer of a real-valued objective function that possesses different local minimizers The feasible set of points in IRn is usually determined by a system of inequalities It is well known that in practically all disciplines where mathematical models are used there are many real-world problems which can be formulated as multi extremal global optimization problems Standard nonlinear programming techniques have not been successful for solving these problems Their deficiency is due to the intrinsic multiextremality of the formulation and not to the lack of smoothness or continuity, for often the latter properties are present One can observe that local· tools such as gradients, subgradients, and second order constructions such as Hessians, cannot be expected to yield more than local solutions One finds, for example, that a stationary point is often detected for which there is even no guarantee of local minimality Moreover, determining the local minimality of such a point is known to be NP-hard in the sense of computational complexity even in relatively simple cases Apart from this deficiency in the local situation, classical methods not recognize conditions for global optimality VI For these reasons global solution methods must be significantly different from standard nonlinear programming techniques, and they can be expected to be - and are - much more expensive computationally Throughout this book our focus will be on typical procedures that respond to the inherent difficulty of multiextremality and which take advantage of helpful specific features of the problem structure In certain sections, methods are presented for solving very general and difficult global problems, but the reader should be aware that difficult large scale global optimization problems cannot be solved with sufficient accuracy on currently available computers For these very general cases our exposition is intended to provide useful tools for transcending local optimality restrictions, in the sense of providing valuable information about the global quality of a given feasible point Typically, such information will give upper and lower bounds for the optimal objective function value and indicate parts of the feasible set where further investigations of global optimality will not be worthwhile On the other hand, in many practical global optimizations, the multiextremal feature involves only a small number of variables Moreover, many problems have additional structure that is amenable to large scale solutions Many global optimization problems encountered in the decision sciences, engineering and operations research have at least the following closely related key properties: (i) convexity is present in a limited and often unusual sense; (ii) a global optimum occurs within a subset of the boundary of the feasible set With the current state of the art, these properties are best exploited by deterministic methods that combine analytical and combinatorial tools in an effective way We find that typical approaches use techniques such as branch and bound, relaxation, outer approximation, and valid cutting planes, whose basic principles have long appeared in the related fields of integer and combinatorial optimization as well as convex minimization We have found, however, that application of these fruitful ideas to global optimization is raising many new interesting theoretical and computational questions whose answers cannot be inferred from previous successes For example, branch and bound methods applied to global optimization problems generate infinite processes, and hence their own convergence theory must be developed In contrast, in integer programming these are finite procedures, and so their convergence properties not directly apply Other examples involve important VII results in convex minimization that reflect the coincidence of local and global solutions Here also one cannot expect a direct application to multi extremal global minimization In an abundant class of global optimizations, convexity is present in a reverse sense In this direction we focus our exposition on the following main topics: (a) minimization of concave functions subject to linear and convex constraints (i.e., "concave minimization"); (b) convex minimization over the intersection of convex sets and complements of convex sets (i.e., "reverse convex programming"); and (c) global optimization of functions that can be expressed as a difference of two convex functions (i.e., "d.c.-programming") Another large class of global optimization that we shall discuss in some detail has been termed "Lipschitz Programming", where now the functions in the formulation are assumed to be Lipschitz continuous on certain subsets of their domains Although neither of the aforementioned properties (i)-(ii) is necessarily satisfied in Lipschitz problems, much can be done here by applying the basic ideas and techniques which we shall develop for the problem classes (a), (b), and (c) mentioned above Finally, we also demonstrate how global optimization problems are related to solving systems of equations and/or inequalities As a by-product, then, we shall present some new solution methods for solving such systems The underlying purpose of this book is to present general methods in such a way as to enhance the derivation of special techniques that exploit frequently encountered additional problem structure The multifaceted approach is manifested occasionally in some computational results for these special but abundant problems However, at the present stage, these computational results should be considered as preliminary The book is divided into three main parts Part A introduces the main global optimization problem classes we study, and develops some of their basic properties and applications It then discusses the fundamental concepts that unify the various general methods of solution, such as outer approximation, concavity cuts, and branch and bound VIII Part B treats concave minimization and reverse convex programming subject to linear and reverse convex constraints In this part we present additional detail on specially structured problems Examples include decomposition, projection, separability, and parametric approaches In Part C we consider rather general global optimization problems We study d.c.-programming and Lipschitz optimization, and present our most recent attempts at solving more general global optimization problems In this part, the specializations most naturally include biconvex programming, indefinite "all-quadratic" optimization, and design centering as encountered in engineering design Each chapter begins with a summary of its contents The technical prerequisites for this book are rather modest, and are within reach of most advanced undergraduate university programs They include a sound knowledge of elementary real analysis, linear algebra, and convexity theory No familiarity with any other branch of mathematics is required In preparing this book, we have received encouragement, advice, and suggestions from a large group of individuals For this we are grateful to Faiz AI-Khayyal, Harold P Benson, Neil Koblitz, Ken Kortanek, Christian Larsen, Janos Pinter, Phan Thien Thach, Nguyen van Thoai, Jakob de Vries, Graham Wood, and to several other friends, colleagues and students We are indebted to Michael Knuth for drawing the figures We acknowledge hospitality and / or financial support given by the following organizations and institutions: the German Research Association (DFG), the Alexander von Humboldt Foundation, the Minister of Science and Art of the State Lower Saxony, the National Scientific Research Center in Hanoi, the Universities of Oldenburg and Trier Our special thanks are due to Frau Rita Feiden for the efficient typing and patient retyping of the many drafts of the manuscript Finally, we thank our families for their patience and understanding December 1989 Reiner Horst Hoang Tuy CONTENTS PART A: INTRODUCTION AND BASIC TECHNIQUES CHAPTER I SOME IMPORTANT CLASSES OF GLOBAL OPTIMIZATION PROBLEMS Global Optimization Concave Minimization 2.1 Definition and Basic Properties 2.2 Brief Survey of Direct Applications 12 2.3 Integer Programming and Concave Minimization 14 2.4 Bilinear Programming and Concave Minimization 19 2.5 Complementarity Problems and Concave Minimization 2.6 Max-Min Problems and Conc1J,ve Minimization 25 23 D.C Programming and Reverse Convex Constraints 3.1 D.C Programming: Basic Properties 26 3.2 D.C Programming: Applications 32 3.3 Reverse Convex Constraints 36 3.4 Canonical D.C Programming Problems 39 Lipschitzian Optimization and Systems of Equations and Inequalities 4.1 Lipschitzian Optimization 42 4.2 Systems of Equations and Inequalities 46 26 42 CHAPTER II OUTER APPROXIMATION 51 Basic Outer Approximation Method 51 Outer Approximation by Convex Polyhedral Sets 56 Constraint Dropping Strategies 65 On Solving the Subproblems (Qk) 4.1 Finding an Initial Polytope Dl and its Vertex Set VI 69 4.2 Computing New Vertices and New Extreme Directions 71 4.3 Identifying Redundant Constraints 82 68 x CHAPTER m CONCAVITY CUTS 85 Concept of a Valid Cut 85 Valid Cuts in the Degenerate Case 91 Convergence of Cutting Procedures 95 Concavity Cuts for Handling Reverse Convex Constraints 100 A Class of Generalized Concavity Cuts 104 Cuts Using Negative Edge Extensions 108 CHAPTER IV BRANCH AND BOUND 111 A Prototype Branch and Bound Method 111 Finiteness and Convergence Conditions 121 Typical Partition Sets and their Refinement 3.1 Simplices 132 3.2 Rectangles and Polyhedral Cones 137 132 Lower Bounds 4.1 Lipschitzian Optimization 140 4.2 Vertex Minima 141 4.3 Convex Subfunctionals 142 4.4 Duality 153 4.5 Consistency 158 139 Deletion by Infeas:ibility 163 Restart; Branch and Bound Algorithm 169 PART B: CONCAVE MINIMIZATION 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Vilnius (in Russian) ZWART, P.B (1973), Nonlinear Programming: Counterexamples to two Global Optimization Algorithms Operations Research, 21, 1260-1266 ZWART, P.B (1974), Global Maximization of a Convex Function with Linear Inequality Constraints Operations Research, 22, 602-{)09 NOTATION IN set of natural numbers IR set of real numbers set of extended real numbers (I = IR U {+IIJ, uJ} ) IR+ set of real n-vectors loJ r01 lower integer part of MeN M (not necessarily strict) subset of N M\N difference of sets M and N M-N algebraic difference of sets M and N M+N sum of sets M and N IMI cardinality of set M lin M linear hull of set M affM affine hull of set M conv M convex hull of set M coneM conical hull of set M O(M) = d(M) diameter of M xy inner product of vectors x,y I identity matrix upper integer part of (n"n) identity matrix 690 diag(a) = diagonal diag( al' ,an ) matrix with entries al' ,an (where a=( al' ,an ) transpose of matrix A inverse of matrix A determinant of A xi ~ Yi for all i (where x system of linear equalities Ax=b Ax~ Q system of linear inequalities b = (zl, ,zn) matrix of columns zl, ,zn cone spanned by the columns of the matrix Q con(Q) conv [xO, ,xn] = (xl' 'xn), y = (Yl'···,yn» = [xO, ,xn] simplex spanned by its n+l affine1y independent ° vert Ices x , ,xn polyhedron, convex polyhedral set set of solutions of a system of linear inequalities polytope bounded polyhedron vert(P), V(P) vertex set of polyhedron (polytope) P extd(P), U(P) set of extreme directions of a polyhedron P R(K) recession cone of convex set K G = (V,A) directed graph epi (f) epigraph of f hypo (f) hypograph of f dom (f) effective domain of a function f: IRn - Vf(x) gradient of f at x Of(x) subdifferential of f at x 11·11 Euclidean norm N(x,c;) open ball centered at x with radius c; int M interior of set M iR 691 clM=M closure of set M 8M boundary of set M w(x) projection of a point x onto a specified set f(M) global minimum of function f over set M argmin f(M) set of global minimizers of f over M (BCP) basic concave programming problem (=linearly constrained concave minimization problem) (BLP) bilinear programming problem (CCP) concave complementarity problem (CDC) canonical dc programming problem (CF) minimum concave cost flow problem (CP) concave minimization problem (concave programming problem) (LCP) linear complementarity problem (LP) linear programming problem (LRCP) linear program with an additional reverse convex constraint (MIP) mixed integer programming problem (PCP) parametric concave programming problem (QCP) quadratic concave programming problem (SBC) special biconvex programming problem (SCP) separable concave programming problem (SUCF) single source uncapacitated minimum concave cost flow problem (UCF) uncapacitated minimum concave cost flow problem (UL) univariate Lipschitz optimization problem NCS normal conical subdivision NRS normal rectangular subdivision NSS normal simplicial subdivision INDEX A all-quadratic problem, 618 approximate relief indicator method, 650 arc, 121, 410 assignment problems, 15, 21 B barycenter, 135 basic solution, 92 basic variables, 92, 187 Bender's decomposition, 373, 390 biconvex programming problem, 20, 35,577 bid evaluation problem, 12 bilinear constraints approach, 625 bilinear programming cut, 217 bilinear programming problem, 19, 212,434,577 bimatrix game, 20 bisection, 136,293, 361 bounded convergence principle, 97 bound improving selection, 125 bounding, 117, 139 branch and bound 112, 286, 539, 602 branch and bound algOrithms for (see also conical alg., rectangular alg., simplicial alg.) - biconvex problems, 582 - concave minimization, 197,299, 314, 335, 357 - d.c problems, 539 - Lipschitz optimization, 600, 605, 616 - branch and bound / outer approximation algorithms, 315, 543 C canonical d.c problem, 36, 41, 506 Caratheodory's Theorem, 10, 145 certain in the limit, 131 complementarity condition, 23, 36 complementarity problem, 23, 456 complete selection, 124 communication network, 13 concave complementarity problem, 456,473 concavity cut, 90, 101, 209 concave minimization, 9, 173, 184, 195 concave minimization over network, 13,410 concave objective network problem, 12,410 concave polyhedral underestimation, 271 concave variable cost, 13 cone of the first category, 464 cone of the second category, 464 conical algorithms, 286 conical algorithm for the - bilinear programming problem, 445 - concave complementarity problem, 473 -concave minimization problem, 196,289,299,315 - d.c programming problem, 539 - linear complementarity problem, 462,464 -linear program with an additional reverse convex constraint, 494, 498 conjugate function, 146 conjunctive cuts, 95 consistent bounding operation, 123, 158 constraint dropping strategies, 90, 101, 209 constraint qualification, 155 convergent branch and bound procedure, 117, 123 convergent cutting procedures, 95 convex envelope, 143, 262, 265, 363, 387, 580, 622, 629 convex inequality, convexity cut, 101 693 convex minimization, 6, 57, 62 convex subfunctional, 142 convex underestimation, 143, 259 cut, 86, 90, 101, 209 cut and split algorithm, 195, 199 cuts using negative edge extensions, 108 cutting plane method, 56, 62 cutting plane methods for the - bilinear programming problem, 216,438 - concave minimization problem, 182, 233 -