II KLUWER ACADEMIC PUBLISHERS. Abstract Convexity and Global Optimization. Alexander Rubinov

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1 Abstract Convexity and Global Optimization by Alexander Rubinov School of Information Technology and Mathematical Sciences, University of Ballarat, Victoria, Australia II KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON

2 Contents Preface Acknowledgment xi xvii 1. AN INTRODUCTION TO ABSTRACT CONVEXITY Preliminaries Abstract convex functions and sets Subdifferentiability Conjugation Abstract concave functions and infima of abstract convex functions ELEMENTS OF MONOTONIC ANALYSIS: IPH FUNCTIONS AND NORMAL SETS Introduction Increasing positively homogeneous functions defined on positive orthant Preliminaries IPH functions Min-type functions and IPH functions Abstract convexity with respect to the set of mintype functions Level sets of IPH functions Polarity for normal sets and IPH functions Support sets Subdifferential Concavity of the polar function Comparison with convex analysis Increasing positively homogeneous functions defined on the non-negative orthant Preliminaries 43 v

3 vi ABSTRACT CONVEXITY L-convex function Support sets Two kinds of normality Properties of the support sets Subdifferentials of IPH functions Abstract concavity and superdifferentials Best approximation by normal sets Preliminaries Distance to a normal set Separation Distance to the union and the intersection of normal sets 3. ELEMENTS OF MONOTONIC ANALYSIS: MONOTONIC FUNCTIONS Introduction Increasing co-radiant functions Definition and properties of ICR functions ICR functions and IPH functions Abstract convexity of ICR functions Increasing convex-along-rays functions ICAR functions: definition, examples and some properties ICAR functions as abstract convex functions Subdifferentiability of ICAR functions Subdifferentiability of strictly ICAR functions Lipschitz function and ICAR functions Decreasing functions Decreasing functions and IPH functions Multiplicative inf-convolution APPLICATION TO GLOBAL OPTIMIZATION: LAGRANGE AND PENALTY FUNCTIONS Introduction Extended Lagrange and penalty functions Preliminaries Extended Lagrange functions Extended penalty functions Examples Support set of the dual function Another approach Extended penalization for problems with one constraint Preliminaries

4 Contents Perturbation functions Exact penalization Penalization by IPH functions pk ELEMENTS OF STAR-SHAPED ANALYSIS Introduction Radiant and co-radiant sets and their gauges Radiant and co-radiant sets Radiative and co-radiative sets Radiant sets with Lipschitz continuous Minkowski gauges Star-shaped sets and co-star-shaped sets Star-shaped sets and their kerneis Sum of star-shaped sets and sum of co-star-shaped sets Separation Cone-separation and Separation by a finite collection of linear functions Separation of star-shaped sets Separation of co-star-shaped sets Abstract convexity with respect to general min-type functions Positively homogeneous abstract convex functions %-convex functions Subdifferentials of % n+ i-convex functions Abstract convex sets Other classes of abstract convex functions SUPREMAL GENERATORS AND THEIR APPLICATIONS Introduction Continuous and lower semicontinuous functions Lower semicontinuous functions Examples ICAR extensions of functions defined on the unit simplex Supremal generators for Spaces of homogeneous functions Preliminaries Homogeneous functions of degree one Symmetrie positively homogeneous functions of degree two Some applications of supremal generators 247 vn

5 viii ABSTRACT CONVEXITY Convergence of sequences of positive functionals The supremal rank of a compact set Application to Hadamard-type inequalities Hadamard-type inequalities for convex functions Quasiconvex functions and P-functions Inequalities of Hadamard type for P-functions and quasiconvex functions Inequality of Hadamard type for ICAR functions FURTHER ABSTRACT CONVEXITY Introduction Abstract Convexity with respect on a subset Basic definitions and properties Subdifferentiability Conjugation and approximate subdifferentials Abstract convexity and global minimization Minimax result for abstract convex functions Positively homogeneous functions Positively homogeneous extension Polarity for functions and sets, which are abstract convex with respect to a conic set of positively homogeneous functions Some classes of abstract convex functions Linear functions - generating sublinearity Affine functions - generating convexity Min-type functions - generating convexity-alongrays Two-step functions - generating quasiconvexity. Abstract convex functions Two-step functions - generating quasiconvexity. Abstract convex sets Two-step functions - generating quasiconvexity. Abstract subdifferentials Infima of families of abstract convex functions Minkowski duality, c 2 -lattices and semilinear lattices The Minkowski duality Minkowski duality for cmattices and semilinear lattices APPLICATION TO GLOBAL OPTIMIZATION: DUALITY Introduction General solvability theorems

6 Contents ix Solvability theorems for Systems of abstract convex functions Further solvability results Sublinear inequality Systems Convex inequality Systems Positively homogeneous Systems Twice continuously differentiable Systems Quasiconvex inequality Systems Systems involving functions expressible as the infima of families of convex functions Convex maximization Maximal elements of Support sets and Toland-Singer formula Maximal elements Existence of maximal elements of support sets Positively homogeneous elementary functions Maximal elements of support sets with respect to conic sets of positively homogeneous functions The Toland-Singer formula and maximal elements of support sets Excess functions Optimization of the difference of abstract convex functions Minimization of the difference of coercive convex functions Minimization of the difference of coercive convex and sublinear functions Dual optimality conditions for the difference of ICAR functions Necessary and sufficient conditions for the minimum of the difference of coercive strictly ICAR functions Minimization of the quotient of convex functions 396 APPLICATION TO GLOBAL OPTIMIZATION: NUMERICAL METHODS Introduction Conceptual schemes of numerical methods Special minorates of abstract convex functions Generalized cutting plane method Branch-and-bound methods Tabu search External centres method Lipschitz programming via abstract convexity Cutting angle method

7 x ABSTRACT CONVEXITY Cutting angle method for ICAR functions The subproblem Numerical results - ICAR objective functions Cutting angle method for increasing co-radiant functions Numerical results - ICR objective functions Cutting angle method for Lipschitz functions Numerical results - Lipschitz functions Branch-and-bound method for Lipschitz functions 9.4 Cutting angle method (continuation) Cutting angle method for the minimization of IPH functions over the unit simplex The subproblem: local minima The subproblem: global minima Results of numerical experiments An exact method for solving the subproblem 9.5 Monotone optimization References Preliminaries Problems of monotonic optimization Basic properties Proposed Solution method Convergence Computational experience Index 489