ASYMPTOTIC CONES AND FUNCTIONS IN OPTIMIZATION AND VARIATIONAL INEQUALITIES
CONTENTS CONVEX ANALYSIS AND SET-VALUED MAPS: A REVIEW CONVEX SETS CONVEX FUNCTIONS SUPPORT FUNCTIONS SET-VALUED MAPS NOTES AND REFERENCES ASYMPTOTIC CONES AND FUNCTIONS DE?NITIONS OF ASYMPTOTIC CONES DUAL CHARACTERIZATION OF ASYMPTOTIC CONES CLOSEDNESS CRITERIA CONTINUOUS CONVEX SETS ASYMPTOTIC FUNCTIONS DI?ERENTIAL CALCULUS AT IN?NITY APPLICATION I: SEMIDE?NITE OPTIMIZATION APPLICATION II: MODELING AND SMOOTHING OPTIMIZATION PROB- LEMS EXISTENCE AND STABILITY IN OPTIMIZATION PROBLEMS COERCIVE PROBLEMS WEAK COERCIVITY ASYMPTOTICALLY LEVEL STABLE FUNCTIONS EXISTENCE OF OPTIMAL SOLUTIONS STABILITY FOR CONSTRAINED PROBLEMS DUAL OPERATIONS AND SUBDI?ERENTIAL CALCULUS ADDITIONAL RESULTS IN THE CONVEX CASE THE FEASIBILITY PROBLEM MINIMIZING AND STATIONARY SEQUENCES OPTIMALITY CONDITIONS IN CONVEX MINIMIZATION ASYMPTOTICALLY WELL-BEHAVED FUNCTIONS ERROR BOUNDS FOR CONVEX INEQUALITY SYSTEMS STATIONARY SEQUENCES IN CONSTRAINED MINIMIZATION DUALITY IN OPTIMIZATION PROBLEMS PERTURBATIONAL-CONJUGATE DUALITY FENCHEL DUALITY LAGRANGIAN DUALITY ZERO DUALITY GAP FOR SPECIAL CONVEX PROGRAMS DUALITY AND ASYMPTOTIC FUNCTIONS LAGRANGIANS AND MINIMAX THEORY DUALITY AND STATIONARY SEQUENCES MAXIMAL MONOTONE MAPS AND VARIATIONAL INEQUALITIES MAXIMAL MONOTONE MAPS MINTY THEOREM CONVEX FUNCTIONALS AND MAXIMAL MONOTONICITY DOMAINS AND RANGES OF MAXIMAL MONOTONE MAPS ASYMPTOTIC FUNCTIONALS OF MAXIMAL MONOTONE MAPS FURTHER PROPERTIES OF MAXIMAL MONOTONE MAPS VARIATIONAL INEQUALITIES PROBLEMS EXISTENCE RESULTS FOR VARIATIONAL INEQUALITIES DUALITY FOR VARIATIONAL INEQUALITIES 6 10 NOTES AND REFERENCES REFERENCES CONVEX SETS THROUGHOUT THIS BOOK WE WILL CONSIDER ONLY ?NITE-DIMENSIONAL VECTOR SPACES THE N-DIMENSIONAL REAL EUCLIDEAN VECTOR SPACE WILL BE DENOTED BY RN FOR VECTORS X = (X1, , XN) ?RN, Y ?RN, THE INNER PRODUCT BETWEEN X AND Y IS DE?NED BY ?X, Y?:= N I=1 XIYI THE NORM ARISING FROM THE INNER PRODUCT WILL BE DENOTED BY ?X?:= ?X, X?1/2 DE?NITION 1 1 1 A SUBSET C OF RN IS CONVEX IF TX + (1 ?T)Y ?C, ?X, Y ?C, ?T ?[0, 1] USING THE ALGEBRAIC SUM OF TWO CONVEX SETS, ?C1 + ?C2 := {X | X = ?C1 + ?C2, C1 ?C1, C2 ?C2}, ??, ? ?R, TI = 1, TI ?0, I = 1, , N HYPERPLANES, CONVEX AND A?NE HULLS A HYPERPLANE IS A SET H := {X ?RN |?X, A?= ?}, WHERE A ?RN, A ?= TI =