The Optimality Theories For Multiobjective Optimization Constrained By Conic Equilibrium Constraints

Multiobjective optimization problems with conic equilibrium constraints are the multiob-jective optimization problems whose equilibrium constraints include parameterized variational inequalities or generalized equations,which are defined by closed convex cones.They are gen-eralizations of mathematical programs with equilibrium constraints(MPECs)and equilibrium problems with equilibrium constraints(EPECs).Such problems are widely used in many fields such as economics,engineering and energies etc.This dissertation is devoted to the study of optimality theories for multiobjective optimiza-tion constrained by conic equilibrium constraints,including the optimality conditions for multi-objective optimization problems constrained by parameterized variational inequalities;the opti-mality conditions for multiobjective optimization problems governed by second-order cone con-strained generalized equations;the asymptotic convergence of KKT points of stochastic multi-objective programs with parametric variational inequality constraint.For the constraint problem of multiobjective optimization constrained by conic equilibrium constraints:nonsmooth equilib-rium problems,we propose an executable numerical algorithm-the approximate bundle method.The main results of this dissertation can be summarized as follows:Chapter 3 focuses on the optimality conditions for the multiobjective optimization problem constrained by parameterized variational inequalities.Under the calmness condition,an efficient upper estimate of coderivatives for a composite normal cone set-valued mapping is derived.By separation theorem for convex sets,we translate the multiobjective optimality problem into a single objective optimality problem.Under the assumption of linear independent constraint qualifications,we obtain the first-order optimality conditions of this problem.Chapter 4 mainly presents the optimality conditions for the multiobjective optimization problem governed by the cone constrained generalized equations.By virtue of variational anal-ysis,the regular(limiting)normal cone of normal cone mappings constrained by second-order cone is obtained.Under the calmness condition,an estimate of coderivative for a composite nor-mal cone set-valued mapping is established.The necessary optimality condition for the problem is established under the linear independent constraint qualification.Furthermore,a simpler op-timality condition is derived if the strict complementarity relaxation condition is satisfied.In Chapter 5,we consider the stochastic multiobjective program with parametric variational inequality constraint.we approximate the true problem by sample average approximation(SAA)method.Under the linear independent constraint qualification and strict complementarity relax-ation condition,we obtain the optimality conditions for the true problem and SAA problem.By graphical convergence of set-valued mappings,the KKT points of the SAA problem converge to the KKT points of the true problem when the sample size tends to infinity.Under the convexity assumption,the convergence of optimal solutions of SAA problems is also derived.Chapter 6 is devoted to the study of approximate bundle methods for solving nonsmooth e-quilibrium problems.The constraint problem of multiobjective optimization problems with conic equilibrium constraints:the variational inequalities or generalized equations,are summarized as a nonsmooth equilibrium problem.The equilibrium problem is changed into an optimization problem by auxiliary principle.For inexact oracles in objective functions and subgradients,we assume that the errors of inexact oracles are bounded and they need not vanish in the limit.A descent criterion adapting the setting of inexact oracles is put forward to measure the current de-scent behavior.The sequence generated by the algorithm converges to approximate solutions of equilibrium problems under proper assumptions.Numerical results show that the approximate bundle method is effective in solving a variety of nonsmooth equilibrium problems.