Study On Stability Of Stochastic Conic Optimization Problems

Stochastic optimization has important theoretical and practical value in financial analysis,engineering design and other fields.It has many mathematical models.Essentially,it is an opti-mization problem whose functions are defined by mathematical expectation or probability,such as constrained optimization problem defined by mathematical expectations,chance-constrained stochastic optimization problem,and two-stage stochastic optimization problem,etc.Most of the stochastic programming models studied in the literature are optimization models on polyhe-dral cones.The stability analysis of cone constrained stochastic optimization problems including semi-definite matrix cone constraints and second-order cone constraints is still blank.Many im-portant problems are stochastic programming problems with matrix variables or stochastic opti-mization problems with non-polyhedral cone constraints.It is undoubtedly of great theoretical and practical value to extend polyhedral cone constraints to non-polyhedral cone constraints,and study the stability of these cone constrained stochastic optimization problems.At present,the stability analysis of stochastic optimization with polyhedral cone constraints is mainly focused on the optimal value function and the set of optimal solutions.However,no work has been pub-lished on the strong regularity of Karush-Kuhn-Tucker system and other important perturbation properties.In this paper,we study cone constrained stochastic optimization problems,including non-linear stochastic programming,second-order cone constrained stochastic optimization,and the stability of semi-positive matrix cone constrained stochastic optimization.We study not only the stability of the optimal solution set when the probability measure perturbs,but also the strong regularity of Karush-Kuhn-Tucker(KKT)systems when the probability measure perturbs.The results can be summarized as follows:1.Chapter 2 focuses on the study for the stability of stochastic nonlinear programming when the probability measure is perturbed.Under the Lipschitz continuity of the objective func-tion and metric regularity of the feasible set-valued mapping,the outer semi-continuity of the optimal solution set and Lipschitz continuity of optimal values are guaranteed.Importantly,it is proved that,if the linear independence constraint qualification and strong second-order suffi-cient condition hold at a local minimum point of the original problem,there exists a Lipschitz continuous solution path satisfying the Karush-Kuhn-Tucker conditions.2.Chapter 3,the stability of stochastic second-order cone programming,under probability measure perturbed,is investigated.Under the Lipschitz continuity of the objective function and metric regularity of the feasible set-valued mapping,the outer semi-continuity of the optimal solution set and Lipschitz continuity of optimal values are demonstrated.Moreover,we prove that,if the constraint non-degeneracy condition and strong second-order sufficient condition hold at a local minimum point of the original problem,there exists a Lipschitz continuous solution path satisfying the Karush-Kuhn-Tucker conditions.3.In Chapter 4,we study stability of stochastic semi-definite programming when the prob-ability measure varies.Under the Lipschitz continuity of the objective function and metric reg-ularity of the set-valued mapping defined by the feasible set,the outer semi-continuity of the optimal solution set and Lipschitz continuity of optimal values are established.Moreover,if the constraint non-degeneracy condition and strong second-order sufficient condition hold at a local minimum point of the original problem,there exists a Lipschitz continuous solution path satisfying the Karush-Kuhn-Tucker conditions,and the constraint non-degeneracy condition and second-order growth condition of the perturbed problem hold.