Optimization Problems With Second-Order Cone Equilibrium Constraints

Optimization problems with second-order cone equilibrium constraints have many wide implications in the fields of industrial engineering, robot manufacture, inverse problem, and so on. This dissertation focuses on the study of variational analysis of the second-order cone com-plementarity set and its applications to study the optimality conditions of optimization problems with second-order cone equilibrium constraints, the Aubin property of the second-order cone complementarity set, as well as the methods for solving inverse linear programming problems and inverse linear second-order cone programming problems. The main results, obtained in this dissertation, may be summarized as follows:1. Chapter 2, based on the classic theory of variational analysis, develops variational analysis of the second-order cone complementarity set, including the formulas of tangent cone, normal cone of the second-order cone complementarity set and the coderivative of the mapping of normal cone associated with second-order cone. These results in this chapter pave a way to carry out future discussion.2. Chapter 3, based on the variational analysis developed in Chapter 2, studies the optimality conditions of optimization problems with second-order cone equilibrium constraints. The constraint qualification of the set of the second-order cone constraints is characterized. Then, the optimality condition of general optimization problems with second-order cone equilibrium constraints is discussed. Finally, the optimality condition of optimization problems with second-order cone complementarity constraints is studied in detail.3. Chapter 4, on the basis of the results obtained in previous two Chapters, studies the Aubin property of the second-order cone complementarity set. This Chapter first derive the for- mulas of the coderivative of the mapping of the normal cone of the second-order com-plementarity set provided that the constraint qualification holds. Based on that, by using Mordukhovich criterion, we discuss the Aubin property of the second-order cone comple-mentarity set. To close this chapter, an example is illustrated to show that the second-order cone complementarity set does not satisfy the Aubin property at the origin.4. In Chapter 5, we propose an inexact Newton smoothing method for solving optimization problems with second-order cone complementarity constraints. In particular, we apply this method to solve a type of inverse linear programming problem and inverse linear second-order cone programming problem. We formulate these problems as a second-order cone complementarity constrained minimization problem. With the help of the smoothing Fischer-Burmeister function, we apply a perturbation approach to solve the inverse problem and an inexact Newton method to solve the perturbed problem. We also demonstrate its global convergence and quadratic convergence. Numerical results are reported to show the effectiveness of the approach.