Study On Theory And Numerical Methods For Inverse Continuous Optimization Problems

Continuous optimization models usually consist of two types of variables, one group of vari-ables are parameters and the other group of variables are decision variables. Many mathematical problems with important practical backgrounds are to estimate problem parameters based on the given information. An inverse optimization problem is to find the problem parameter with the smallest distance to the given parameter estimate when the optimal solution to the optimization problem is available.The dissertation focuses on the study of inverse nonlinear programming problems and in-verse semidefinite programming problems. The results obtained in this dissertation are summa-rized as follows:1. Chapter 3 proposes the inverse nonlinear programming model and studies its optimality conditions. The MPEC reformulation is developed for the inverse problem arising from a con-vex nonlinear programming problem, and formulas for the tangent cone, the regular normal cone and the normal cone of the feasible set for this MPEC problem, are presented. Based on these formulas we establish the first-order necessary optimality conditions and the second-order nec-essary optimality conditions and the second-order sufficient optimality conditions for the inverse nonlinear programming problem when the set of parameter set is given by Θ= S+l.2. Chapter 4 discusses a smoothing method for the inverse nonlinear programming problem and its convergence properties. Using the smoothed Fischer-Burmeister function, a set-valued mapping is constructed to approximate the complementarity set, and the linear independence con-straint qualification for the feasible set of the smoothing problem is proved to be satisfied, and with this condition formulas for its tangent cone and normal cone are established. It is demon-strated that, when the parameter ε↘0, the feasible set for the smoothing problem converges to the feasible set of the inverse nonlinear programming problem, the outer limit of the solution mapping is contained in the solution set of the inverse problem, and the outer limit of the KKT-point mapping is contained in the set of Clarke stationary points associated with corresponding multipliers.3. Chapter 5 applies the results obtained in the previous two chapters to analyze the inverse continuous optimization models studied in the literature. The models discussed here include inverse linear programming problems, inverse quadratic programming problems, second-order cone constrained inverse linear programming and quadratic programming problems, and inverse semidefinite cone constrained quadratic programming problems, and it is pointed out that the topic about the inverse semidefinite programming problems is worth investigating.4. Chapter 6 consists of two parts for exploring numerical methods for inverse semidefinite programming problems. The first part proposes a bilinear penalty function method, proves the global convergence of the penalty function method; and a sequential convex approximation ap-proach is presented for solve the penalized problem and it is proved that any cluster point of the generated sequence is a stationary point of the penalized problem. The second part of this chapter gives the "Jacobian uniqueness" theorem for the nonlinear semidefinite programming problem, based on which the optimality conditions for a bilevel semidefinite programming problem de-pending the parameter in a simple way are studied, which provides a theoretical foundation for the implicit programming method to solve inverse semidefinite programming problems.