Effective Numerical Methods For Several Inverse Quadratic Programming Problems

For a continuous optimization model,it usually consists of two parts of variables,one part is the parameters and the other part is the decision variables.A forward problem is to solve the values of the optimal decision variables when the parameters are assigned.However,in practice,We can often obtain the estimated values of the parameters and the values of the optimal decision variables.The inverse optimization problem is to get the values of parameters,which were adjusted as little as possible so that the known optimal decision variables are still optimal.This dissertation is devoted to the study on numerical algorithms for two types of inverse quadratic programming problems and a type of inverse semidefinite quadratic programming problem.The main results of this dissertation are summarized as follows:1.In Chapter 3,we consider a class of inverse quadratic programming problem with l1 norm measure.We formulate this problem as a minimization problem with a positive semidefinite cone constraint.By utilizing convex optimization theory,we rewrite its first order optimality condition as a generalized equation.Under extremely simple assumptions,we prove that any element of the generalized Jacobian of the equation at its solution is nonsingular.Based on this,we construct an inexact Newton method with Armijo line search to solve the equation and demonstrate its global convergence.Finally,we report the numerical results illustrating effectiveness of the Newton methods.2.In Chapter 4,we focus on an inverse quadratic programming problem with matrix spec-tral norm and vector l?norm measure.We transform the problem into a convex optimization problem with objective function separable and propose Gauss back substitution alternating di-rection method to solve it.we use the singular value threshold method,Moreau-Yosida regu-larization algorithm and the quadprog function in MATLAB optimization toolbox to solve the corresponding subproblem accurately.It is found that one subproblem is still a convex opti-mization problem with separable variables objective function.So we use the inexact and exact alternating direction methods to solve this subproblem.Finally,the convergence analysis and numerical experiments of the Gauss back substitution alternating direction method are given.The numerical experiments show that this method can solve the inverse quadratic programming problem efficiently.3.In Chapter 5,We consider an inverse problem arising from a semidefinite quadratic pro-gramming problem,which is a minimization problem involving l1 vector norm with two positive semidefinite cone constraint.By using convex optimization theory,the first order optimality con-dition of the problem can be formulated as a semismooth equation.Under two assumptions,we can prove that any element of the generalized Jacobian of the equation at its solution is non-singular.Based on this,a smoothing approximation operator is given and a smoothing Newton method is proposed for solving the solution of the semismooth equation.Finally,we give the numerical results to show the effectiveness of the smoothing Newton method for this inverse problem.