An Augmented Lagrange Method For A Type Of Inverse Quadratic Programming Problems With Second-order Cone Constraints

In an optimization model, there are many parameters associated with decision variables in the objective function or in the constraint set. When solving the optimization problem ,we ususlly assume that these parameter values are known and we need to find an optimal solution to it. But in many instances, we only know some estimates for parameter values and have certain optimal solutions from experiences, observations or experiments. We try to find the values of parameters which make the known solutions optimal and differ from the given estimates as little as possible. This is the inverse problem. As the inverse problems can be widely applied in many fields, it gradually attracts much attention from scholars in recent years.In this paper, we consider an inverse quadratic programming problem with second order cone constraint in which the parameters in the objective function are adjusted as little as possible to make a known feasible solution become the optimal one. We transform this problem to a minimization problem with a positive semidefinite cone constraint and derive the expression of the dual of this minimization problem. Its dual is a linearly constrained and second order cone constrained convex programming problem with fewer variables than the original one and the gradient of the objective function is semi-smooth. We solve the dual problem with augmented Lagrange method and analyze the local convergence and the rate of convergence of the augmented Lagrange method. The objective function of the dual problem is a semismoothly differentiable function which involve the projection opeator onto the cone of symmetrically semi-definite matrices and second-order cone. In the analysis, we use extensive tools such as semismooth implicit function theorems and differential properties of the pojection opterators onto the SDP cone and second order cone. Furthermore, the semi-smooth Newton method with Armijo line search is applied to solve the subproblems in the augmented Lagrange approach, which is proved to have global convergence and local quadratic rate. Finally we compile an Matlab code to solve the inverse quadratic programming problems.