Smoothing Newton Method For Inverse Semi-definite Quadratic Programming Problems

The l1-norm is widely used in our lives, such as pattern recognition, data process-ing, the problem of optimal robust tracking control, etc. So it is significant to discuss optimization models under l1-norm.This paper focuses on an inverse optimization problem under l1-norm raised from the semi-definite quadratic programming (SDQP) problem. The inverse problem is to adjust the parameters in the objective function of a given SDQP problem as little as possible so that a known feasible solution becomes the optimal one. We formulate the inverse problem of a given SDQP as a problem with cone constraints. And its dual problem is a semismoothly differentiable convex programming problem. When the dimension of SDQP problem is large, the variables of its dual problem are fewer than that of the original one. With the help of duality theory, we turn to solve the dual problem of the inverse problem of the SDQP problem under l1-norm.Using the results in iterature, we get the expression of the inverse problem of SDQP problem and derive its dual problem. The KKT system of the dual problem consist of a series of semismooth equations. We introduce two smoothing functions to transform the KKT system into a series of smoothing equations. We employ the smoothing New-ton method with Armijo line search to solve the smoothing equations. We prove that the algorithm has global convergence and local quadratic convergence rate. Finally, the numerical experiments are reported to verify the effectiveness of the smoothing Newton method.