Inverse semidefinite quadratic programming problem with [formula omitted] norm measure

Abstract

We consider an inverse problem arising from a semidefinite quadratic programming (SDQP) problem, which is a minimization problem involving l1 vector norm with positive semidefinite cone constraint. By using convex optimization theory, the first order optimality condition of the problem can be formulated as a semismooth equation. Under two assumptions, we prove that any element of the generalized Jacobian of the equation at its solution is nonsingular. Based on this, a smoothing approximation operator is given and a smoothing Newton method is proposed for solving the solution of the semismooth equation. We need to compute the directional derivative of the smoothing operator at the corresponding point and to solve one linear system per iteration in the Newton method and its global convergence is demonstrated. Finally, we give the numerical results to show the effectiveness and stability of the smoothing Newton method for this inverse problem.