Inverse quadratic programming problem with \begin{document}$ l_1 $\end{document} norm measure

We consider an inverse quadratic programming (QP) problem in which the parameters in the objective function of a given QP problem are adjusted as little as possible so that a known feasible solution becomes the optimal one. We formulate this problem as a minimization problem involving $l_1$ vector norm 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.