We consider an inverse problem arising from a semidefinite quadratic programming (SDQP) problem, which is a minimization problem involving $l_1$ 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.

我们考虑由半定二次规划（SDQP）问题引起的逆问题，该问题是最小化问题，涉及 ${升}_{\mathrm{1个}}$具有正半定锥约束的向量范数。通过使用凸优化理论，可以将问题的一阶最优条件公式化为半光滑方程。在两个假设下，我们证明方程方程组的广义雅可比行列式的任何元素在其解中都是非奇异的。在此基础上，给出了一个光滑近似算子，并提出了一个光滑牛顿法来求解半光滑方程的解。我们需要计算相应点的平滑算子的方向导数，并在牛顿法中每次迭代求解一个线性系统，并证明其全局收敛性。最后，我们给出数值结果以表明平滑牛顿法对这个反问题的有效性和稳定性。