We consider an inverse problem arising from a semidefinite quadratic programming (SDQP) problem, which is a minimization problem involving h vector norm with positive semidefinite cone constraint. By using convex opti...
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We consider an inverse problem arising from a semidefinite quadratic programming (SDQP) problem, which is a minimization problem involving h 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. (C) 2020 Elsevier B.V. All rights reserved.
A problem of minimizing a sum of many convex piecewise-linear functions is considered. In view of applications to two-stage linear programming, where objectives are marginal values of lower level problems, it is assum...
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A problem of minimizing a sum of many convex piecewise-linear functions is considered. In view of applications to two-stage linear programming, where objectives are marginal values of lower level problems, it is assumed that domains of objectives may be proper polyhedral subsets of the space of decision variables and are defined by piecewise-linear induced feasibility constraints. We propose a new decomposition method that may start from an arbitrary point and simultaneously processes objective and feasibility cuts for each component. The master program is augmented with a quadratic regularizing term and comprises an a priori bounded number of cuts. The method goes through nonbasic points, in general, and is finitely convergent without any nondegeneracy assumptions. Next, we present a special technique for solving the regularized master problem that uses an active set strategy and QR factorization and exploits the structure of the master. Finally, some numerical evidence is given.
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