Numerical results for a globalized active-set newton method for mixed complementarity problems

We discuss a globalization scheme for a class of active-set Newton methods for solving the mixed complementarity problem (MCP), which was proposed by the authors in[3]. The attractive features of the local phase of the method are that it requires solving only one system of linear equations per iteration, yet the local superlinear convergence is guaranteed under extremely mild assumptions, in particular weaker than the property of semistability of anMCPsolution. Thus the local superlinear convergence conditions of the method are weaker than conditions required for the semismooth (generalized) Newton methods and also weaker than convergence conditions of the linearization (Josephy–Newton) method. Numerical experiments on some test problems are presented, including results on the MCPLIB collection for the globalized version. © 2005 SBMAC

Mixed complementarity problems: Regularity, error bounds, and Newton-type methods

This paper is devoted to mixed complementarity problems (variational inequalities on a box). This class includes many important problem statements, for example, systems of equations, conventional complementarity problems, and Karush-Kuhn-Tucker systems. Error bounds and Newton-type methods for these problems are discussed. A new family of Newton-type methods is suggested that are globally convergent and the rate of local convergence is superlinear; these methods are superior to the available methods in certain respects. The presentation is accompanied by a detailed comparison of various relevant regularity conditions. Copyright © 2004 by MAIK "Nauka/ Interperiodica"

A class of active-set newton methods for mixed complementarity problems

Based on the identification of indices active at a solution of the mixed complementarity problem (MCP), we propose a class of Newton methods for which local superlinear convergence holds under extremely mild assumptions. In particular, the error bound condition needed for the identification procedure and the nondegeneracy condition needed for the convergence of the resulting Newton method are individually and collectively strictly weaker than the property of semistability of a solution. Thus the local superlinear convergence conditions of the presented method are weaker than conditions required for the semismooth (generalized) Newton methods applied to MCP reformulations. Moreover, they are also weaker than convergence conditions of the linearization (Josephy-Newton) method. For the special case of optimality systems with primal-dual structure, we further consider the question of superlinear convergence of primal variables. We illustrate our theoretical results with numerical experiments on some specially constructed MCPs whose solutions do not satisfy the usual regularity assumptions. © 2004 Society for Industrial and Applied Mathematics

Numerical results for a globalized active-set newton method for mixed complementarity problems

We discuss a globalization scheme for a class of active-set Newton methods for solving the mixed complementarity problem (MCP), which was proposed by the authors in[3]. The attractive features of the local phase of the method are that it requires solving only one system of linear equations per iteration, yet the local superlinear convergence is guaranteed under extremely mild assumptions, in particular weaker than the property of semistability of anMCPsolution. Thus the local superlinear convergence conditions of the method are weaker than conditions required for the semismooth (generalized) Newton methods and also weaker than convergence conditions of the linearization (Josephy–Newton) method. Numerical experiments on some test problems are presented, including results on the MCPLIB collection for the globalized version. © 2005 SBMAC

A class of active-set newton methods for mixed complementarity problems

Based on the identification of indices active at a solution of the mixed complementarity problem (MCP), we propose a class of Newton methods for which local superlinear convergence holds under extremely mild assumptions. In particular, the error bound condition needed for the identification procedure and the nondegeneracy condition needed for the convergence of the resulting Newton method are individually and collectively strictly weaker than the property of semistability of a solution. Thus the local superlinear convergence conditions of the presented method are weaker than conditions required for the semismooth (generalized) Newton methods applied to MCP reformulations. Moreover, they are also weaker than convergence conditions of the linearization (Josephy-Newton) method. For the special case of optimality systems with primal-dual structure, we further consider the question of superlinear convergence of primal variables. We illustrate our theoretical results with numerical experiments on some specially constructed MCPs whose solutions do not satisfy the usual regularity assumptions. © 2004 Society for Industrial and Applied Mathematics

Mixed complementarity problems: Regularity, error bounds, and Newton-type methods

This paper is devoted to mixed complementarity problems (variational inequalities on a box). This class includes many important problem statements, for example, systems of equations, conventional complementarity problems, and Karush-Kuhn-Tucker systems. Error bounds and Newton-type methods for these problems are discussed. A new family of Newton-type methods is suggested that are globally convergent and the rate of local convergence is superlinear; these methods are superior to the available methods in certain respects. The presentation is accompanied by a detailed comparison of various relevant regularity conditions. Copyright © 2004 by MAIK "Nauka/ Interperiodica"

A Newton-type method for quadratic programming problem and variational equilibrium problem

The active-set Newton method developed earlier by the author and her supervisor for mixed complementarity problems is applied to solving the quadratic programming problem with a positive definite matrix of the objective function and for variational equilibrium problem. A theoretical justification is given to the fact that the method is guaranteed to find the exact solution in a finite number of steps. Numerical results indicate that this approach is competitive with other available methods as for quadratic programming problems and for variational equilibrium problem. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the 13th International Symposium “Intelligent Systems” (INTELS'18)