Semidefinite Complementarity Problem Research Articles

A Squared Smoothing Newton Method for Nonsmooth Matrix Equations and Its Applications in Semidefinite Optimization Problems

We study a smoothing Newton method for solving a nonsmooth matrix equation that includes semidefinite programming and the semidefinite complementarity problem as special cases. This method, if specialized for solving semidefinite programs, needs to solve only one linear system per iteration and achieves quadratic convergence under strict complementarity and nondegeneracy. We also establish quadratic convergence of this method applied to the semidefinite complementarity problem under the assumption that the Jacobian of the problem is positive definite on the affine hull of the critical cone at the solution. These results are based on the strong semismoothness and complete characterization of the B-subdifferential of a corresponding squared smoothing matrix function, which are of general theoretical interest.

The bounded smooth reformulation and a trust region algorithm for semidefinite complementarity problems

In this paper, we present a new merit function for the semidefinite complementarity problem (SDCP) by extending the bounded smooth reformulation for variational inequality problems. We prove that the merit function has a bounded level set and any stationary point of it is a global minimizer without the assumption of monotonicity. Moreover, we present a trust region algorithm for solving the minimization problem with semidefinite constraints. The trust region subproblem is solved by the truncated conjugate gradient method and the global convergence is established even without requiring the existence of an accumulation point of the generated sequence.

Non-Interior Continuation Method for Solving the Monotone Semidefinite Complementarity Problem

Recently, Chen and Tseng extended non-interior continuation/ smooth- ing methods for solving linear/ nonlinear complementarity problems to semidefinite complementarity problems (SDCP). In this paper we propose a non-interior continuation method for solving the monotone SDCP based on the smoothed Fischer—Burmeister function, which is shown to be globally linearly and locally quadratically convergent under suitable assumptions. Our algorithm needs at most to solve a linear system of equations at each iteration. In addition, in our analysis on global linear convergence of the algorithm, we need not use the assumption that the Frechet derivative of the function involved in the SDCP is Lipschitz continuous. For non-interior continuation/ smoothing methods for solving the nonlinear complementarity problem, such an assumption has been used widely in the literature in order to achieve global linear convergence results of the algorithms.

Non-Interior continuation methods for solving semidefinite complementarity problems

There recently has been much interest in non-interior continuation/smoothing methods for solving linear/nonlinear complementarity problems. We describe extensions of such methods to complementarity problems defined over the cone of block-diagonal symmetric positive semidefinite real matrices. These extensions involve the Chen-Mangasarian class of smoothing functions and the smoothed Fischer-Burmeister function. Issues such as existence of Newton directions, boundedness of iterates, global convergence, and local superlinear convergence will be studied. Preliminary numerical experience on semidefinite linear programs is also reported.

Analysis of Nonsmooth Symmetric-Matrix-Valued Functions with Applications to Semidefinite Complementarity Problems

For any function f from $\mathbb R$ to $\mathbb R$, one can define a corresponding function on the space of n × n (block-diagonal) real symmetric matrices by applying f to the eigenvalues of the spectral decomposition. We show that this matrix-valued function inherits from f the properties of continuity, (local) Lipschitz continuity, directional differentiability, Fréchet differentiability, continuous differentiability, as well as ($\rho$-order) semismoothness. Our analysis uses results from nonsmooth analysis as well as perturbation theory for the spectral decomposition of symmetric matrices. We also apply our results to the semidefinite complementarity problem, addressing some basic issues in the analysis of smoothing/semismooth Newton methods for solving this problem.

Semidefinite Programs: New Search Directions, Smoothing-Type Methods, and Numerical Results

Motivated by some results for linear programs and complementarity problems, this paper gives some new characterizations of the central path conditions for semidefinite programs. Exploiting these characterizations, some smoothing-type methods for the solution of semidefinite programs are derived. The search directions generated by these methods are automatically symmetric, and the overall methods are shown to be globally and locally superlinearly convergent under suitable assumptions. Some numerical results are also included which indicate that the proposed methods are very promising and comparable to several interior-point methods. Moreover, the current method seems to be superior to the smoothing method recently proposed by Chen and Tseng [Non-interior continuation methods for solving semidefinite complementarity problems, {Technical report}, Department of Mathematics, University of Washington, Seattle, 1999].

Application of topological degree theory to semi-definite complementarity problem*

The topological degree theory is applied to study the problem of existence of solutions to the semi-definite complementarity problem (SDCP). A notion of an exceptional family of matrices is introduced, and assertions of a non-strict alternative type are obtained. Namely, for a continuous mapping, there exists at least one of the following two items: either a solution to the SDCP, or an exceptional family of matrices. Hence, if there is no exceptional family, then at least one solution exists

General Interior-Point Maps and Existence of Weighted Paths for Nonlinear Semidefinite Complementarity Problems

Extending the previous work of Monteiro and Pang (1998), this paper studies properties of fundamental maps that can be used to describe the central path of the monotone nonlinear com-plementarity problems over the cone of symmetric positive semidefinite matrices. Instead of focusing our attention on a specific map as was done in the approach of Monteiro and Pang (1998), this paper considers a general form of a fundamental map and introduces conditions on the map that allow us to extend the main results of Monteiro and Pang (1998) to this general map. Each fundamental map leads to a family of “weighted” continuous trajectories which include the central trajectory as a special case. Hence, for complementarity problems over the cone of symmetric positive semidefinite matrices, the notion of weighted central path depends on the fundamental map used to represent the central path.