We study interior-point methods for optimization problems in the case of infeasibility or unboundedness. While many such methods are designed to search for optimal solutions even when they do not exist, we show that they can be viewed as implicitly searching for well-defined optimal solutions to related problems whose optimal solutions give certificates of infeasibility … Read more
The search directions in an interior-point method for large scale semidefinite programming (SDP) can be computed by applying a Krylov iterative method to either the Schur complement equation (SCE) or the augmented equation. Both methods suffer from slow convergence as interior-point iterates approach optimality. Numerical experiments have shown that diagonally preconditioned conjugate residual method on … Read more
In this paper, we survey the most recent methods that have been developed for the solution of semidefinite programs. We first concentrate on the methods that have been primarily motivated by the interior point (IP) algorithms for linear programming, putting special emphasis in the class of primal-dual path-following algorithms. We also survey methods that have … Read more
We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems. Citation Preliminary version appeared as Report PNA-R0210, CWI, Amsterdam, April 2002. To appear as Chapter in the Handbook on Discrete Optimization, K. Aardal, G. Nemhauser, R. Weismantel, eds., Elsevier Publishers. Article Download View Semidefinite programming and integer … Read more
We study a smoothing Newton method for solving a nonsmooth matrix equation that includes semidefinite programming and the semidefinte 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. We also establish quadratic convergence of this … Read more
The presence of complementarity constraints brings a combinatorial flavour to an optimization problem. A quadratic programming problem with complementarity constraints can be relaxed to give a semidefinite programming problem. The solution to this relaxation can be used to generate feasible solutions to the complementarity constraints. A quadratic programming problem is solved for each of these … Read more
Cutting plane methods provide the means to solve large scale semidefinite programs (SDP) cheaply and quickly. They can also conceivably be employed for the purposes of re-optimization after branching, or the addition of cutting planes. We give a survey of various cutting plane approaches for SDP in this paper. These cutting plane approaches arise from … Read more
Consider the problem of finding optimal bounds on the expected value of piece-wise polynomials over all measures with a given set of moments. We show that this problem can be studied within the framework of conic programming. Relying on a key approximation result for conic programming, we show that these bounds can be numerically computed … Read more
This is the user’s guide for version 5.0 of CSDP, a C library for semidefinite programming. The code is available at http://www.nmt.edu/~borchers/csdp.html Article Download View CSDP 5.0 User's Guide
Abstract: The SDPA (SemiDefinite Programming Algorithm) is known as efficient computer software based on primal-dual interior-point method for solving SDPs (Semidefinite Programs). In many applications, however, some SDPs become larger and larger, too large for the SDPA to solve on a single processor. In execution of the SDPA applied to large scale SDPs, the computation … Read more