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
We perform a smoothed analysis of the termination phase of an interior-point method. By combining this analysis with the smoothed analysis of Renegar’s interior-point algorithm by Dunagan, Spielman and Teng, we show that the smoothed complexity of an interior-point algorithm for linear programming is $O (m^{3} \log (m/\sigma ))$. In contrast, the best known bound … Read more
A class of matrix valued functions defined by singular values of nonsymmetric matrices is shown to have many properties analogous to matrix valued functions defined by eigenvalues of symmetric matrices. In particular, the (smoothed) matrix valued Fischer-Burmeister function is proved to be strongly semismooth everywhere. This result is also used to show the strong semismoothness … 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 study several issues related to the characterization of specific classes of multivariate quadratic mappings that are nonnegative over a given domain, with nonnegativity defined by a pre-specified conic order. In particular, we consider the set (cone) of nonnegative quadratic mappings defined with respect to the positive semidefinite matrix cone, and study … 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
This paper considers robust optimization to cope with uncertainty about the stock return process in one period portfolio selection problems involving options. The ro- bust approach relates portfolio choice to uncertainty, making more cautious portfolios when uncertainty is high. We represent uncertainty by a set of plausible expected returns of the underlyings and show that … 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