We propose the use of controlled perturbations to address the challenging question of optimal active-set prediction for interior point methods. Namely, in the context of linear programming, we consider perturbing the inequality constraints/bounds so as to enlarge the feasible set. We show that if the perturbations are chosen appropriately, the solution of the original problem … Read more
A symmetric matrix is called copositive if it generates a quadratic form taking no negative values over the nonnegative orthant, and the linear optimization problem over the set of copositive matrices is called the copositive programming problem. Recently, many studies have been done on the copositive programming problem (see, for example, \cite{aDUR10, aBOMZE12}). Among others, … Read more
We present an improved version of an infeasible interior-point method for linear optimization published in 2006. In the earlier version each iteration consisted of one so-called infeasibility step and a few – at most three – centering steps. In this paper each iteration consists of only a infeasibility step, whereas the iteration bound improves the … Read more
We present a strongly polynomial algorithm for linear optimization problems of the form min{cx|Ax = b, x >= 0} having 0-1 vectors among their optimal solutions. The algorithm runs in time O(n^4*max\{m,log n}), where n is the number of variables and m is the number of equations. The algorithm also constructs necessary and sufficient optimality … Read more
Many practical problems can be formulated as $\ell_0$-minimization problems with nonnegativity constraints, which seek the sparsest nonnegative solutions to underdetermined linear systems. Recent study indicates that $\ell_1$-minimization is efficient for solving some classes of $\ell_0$-minimization problems. From a mathematical point of view, however, the understanding of the relationship between $\ell_0$- and $\ell_1$-minimization remains incomplete. In … Read more
Many practical problems can be formulated as $\ell_0$-minimization problems with nonnegativity constraints, which seek the sparsest nonnegative solutions to underdetermined linear systems. Recent study indicates that $\ell_1$-minimization is efficient for solving some classes of $\ell_0$-minimization problems. From a mathematical point of view, however, the understanding of the relationship between $\ell_0$- and $\ell_1$-minimization remains incomplete. In … Read more
We treat uncertain linear programming problems by utilizing the notion of weighted analytic centers and notions from the area of multi-criteria decision making. In addition to many practical advantages, due to the flexibility of our approach, we are able to prove that the robust optimal solutions generated by our algorithms are at least as desirable … Read more
This is the “front matter” of a new open-source book on Linear Optimization. The book and associated Matlab/AMPL/Mathematica programs are freely available from: https://sites.google.com/site/jonleewebpage/home/publications/#book CitationJon Lee, “A First Course in Linear Optimization”, Third Edition, Reex Press, 2013-2017.ArticleDownload View PDF
This article aims to explain the Nested Benders algorithm for the solution of large-scale stochastic programming problems in a way that is intelligible to someone coming to it for the first time. In doing so it gives an explanation of Benders decomposition and of its application to two-stage stochastic programming problems (also known in this … Read more
We introduce and study conic geometric programs (CGPs), which are convex optimization problems that unify geometric programs (GPs) and conic optimization problems such as linear programs (LPs) and semidefinite programs (SDPs). A CGP consists of a linear objective function that is to be minimized subject to affine constraints, convex conic constraints, and upper bound constraints … Read more