We present a new implementation of a widely used swap-based local search procedure for the P-median problem, proposed in 1968 by Teitz and Bart. Our method produces the same output as the best alternatives described in the literature and, even though its worst-case complexity is similar, it can be significantly faster in practice: speedups of … Read more
This paper attempts to extend the notion of duality for convex cones, by basing it on a pre-described conic ordering and a fixed bilinear mapping. This is an extension of the standard definition of dual cones, in the sense that the {\em nonnegativity}\/ of the inner-product is replaced by a pre-specified conic ordering, defined by … Read more
Given a real function on a Euclidean space, we consider its “robust regularization”: the value of this new function at any given point is the maximum value of the original function in a fixed neighbourhood of the point in question. This construction allows us to impose constraints in an optimization problem *robustly*, safeguarding a constraint … Read more
We propose the elastic string algorithm for computing mountain passes in finite-dimensional problems. We analyze the convergence properties and numerical performance of this algorithm for benchmark problems in chemistry and discretizations of infinite-dimensional variational problems. We show that any limit point of the elastic string algorithm is a path that crosses a critical point at … Read more
Given N customers and a set F of M potential facilities, the P-median problem consists in finding a subset of F with P facilities such that the cost of serving all customers is minimized. This is a well-known NP-complete problem with important applications in location science and classification (clustering). We present here a GRASP (Greedy … Read more
We prove a sufficient global optimality condition for quadratic optimization with quadratic constraints where the variables are allowed to take -1 and 1 values. We extend the condition to quadratic programs with matrix variables and orthogonality conditions, and in particular, to the quadratic assignment problem. CitationBilkent University Technical Report, September 2002.ArticleDownload View PDF
The SDPA (SemiDefinite Programming Algorithm) is a software package for solving general SDPs(SemiDefinite Programs). It is written in C++ with the help of {\it LAPACK} for numerical linear algebra for dense matrix computation. The purpose of this paper is to present a brief description of the latest version of the SDPA and its high performance … Read more
The affine-scaling algorithm was initially developed for linear programming problems. Its extension to problems with a nonlinear objective performs at each iteration a scaling followed by a line search along the steepest descent direction. In this paper we prove that any accumulation point generated by this algorithm when applied to a convex function is an … Read more
The trust region subproblem (the minimization of a quadratic objective subject to one quadratic constraint and denoted TRS) has many applications in diverse areas, e.g. function minimization, sequential quadratic programming, regularization, ridge regression, and discrete optimization. In particular, it determines the step in trust region algorithms for function minimization. Trust region algorithms are popular for … Read more
Quasi-Newton methods in conjunction with the piecewise sequential quadratic programming are investigated for solving mathematical programming with equilibrium constraints, in particular for problems with complementarity constraints. Local convergence as well as superlinear convergence of these quasi-Newton methods can be established under suitable assumptions. In particular, several well-known quasi-Newton methods such as BFGS and DFP are … Read more