arXiv subjects

Angelika Wiegele

Publications and source records attributed to Angelika Wiegele.

21 records · Page 2Linked to original sources

arXiv2016

A coordinate ascent method for solving semidefinite relaxations of non-convex quadratic integer programs

We present a coordinate ascent method for a class of semidefinite programming problems that arise in non-convex quadratic integer optimization. These semidefinite programs are characterized by a small total number of active constraints and by low-rank constraint matrices. We exploit this special structure by solving the dual problem, using a barrier method in combination with a coordinate-wise exact line search. The main ingredient of our algorithm is the computationally cheap update at each iteration and an easy computation of the exact step size. Compared to interior point methods, our approach is much faster in obtaining strong dual bounds. Moreover, no explicit separation and reoptimization is necessary even if the set of primal constraints is large, since in our dual approach this is covered by implicitly considering all primal constraints when selecting the next coordinate.

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arXiv2016

Exact Solution Methods for the $k$-item Quadratic Knapsack Problem

The purpose of this paper is to solve the 0-1 $k$-item quadratic knapsack problem $(kQKP)$, a problem of maximizing a quadratic function subject to two linear constraints. We propose an exact method based on semidefinite optimization. The semidefinite relaxation used in our approach includes simple rank one constraints, which can be handled efficiently by interior point methods. Furthermore, we strengthen the relaxation by polyhedral constraints and obtain approximate solutions to this semidefinite problem by applying a bundle method. We review other exact solution methods and compare all these approaches by experimenting with instances of various sizes and densities.

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arXiv2015

Another pedagogy for mixed-integer Gomory

We present a version of GMI (Gomory mixed-integer) cuts in a way so that they are derived with respect to a "dual form" mixed-integer optimization problem and applied on the standard-form primal side as columns, using the primal simplex algorithm. This follows the general scheme of He and Lee, who did the case of Gomory pure-integer cuts. Our input mixed-integer problem is not in standard form, and so our cuts are derived rather differently from how they are normally derived. A convenient way to develop GMI cuts is from MIR (mixed-integer rounding) cuts, which are developed from 2-dimensional BMI (basic mixed integer) cuts, which involve a nonnegative continuous variable and an integer variable. The nonnegativity of the continuous variable is not the right tool for us, as our starting point (the "dual form" mixed-integer optimization problem) has no nonnegativity. So we work out a different 2-dimensional starting point, a pair of somewhat arbitrary inequalities in one continuous and one integer variable. In the end, we follow the approach of He and Lee, getting now a finitely converging primal-simplex column-generation algorithm for mixed-integer optimization problems.

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