A note on completely positive relaxations of quadratic problems in a multiobjective framework

Gabriele Eichfelder,Patrick Groetzner

doi:10.1007/s10898-021-01091-2

Gabriele Eichfelder, Patrick Groetzner

Open Access

https://doi.org/10.1007/s10898-021-01091-2

Copy DOI

Abstract

In a single-objective setting, nonconvex quadratic problems can equivalently be reformulated as convex problems over the cone of completely positive matrices. In small dimensions this cone equals the cone of matrices which are entrywise nonnegative and positive semidefinite, so the convex reformulation can be solved via SDP solvers. Considering multiobjective nonconvex quadratic problems, naturally the question arises, whether the advantage of convex reformulations extends to the multicriteria framework. In this note, we show that this approach only finds the supported nondominated points, which can already be found by using the weighted sum scalarization of the multiobjective quadratic problem, i.e. it is not suitable for multiobjective nonconvex problems.

Highlights

Even combinatorial optimization problems can equivalently be reformulated as convex problems over the so-called completely positive matrix cone
Especially in small dimensions, an optimal solutions for the nonconvex problem can be obtained by solving the convex reformulation
We will show that considering multiobjective completely positive reformulations for this problem returns just a subset of the optimal solution set: only those efficient solutions which can already be found by using a weighted sum scalarization

Summary

Introduction

Even combinatorial optimization problems can equivalently be reformulated as convex problems over the so-called completely positive matrix cone. We will show that considering multiobjective completely positive reformulations for this problem returns just a subset of the optimal solution set: only those efficient solutions which can already be found by using a weighted sum scalarization. With our results we show that first applying the weighted sum method and studying its completely positive reformulation, as proposed by [1], finds exactly the same efficient solutions as solving the direct multiobjective completely positive reformulation. This convexifying approach using the cone of completely positive matrices seems not to be a suitable tool for nonconvex problems with multiple objectives.

Convex reformulations of nonconvex quadratic problems

Multiobjective optimization: efficiency and supported points

Conclusion