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arXiv · 2110.12547

A Graph-based Decomposition Method for Convex Quadratic Optimization with Indicators

Peijing Liu·Salar Fattahi·Andrés Gómez·Simge Küçükyavuz
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Abstract

In this paper, we consider convex quadratic optimization problems with indicator variables when the matrix $Q$ defining the quadratic term in the objective is sparse. We use a graphical representation of the support of $Q$, and show that if this graph is a path, then we can solve the associated problem in polynomial time. This enables us to construct a compact extended formulation for the closure of the convex hull of the epigraph of the mixed-integer convex problem. Furthermore, we propose a novel decomposition method for general (sparse) $Q$, which leverages the efficient algorithm for the path case. Our computational experiments demonstrate the effectiveness of the proposed method compared to state-of-the-art mixed-integer optimization solvers.

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Peijing Liu, Salar Fattahi, Andrés Gómez, Simge Küçükyavuz. 2021-10-24. A Graph-based Decomposition Method for Convex Quadratic Optimization with Indicators. https://arxiv.org/abs/2110.12547

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