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

Peijing Liu(peijingl***at***usc.edu)
Salar Fattahi(fattahi***at***umich.edu)
Andres Gomez(gomezand***at***usc.edu)
Simge Kucukyavuz(simge***at***northwestern.edu)

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.

Keywords: Quadratic optimization, indicator variables, sparsity, decomposition, graphical models, Fenchel dual, convex hull

Category 1: Integer Programming ((Mixed) Integer Nonlinear Programming )

Category 2: Applications -- Science and Engineering (Statistics )

Citation: Technical report, USC, October 2021

Download: [PDF]

Entry Submitted: 10/21/2021
Entry Accepted: 10/21/2021
Entry Last Modified: 10/21/2021

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