A graph-based decomposition method for convex quadratic optimization with indicators

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, motivated by inference problems with graphical models, we propose a novel decomposition method for a class of general (sparse) strictly diagonally dominant 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.