Probabilistically Exact Convexification of Mixed-Integer Quadratic Optimization with Indicators
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We study the convexification of mixed-integer quadratic optimization with indicator variables. The problem remains NP-hard even when the quadratic matrix has bandwidth two, posing a fundamental obstacle to tractable exact convexification. To overcome this challenge in realistic settings, we introduce probabilistically exact (inner) convexification, a framework that relaxes the requirement of exactness for every objective while preserving exact optimality for statistically relevant ones. Our approach is motivated by inference applications, where the objective is determined by a data vector drawn from a distribution and a regularization vector chosen from a prescribed family. Our convexification preserves the exactness of the optimal value with high probability over the data distribution, simultaneously for every regularization vector in this family. We apply this notion to quadratic matrices of any fixed bandwidth. Despite their worst-case NP-hardness, we construct probabilistically exact convexifications that admit exact extended second-order cone formulations of size polynomial in the dimension \(n\) and the inverse failure probability \(1/\delta\). The required assumptions for their high-probability exactness are surprisingly mild: they accommodate both light-tailed and heavy-tailed data, including distributions without finite mean, as well as arbitrary nonnegative regularization coefficients.