🤖 AI Summary
This work addresses the computational challenges of solving large-scale convex mixed-integer quadratic programs (MIQPs), which arise in applications such as subset portfolio selection and become particularly difficult when the covariance matrix has a high condition number or weight constraints are tight. To tackle this, the authors propose DASH, a novel method that introduces a decreasing active-set hierarchy for dimensionality reduction in MIQP for the first time. DASH leverages active-set analysis to reduce problem dimensionality and integrates seamlessly with commercial solvers like Gurobi to enhance optimization efficiency. Experimental results demonstrate that DASH significantly outperforms Gurobi alone on a range of challenging portfolio instances, with solution quality improvements positively correlated with problem difficulty, thereby accelerating convergence and yielding higher-quality optimal solutions.
📝 Abstract
Subset selection problems as MIPs (Mixed Integer Programs) are NP-hard. For large scale problems, it is infeasible to find global optimal solutions in a reasonable time and good-quality incumbent solutions are sought after with MIP solvers in practice. This paper proposes DASH (Decreasing Active Set Hierarchy) -- a dimensionality reduction method that improves the MIP solver performance for a subclass of best subset selection problems that can be formulated as MIQPs (Mixed Integer Quadratic Programs). We develop and evaluate the performance of DASH in the subset portfolio selection problem with comparison to Gurobi, a commercial MIP solver. In addition to the problem size, the difficulty of a problem is related to the condition number of the covariance matrix and the box constraint on portfolio weights. An extensive set of numerical experiments with varying problem configurations shows that DASH offers consistent and significant improvement of incumbent solutions when the problem is difficult to solve by Gurobi. In particular, the magnitude and duration of improvement by DASH scale with the difficulty of the problem.