A Frank-Wolfe Based Branch-and-Bound Algorithm for Mean-Risk Optimization

TL;DR

This paper introduces an exact branch-and-bound algorithm utilizing a Frank-Wolfe based method to solve mean-risk portfolio optimization problems with mixed-integer variables and customizable risk measures.

Contribution

It develops a novel Frank-Wolfe based branch-and-bound algorithm for convex mixed-integer mean-risk optimization with flexible risk modeling.

Findings

Outperforms CPLEX MISOCP solver on portfolio instances with linear risk functions.

Effective for problems with continuous and integer decision variables.

Handles arbitrary monotone risk-weighting functions.

Abstract

We present an exact algorithm for mean-risk optimization subject to a budget constraint, where decision variables may be continuous or integer. The risk is measured by the covariance matrix and weighted by an arbitrary monotone function, which allows to model risk-aversion in a very individual way. We address this class of convex mixed-integer minimization problems by designing a branch-and-bound algorithm, where at each node, the continuous relaxation is solved by a non-monotone Frank-Wolfe type algorithm with away-steps. Experimental results on portfolio optimization problems show that our approach can outperform the MISOCP solver of CPLEX 12.6 for instances where a linear risk-weighting function is considered.