A branch-and-cut approach to portfolio selection with marginal risk control in a linear conic programming framework

Abstract

Marginal risk represents the risk contribution of an individual asset to the risk of the entire portfolio. In this paper, we investigate the portfolio selection problem with direct marginal risk control in a linear conic programming framework. The optimization model involved is a nonconvex quadratically constrained quadratic programming (QCQP) problem. We first transform the QCQP problem into a linear conic programming problem, and then approximate the problem by semidefinite programming (SDP) relaxation problems over some subrectangles. In order to improve the lower bounds obtained from the SDP relaxation problems, linear and quadratic polar cuts are introduced for designing a branch-and-cut algorithm, that may yield an ∈-optimal global solution (with respect to feasibility and optimality) in a finite number of iterations. By exploring the special structure of the SDP relaxation problems, an adaptive branch-and-cut rule is employed to speed up the computation. The proposed algorithm is tested and compared with a known method in the literature for portfolio selection problems with hundreds of assets and tens of marginal risk control constraints.