Smoothing and Regularization for Mixed-Integer Second-Order Cone Programming with Applications in Portfolio Optimization

Abstract

Second-order cone programming problems (SOCPs) have been well studied in literature, and computationally efficient implementations of solution algorithms exist. In this paper, we study an extension: mixed-integer second-order cone programming problems (MISOCPs). Our focus is on designing an algorithm for solving the underlying SOCPs as nonlinear programming problems (NLPs) within two existing frameworks, branch-and-bound and outer approximation, for mixed-integer nonlinear programming (MINLP). We pay particular attention to resolving the nondifferentiability of the underlying SOCPs via a smoothing reformulation, as well as warmstarting and infeasibility detection using a regularization. We investigate the application of our proposed techniques to portfolio optimization problems that can be formulated as MISOCPs, and preliminary numerical results using the Matlab-based optimization package MILANO (Benson, MILANO-a Matlab-based code for mixed-integer linear and nonlinear optimization. http://www. pages.drexel.edu/~hvb22/milano) are provided.

Optimal Learning in Linear Regression with Combinatorial Feature Selection

“…While problem (27) provides an exact solution of the quantized problem (18), it includes binary variables and thus may be time-consuming to solve. There are specialized algorithms for this problem class (Benson and Saglam 2013b), based on branch-and-bound techniques similar to those used to solve mixed-integer linear programs. As such, bounds on the optimality gap are obtained through the construction of the branch-and-bound tree.…”