A simultaneous diagonalization based SOCP relaxation for portfolio optimization with an orthogonality constraint

• Published in: Computational optimization and applications Vol. 85; no. 1; pp. 247 - 261
• Main Authors: Xu, Zhijun, Zhou, Jing
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.05.2023; Springer Nature B.V
• Subjects: Algorithms; Branch and bound methods; Convex and Discrete Geometry; Cost analysis; Efficiency; Management Science; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrices (mathematics); Operations Research; Operations Research/Decision Theory; Optimization; Orthogonality; Quadratic programming; Semidefinite programming; Statistics; Portfolio optimization; 90C33; 90C34; Semi-definite programming relaxation; Branch and bound algorithm; 90C26; Second order cone programming relaxation; Simultaneous diagonalization; Orthogonality constraint
• ISSN: 0926-6003; 1573-2894
• DOI: 10.1007/s10589-023-00452-9

The portfolio rebalancing with transaction costs plays an important role in both theoretical analyses and commercial applications. This paper studies a standard portfolio problem that is subject to an additional orthogonality constraint guaranteeing that buying and selling a same security do not occur at the same time point. Incorporating the orthogonality constraint into the portfolio problem leads to a quadratic programming problem with linear complementarity constraints. We derive an enhanced simultaneous diagonalization based second order cone programming (ESDSOCP) relaxation by taking advantage of the feature that the objective and constraint matrices are commutative. The ESDSOCP relaxation has lower computational complexity than the semi-definite programming (SDP) relaxation, and it is proved to be as tight as the SDP relaxation. It is worth noting that the original simultaneous diagonalization based second order cone programming relaxation (SDSOCP) is only guaranteed to be as tight as the SDP relaxation on condition that the objective matrix is positive definite. Note that the objective matrix in this paper is positive semidefinite (while not positive definite), thus the ESDSOCP relaxation outperforms the original SDSOCP relaxation. We further design a branch and bound algorithm based on the ESDSOCP relaxation to find the global optimal solution and computational results illustrate the effectiveness of the proposed algorithm.