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<p>We consider in this paper an optimal portfolio selection with an additional objective of minimizing the maximum relative marginal risk, a novel measure of risk diversification. Its optimization model is to minimize the sum of a quadratic form and a maximum of quadratic fractional functions subject to linear constraints, which is an NP-hard non-convex and non-smooth optimization problem. First, we reformulate this non-convex optimization problem as an equivalent non-convex quadratically constrained quadratic programming (QCQP). We then propose a successive convex optimization (SCO) algorithm for this non-convex QCQP based on the second-order cone programming (SOCP) approximation and show that it either converges to or terminates finitely to a quasi-<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\epsilon \)</EquationSource> </InlineEquation>-local solution of this non-convex QCQP. Second, we develop a novel branch-and-bound algorithm to find a globally optimal solution to this non-convex QCQP within a pre-specified <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\epsilon \)</EquationSource> </InlineEquation>-tolerance by integrating the SCO approach, the SOCP relaxation and the adaptive branch-and-cut rule. We establish the convergence and the complexity of the proposed algorithm. Finally, we conduct numerical experiments to demonstrate the effectiveness of the proposed algorithm in finding a globally optimal solution to medium and large-scale random instances.</p>

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A novel global algorithm for optimal portfolio selection with maximum relative marginal risk via SCO method and SOCP relaxation

  • 期刊论文

  • 发表日期：

    2025年7月23日

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  • Hezhi Luo,
  • Tianxing Gou,
  • Huixian Wu,
  • Qian Li

摘要

We consider in this paper an optimal portfolio selection with an additional objective of minimizing the maximum relative marginal risk, a novel measure of risk diversification. Its optimization model is to minimize the sum of a quadratic form and a maximum of quadratic fractional functions subject to linear constraints, which is an NP-hard non-convex and non-smooth optimization problem. First, we reformulate this non-convex optimization problem as an equivalent non-convex quadratically constrained quadratic programming (QCQP). We then propose a successive convex optimization (SCO) algorithm for this non-convex QCQP based on the second-order cone programming (SOCP) approximation and show that it either converges to or terminates finitely to a quasi- \(\epsilon \) -local solution of this non-convex QCQP. Second, we develop a novel branch-and-bound algorithm to find a globally optimal solution to this non-convex QCQP within a pre-specified \(\epsilon \) -tolerance by integrating the SCO approach, the SOCP relaxation and the adaptive branch-and-cut rule. We establish the convergence and the complexity of the proposed algorithm. Finally, we conduct numerical experiments to demonstrate the effectiveness of the proposed algorithm in finding a globally optimal solution to medium and large-scale random instances.