[Paper Review] Doubly nonnegative relaxation method for solving multiple objective quadratic programming problems

Yanqin Bai, Chuan-Hao Guo|arXiv (Cornell University)|Nov 20, 2012

Advanced Optimization Algorithms Research10 references3 citations

TL;DR

This paper proposes a doubly nonnegative relaxation method to solve multiple objective quadratic programming (MOQP) problems by transforming them into a single-objective problem via the linear weighted sum method, then relaxing the nonconvex problem into a convex doubly nonnegative program (DNNP). The optimal solutions of the DNNP yield (weakly) Pareto optimal solutions under mild conditions, and numerical results on portfolio selection demonstrate the method's effectiveness and promise in achieving lower risk and higher return with sparse investment allocations.

ABSTRACT

Multicriterion optimization and Pareto optimality are fundamental tools in economics. In this paper we propose a new relaxation method for solving multiple objective quadratic programming problems. Exploiting the technique of the linear weighted sum method, we reformulate the original multiple objective quadratic programming problems into a single objective one. Since such single objective quadratic programming problem is still nonconvex and NP-hard in general. By using the techniques of lifting and doubly nonnegative relaxation, respectively, this single objective quadratic programming problem is transformed to a computable convex doubly nonnegative programming problem. The optimal solutions of this computable convex problem are (weakly) Pareto optimal solutions of the original problem under some mild conditions. Moreover, the proposed method is tested with two examples and a practical portfolio selection problem. The test problems are solved by exttt{CVX} package which is a solver for convex optimization. The numerical results show that the proposed method is effective and promising.

Motivation & Objective

• To address the challenge of solving nonconvex, NP-hard multiple objective quadratic programming (MOQP) problems in practical applications.
• To develop a computationally tractable method that identifies (weakly) Pareto optimal solutions for nonconvex MOQP problems.
• To integrate the linear weighted sum method with lifting and doubly nonnegative relaxation techniques to transform the original nonconvex problem into a convex, solvable form.
• To validate the method’s effectiveness on benchmark and real-world portfolio selection problems.
• To demonstrate that the method produces sparse optimal solutions, indicating focused investment on key assets.

Proposed method

• The linear weighted sum method is applied to convert the multi-objective problem into a single-objective quadratic program (SOQP).
• The SOQP is reformulated as a completely positive programming (CPP) problem using lifting techniques, preserving the original problem's structure.
• A doubly nonnegative relaxation (DNNP) is applied to the CPP problem, transforming it into a convex, computationally tractable optimization problem.
• The optimal solutions of the DNNP problem are shown to yield (weakly) Pareto optimal solutions for the original MOQP under mild conditions.
• The DNNP problem is solved using the CVX package, a convex optimization solver, enabling numerical validation.
• The method leverages the structure of the nonnegative orthant and linear constraints to maintain feasibility throughout the relaxation process.

Experimental results

Research questions

• RQ1 Can the doubly nonnegative relaxation method effectively approximate Pareto optimal solutions for nonconvex MOQP problems?
• RQ2 How does the proposed method compare to existing approaches in terms of solution quality and sparsity in portfolio selection?
• RQ3 Under what conditions do the solutions of the relaxed DNNP problem correspond to (weakly) Pareto optimal solutions of the original MOQP?
• RQ4 Can the method produce sparse investment allocations that reflect strategic focus on key assets?
• RQ5 How sensitive are the results to the choice of weighting vectors in the linear weighted sum method?

Key findings

• For Example 5, the method achieved a risk of 0.1894 and a return of 15.6800 with λ = (0.3482, 0.1655, 0.4863), outperforming the results in [16] in terms of risk reduction and return improvement.
• The optimal solution vector x* was sparse, with only three non-zero components, indicating focused investment on a few key stocks.
• When λ = (0.5141, 0.1931, 0.2927), the method yielded a risk of 0.1320 and return of 15.5462, further confirming the method’s ability to balance risk and return.
• The method produced comparable or better results than the benchmark in [16], with lower risk (0.1894 vs. 0.6823) and higher return (15.6800 vs. 7.5079).
• The DNNP relaxation effectively captured the structure of the original nonconvex problem, enabling convergence to high-quality Pareto solutions.
• The method demonstrated robustness across different weight vectors, allowing investors to adjust preferences and obtain tailored risk-return trade-offs.