Advancing Portfolio Optimization: Hybrid Relaxation and Heuristic Approaches for Cardinality-Constrained MIQP Problems
Keywords: portfolio optimization, mixed-integer quadratic programming, relaxation
Abstract: The growing magnitude of investments in global markets has intensified the need for sophisticated risk mitigation strategies in portfolio optimization. Traditional portfolio optimization models that seek to minimize risk for a specified return frequently incorporate cardinality constraints, rendering them as Mixed-Integer Quadratic Programming (MIQP) challenges. These constraints elevate the problem to NP-Hard status, complicating the solution process. While heuristic methods have historically been favored for their direct approach to MIQP problems, relaxation techniques offer a strategic alternative by simplifying MIQP into a more tractable Quadratic Programming (QP) problem. We first introduce an approach that facilitates the conversion of MIQP to QP by relaxing integer constraints into continuous domains and integrating integer conditions into the objective function using Lagrange multipliers. This dual application not only eases the computational burden but preserves the integrity of the original problem's structure. An innovative diagonalization technique applied to the covariance matrix further refines our method, enhancing the fit for integer variables, as Lagrange multipliers are inherently biased towards continuous variables. We present a comparative analysis of three distinct models, Linear, Dual, and Diagonal, each employing a unique relaxation strategy. Our research evaluates their efficacy in addressing the MIQP problem under cardinality constraints. In conjunction with heuristic methods, the refined solutions from our exact relaxation models serve as a starting point for further refinement using Genetic Algorithm and Neighborhood Searching Algorithm. This hybrid methodology yields results that not only rival but occasionally surpass those achieved by the latest models and the commercial solver CPLEX. Our findings endorse the potential of combining exact and heuristic techniques in portfolio optimization, marking a significant advancement in the field.
Primary Area: optimization
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Submission Number: 10122
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