Portfolio Optimization with 'Physical' Decision Variables and Non-Linear Performance Metrics: Diversification Challenge and Proposals

Abstract: Portfolio optimization (PO) is a core tool in financial and operational decision-making, typically balancing expected profit and risk. In real-world applications, particularly in the energy sector, decision variables can be expressed as physical quantities (e.g., production volumes), and nonlinear performance metrics such as Return on Investment (ROI) may be requested. These modeling choices introduce challenges, including the non-additivity of the objective function. This often results in highly concentrated optimized portfolios and thus limited diversification, which can be problematic for decision-makers seeking balanced investment strategies. This paper proposes two strategies to enhance diversification in ROI-based PO models, both based on the Herfindahl-Hirschman Index (HHI). The first incorporates an HHI term directly into the objective function, with its corresponding weight allowing control over diversification. The second directly maximizes diversification while controlling expected profit and risk degradation around the optimum portfolio (obtained through conventional PO). Both strategies are evaluated using synthetic data (energy assets) to illustrate their behavior and practical trade-offs. The results highlight how each method can support different decision-making needs and enhance portfolio robustness.