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

TL;DR

This work addresses diversification challenges in portfolio optimization when decision variables are physical quantities and performance is nonlinear in ROI, which can lead to highly concentrated portfolios. It introduces two diversification strategies built on the Herfindahl–Hirschman Index: (i) an objective-function extension that adds an HHI term with a controllable weight, and (ii) a constraint-based approach that searches for diversified portfolios within acceptable degradations of ROI and risk relative to a baseline optimal portfolio. Both strategies are demonstrated on synthetic energy-asset data, illustrating how diversification interacts with ROI and risk and offering practitioners a choice between direct diversification control and diversification under performance tolerances. The results highlight trade-offs and robustness benefits, suggesting these methods can support different decision-making needs in energy portfolio management and can be extended with parallelization and systematic comparisons of PO variants.

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.

Figures (5)

• Figure 1: Illustration of the heuristic for generating tolerance pairs $(\Delta_p, \Delta_r)$ around an optimal portfolio $\mathbf{x}^*$ (blue point). The figure represents the efficient frontier (dark blue curve) in the Profit ($\mathrm{ROI}$ in our case) versus $\mathrm{Risk}$ (CVaR deviation in our case) plane, with $\mathbf{x}^*$ denoting the optimal solution obtained from the baseline PO, for a given $w$.
• Figure 2: Results of diversification via objective function extension. Panel (a) shows the baseline optimal portfolio $\mathbf{x}^*$ ($w_d = 0$). Panels (b)-(d) illustrate the effect of increasing $w_d$ (diversification weight). Each asset is represented by a distinct color.
• Figure 3: Comparison of efficient frontiers for different values of the diversification weight $w_d$ in the $\mathrm{ROI(\mathbf{x})}$ versus $\mathrm{Risk_\beta(\mathbf{x})}$ plane. Each curve corresponds to a specific $w_d$, with points representing solutions for various $w$. The axes were normalized to the interval [0, 1].
• Figure 4: Optimal and diversified suboptimal points in the $\mathrm{ROI(\mathbf{x})}$ versus $\mathrm{Risk_\beta(\mathbf{x})}$ plane. The black points indicate the optimal portfolios on the efficient frontier for each value of $w$, while the colored points represent diversified suboptimal portfolios generated for different tolerance pairs $(\Delta_p, \Delta_r)$. The axes were normalized to the interval [0, 1].
• Figure 5: Portfolio share distributions for optimal and suboptimal points. In each subfigure (corresponding to a different $w$), the optimal portfolio $\mathbf{x}^*$ is highlighted with a black dashed outline. Each asset is represented by a distinct color.