• Introduces monotonically decreasing lower bounds on asset weight. • Lower bounds adjust dynamically based on the number of assets selected. • Models asset relationships as networks. • Develops an exact branch-and-bound solution algorithm. • Balances diversification and concentration in portfolios. Portfolio optimization models have been widely employed to allocate resources across various alternatives. This study introduces a novel approach where user-defined dynamic lower bounds on asset weights are adjusted based on the number of selected assets. These bounds are positive only for selected assets and monotonically decreasing with respect to nested set expansions (where the asset is included in the smaller set). This method contrasts with static lower bounds by allowing for decreasing weight thresholds as more assets are included, thus balancing concentration and diversification. However, the introduction of such dynamic bounds can pose significant computational challenges, potentially transforming the problem into a nonlinear or nonconvex optimization scenario. To address these issues, the problem is reformulated using network models based on asset relationships, then leveraging graph theory to solve for optimal sets with specific structural properties. Several theoretical attributes of the model are identified and used to develop an exact combinatorial branch-and-bound solution algorithm for finding sets of assets that form independent sets or cliques. A major advantage of the proposed solution method is that its complexity does not necessarily increase for different monotonically decreasing lower bound set functions. Experiments demonstrating the computational performance on several graph types are conducted for model instances with exponentially decaying lower bounds on asset weights and higher-moment coherent risk measures. This work contributes to the broader understanding and application of risk-averse optimization by exploring the effects of dynamic lower bounds on asset allocation strategies.
Maciej Rysz (Sun,) studied this question.