Discrete Choquet Integral Optimization With Positive and Negative Interactions

Our study focuses on using fuzzy measures and integrals in optimization with interacting decision variables. We optimize piecewise linear objectives through the use of discrete Choquet integral, while s.t. a set of linear constraints. These optimization models are relevant in scenarios where decision variables interact in complex ways, with potential synergies or redundancies among them. Fuzzy measures, which are typically nonadditive and nonmodular set functions, provide a solid mathematical foundation for modeling such interactions. By extending traditional linear objectives, the discrete Choquet integral is a versatile tool for addressing these optimization problems. We decompose fuzzy measures with two opposite kinds of interactions, into submodular and supermodular parts, which accordingly transfer into the convex and concave components of optimized objectives. We customize the difference of convex algorithm for our specific objective functions that involve positive and negative interactions, and investigate its numerical scalability.