Expected Maximization of a Concave Utility Function Under Threshold-Based Activation

Maximizing the expected value of a concave and strictly increasing utility function defines a fundamental class of discrete optimization problems. Among them, coverage decision problems with diminishing marginal returns under uncertainty, typically modeled via a set-union operator, have been extensively studied. In the classical framework, an item becomes active once it is covered by at least one chosen meta-item. Motivated by increasing robustness requirements in applications such as automated systems, social networks, and emergency response planning, we extend this setting by introducing threshold-based activation. The resulting generalized problem can be formulated as a mixed-integer nonlinear programming problem, for which we further propose three exact algorithms. The first two methods linearize the utility function using submodular cuts (SC) and outer-approximation (OA) techniques, respectively, resulting in formulations that can be solved exactly by off-the-shelf mixed-integer linear programming solvers. The third method builds upon the OA framework and further employs Benders decomposition (BD) to project out the item-related variables, which enables superior performance on ultra-large-scale instances. Extensive computational experiments show that, compared with the SC and BD methods, the OA method exhibits a substantial speed advantage on instances with a size of around 40,000, which can be solved within 100 s. In contrast, for ultra-large-scale instances with more than 100,000 items, the BD method demonstrates superior computational efficiency. These results provide practical guidance for algorithmic strategy selection and further demonstrate the computational tractability of this broader class of utility maximization problems under threshold-based activation.