A Novel Logic Model and Solution Paradigm of Optimization under Uncertainty

This paper defines a logic model of optimization under uncertainty which optimizes the expectation of a uncertainty-perturbed objective function and subjects to a new type of constraints--the probabilistically robust constraints (PRC). This novel logic model is an alternative to, but more general than, the existing ones like the stochastic, robust, and chance-constrained models. The novelty mainly resides in the newly defined PRC which logically requires that an optimal solution should be feasible to high-probability realizations of the uncertain variables. Given that the existing methods of obtaining deterministic approximations are either inapplicable or inefficient to this new logic model, we propose a novel solution paradigm. First, the logic model is approximated by a data-driven deterministic program following an alpha-process of the input data (scenario) set. A sufficient condition on the relation between the accuracy of deterministic approximation and the needed number of input scenarios is provided. Second, the concept of strategic scenario selection (S^3) is developed to figure out a limited number of active scenarios as the input to the data-driven deterministic approximation. Three S^3 algorithms are designed for the cases of discrete, continuous, and mixed-integer uncertain variables respectively, which are not sensitive to the continuity of the decision variables. Numerical experiments showed that the S^3-based data-driven program can accurately approximate the new logic model with low computational complexity.Comment: 36 page