Addressing Discrete Dynamic Optimization via a Logic-Based Discrete-Steepest Descent Algorithm

Abstract: Dynamic optimization problems involving discrete decisions have several applications, yet lead to challenging optimization problems that must be addressed efficiently. Combining discrete variables with potentially nonlinear constraints stemming from dynamics within an optimization model results in mathematical programs for which off-the-shelf techniques might be insufficient. This work uses a novel approach, the Logic-based Discrete-Steepest Descent Algorithm (LD-SDA), to solve Discrete Dynamic Optimization problems. The problems are formulated using Boolean variables that enforce differential systems of constraints and encode logic constraints that the optimization problem needs to satisfy. By posing the problem as a generalized disjunctive program with dynamic equations within the disjunctions, the LD-SDA takes advantage of the problem's inherent structure to efficiently explore the combinatorial space of the Boolean variables and selectively include relevant differential equations to mitigate the computational complexity inherent in dynamic optimization scenarios. We rigorously evaluate the LD-SDA with benchmark problems from the literature that include dynamic transitioning modes and find it to outperform traditional methods, i.e., mixed-integer nonlinear and generalized disjunctive programming solvers, in terms of efficiency and capability to handle dynamic scenarios. This work presents a systematic method and provides an open-source software implementation to address these discrete dynamic optimization problems by harnessing the information within its logical-differential structure.