(250a) Addressing Discrete Dynamic Optimization Via a Logic-Based Discrete-Steepest Descent Algorithm

Control tasks with high-level discrete or logical decisions, such as integrated process design and control[1], multi-stage dynamic model switching, and hybrid operation of reactors [2], can be modeled using mixed-integer dynamic optimization (MIDO). However, MIDO problems are computationally challenging to solve, and standard optimization solvers cannot simultaneously handle discrete variables and differential algebraic equations (DAE). The standard routine for solving MIDO problems is to discretize differential equations, resulting in a mixed-integer nonlinear programming (MINLP) problem [3], [4], [5]. Since the discrete behaviors in mixed-integer control problems can sometimes be modeled as logic variables and constraints, an alternative to MIDO is differential-algebraic generalized disjunctive programming (DAGDP). Compared to the MIDO formulation, the disjunctions in DAGDP problems allow for a more compact representation of the logic-induced dynamic problem, where certain operations or configurations are only feasible or relevant under specific discrete conditions. Moreover, another benefit of the DAGDP formulation is that it facilitates the use and development of alternative solution strategies in addition to the MINLP reformulation, which more effectively leverages the structure of the optimization model.

This work uses a novel approach, the Logic-based Discrete-Steepest Descent Algorithm (LD-SDA) [6], [7], [8], 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 found 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. For the most complicated instances, LD-SDA finds the optimal solution within 10 seconds, while both the MINLP and GDP solver take more than 90 seconds to converge.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.

Reference

[1] Flores-Tlacuahuac, A., & Biegler, L. T. (2007). Simultaneous mixed-integer dynamic optimization for integrated design and control. Computers & chemical engineering, 31(5-6), 588-600.

[2] Ruiz-Femenia, R., Flores-Tlacuahuac, A., & Grossmann, I. E. (2014). Logic-Based Outer-Approximation Algorithm for Solving Discrete-Continuous Dynamic Optimization Problems. Industrial & Engineering Chemistry Research, 53(13), 5067-5080.

[3] Bansal, V., Perkins, J. D., & Pistikopoulos, E. N. (2002). A case study in simultaneous design and control using rigorous, mixed-integer dynamic optimization models. Industrial & Engineering Chemistry Research, 41(4), 760-778.

[4] Biegler, L. T. (2010). Nonlinear programming: concepts, algorithms, and applications to chemical processes. Society for Industrial and Applied Mathematics.

[5] Chachuat, B., Singer, A. B., & Barton, P. I. (2006). Global methods for dynamic optimization and mixed-integer dynamic optimization. Industrial & Engineering Chemistry Research, 45(25), 8373-8392.

[6] Linán, D. A., Bernal, D. E., Ricardez-Sandoval, L. A., & Gómez, J. M. (2020). Optimal design of superstructures for placing units and streams with multiple and ordered available locations. Part I: A new mathematical framework. Computers & Chemical Engineering, 137, 106794.

[7] Bernal, D. E., Ovalle, D., Liñán, D. A., Ricardez-Sandoval, L. A., Gómez, J. M., & Grossmann, I. E. (2022). Process superstructure optimization through discrete steepest descent optimization: a GDP analysis and applications in process intensification. In Computer Aided Chemical Engineering (Vol. 49, pp. 1279-1284). Elsevier.

[8] Palma‐Flores, O., Ricardez‐Sandoval, L. A., & Biegler, L. T. (2023). Simultaneous design and NMPC control under uncertainty and structural decisions: A discrete‐steepest descent algorithm. AIChE Journal, 69(11), e18188.