Abstract
The inevitable presence of uncertain parameters in critical applications of process optimization can lead to undesirable or infeasible solutions. For this reason, optimization under parametric uncertainty was, and continues to be a core area of research within Process Systems Engineering. Multiparametric programming is a strategy that offers a holistic perspective for the solution of this class of mathematical programming problems. Specifically, multiparametric programming theory enables the derivation of the optimal solution as a function of the uncertain parameters, explicitly revealing the impact of uncertainty in optimal decision-making. By taking advantage of such a relationship, new breakthroughs in the solution of challenging formulations with uncertainty have been created. Apart from that, researchers have utilized multiparametric programming techniques to solve deterministic classes of problems, by treating specific elements of the optimization program as uncertain parameters. In the past years, there has been a significant number of publications in the literature involving multiparametric programming. The present review article covers recent theoretical, algorithmic, and application developments in multiparametric programming. Additionally, several areas for potential contributions in this field are discussed, highlighting the benefits of multiparametric programming in future research efforts.
Highlights
What is the impact of varying parameters in mathematical optimization problems? Optimal decision-making under parametric uncertainty is a fundamental area of research within the Process Systems Engineering community since its presence in practical applications is unavoidable
Research efforts could be categorized into theoretical developments to derive and store the solution of new classes of multiparametric programming problems, the construction of efficient algorithms for the exploration of the parameter space which is described by the optimal active sets, and their application in engineering problems. multiparametric/explicit model predictive control (mpMPC) is by far the most well-studied and explored application of multiparametric programming
Among the computational algebra techniques, they have shown to be effective in solving nonlinear equations analytically, and further developments in parallel computing will encourage their wider adoption in multiparametric programming
Summary
What is the impact of varying parameters in mathematical optimization problems? Optimal decision-making under parametric uncertainty is a fundamental area of research within the Process Systems Engineering community since its presence in practical applications is unavoidable. Multiparametric programming attracted significant interest, most notably following the efforts of Gal and Nedoma (1972), who considered multiple parameters, and introduced a systematic strategy to solve multiparametric linear programming problems (mpLPs), and explore all optimal active sets In their formulation, the uncertain parameters appear solely in the right-hand side of the constraints, and the solution of the aforementioned program offered an explicit relationship between the decision variables and the varying parameters. Research efforts could be categorized into theoretical developments to derive and store the solution of new classes of multiparametric programming problems, the construction of efficient algorithms for the exploration of the parameter space which is described by the optimal active sets, and their application in engineering problems.
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