Brief paperOptimistic optimization for continuous nonconvex piecewise affine functions☆
Introduction
Piecewise affine (PWA) functions are widely used in various fields for approximating nonlinearities, see Azuma et al., 2010, Breschi et al., 2016, Piga et al., 2020 and Sridhar, Linderoth, and Luedtke (2014); they also appear as cost functions of numerous optimization problems, see Croxton, Gendron, and Magnanti (2003b) and Rizk, Martel, and Ramudhin (2006). The optimization of nonconvex PWA functions are often described as mixed integer linear programming (MILP) problems (Croxton et al., 2003a, Vielma et al., 2010). However, the worst-case complexity of MILP solvers grows exponentially with the number of polyhedral subregions of the PWA functions, which usually make the problem solving process less efficient.
We focus on the optimization problem of a continuous and nonconvex PWA function over a given polytope and propose to apply optimistic optimization to seek the global optimal solution. Optimistic optimization (Munos, 2011, Munos, 2014) is a class of algorithms that start from a hierarchical partition of the feasible set and gradually focuses on the most promising area until they eventually perform a local search around the global optimum of the function. The gap between the best value returned by the algorithm and the real global optimum can be expressed as a function of the number of iterations, which can be specified in advance. Optimistic optimization can be applied to the general problem of black-box optimization of a function given evaluations of the functions over general search spaces. Until now, in the literature on optimistic optimization, the feasible set is often assumed to be a hypercube or a hyperbox. In our previous work (Xu, van den Boom, Buşoniu et al., 2016, Xu, van den Boom, De Schutter, 2016), we have extended optimistic optimization to solve the model predictive control problem for max-plus linear and continuous PWA systems. In Xu, van den Boom, Buşoniu et al. (2016), the PWA-MPC problem is recast as an optimization problem of a continuous PWA objective function. Particularly, the linear constraints on states and inputs are treated as soft constraints and are replaced by adding a penalty function to the objective function. As a result, the feasible set becomes a hyberbox for which the hierarchical partitioning can be efficiently developed.
In this paper, the linear constraints on decision variables are considered as hard constraints and, for the first time in the literature on optimistic optimization, a polytopic feasible set is considered. This extension from a hyperbox feasible set to a polytopic one is not trivial but useful because a polytopic feasible set allows to include general affine constraints on the control variables rather than only single bound constraints. A partition of the given polytope is required to perform the search process. The partitioning should generate well-shaped cells that shrink with the depth. We first employ Delaunay triangulation to divide the polytope into a mesh of simplices and next repeatedly use edgewise subdivision to subdivide the simplices into smaller simplices that satisfy the requirements for optimistic optimization. For this partitioning approach, we develop analytic expressions for the core parameters of optimistic optimization based on the knowledge of the Lipschitz constants of the PWA function. The numerical example shows that using optimistic-optimization-based algorithms for the optimization of a continuous and nonconvex PWA function over a given polytope is more efficient than transforming into an MILP problem if the number of polyhedral subregions of the PWA function is large.
This paper is organized as follows. In Sections 2 Preliminaries, 3 Problem statement, we give some definitions and describe the optimization problem of continuous PWA functions. In Section 4, we introduce background of an optimistic optimization algorithm. In Section 5, we propose a partitioning approach for which we develop the analytic expressions for the core parameters of optimistic optimization. In Section 6, the proposed approach is assessed with a numerical example. Finally, Section 7 concludes the paper.
Section snippets
Preliminaries
For any , define . This section presents some necessary definitions, which are based on Borrelli (2003) and Preparata and Shamos (1985).
Definition 1 Polyhedron A polyhedron is a convex set given as the intersection of a finite number of half-spaces.
Definition 2 Polytope A bounded polyhedron is called a polytope, for some matrix and some vector .
Definition 3 Simplex An -simplex with is the convex hull of affinely independent points which are its vertices. If , the set is simply called a
Problem statement
Consider the following optimization problem where and are the constraint matrix and vector, and is a scalar-valued continuous PWA function given by , , with , , . We assume that the feasible set is nonempty and bounded. From Definition 2, is a polytope.
In this paper, we consider the case that is continuous and nonconvex and that the number of polyhedral subregions is much larger than . For
Deterministic optimistic optimization
In this section, we introduce the background of the deterministic optimistic optimization (DOO) algorithm (Munos, 2011). The notations and remain generic in this section.
DOO algorithm is based on a given partitioning of . For any integer , the space is recursively split into cells where is a finite positive integer denoting the maximum number of child cells of a parent cell. The partitioning may be represented by a tree structure. The whole set is denoted as and
Optimistic optimization of PWA functions
In this section, we first develop a partitioning approach for the polytopic feasible set .
Numerical example
In this section, we evaluate the proposed optimistic-optimization-based approach for continuous PWA functions and compare it with other methods. The instances considered include 60 randomly generated continuous PWA functions in which the vector pairs , contain pseudorandom values drawn from the standard normal distribution with where is also random. Below we compare the efficiency of the DOO algorithm, the MILP method, and the DIRECT algorithm (Jones,
Conclusions
In this paper, we have considered the optimization of a continuous nonconvex PWA function over a polytope. We have proposed an optimistic-optimization-based approach to solve the given problem. In particular, by employing Delaunay triangulation and edgewise subdivision, we have constructed a partition of the feasible set satisfying the requirements for optimistic optimization. We have also derived the analytic expressions for the core parameters. Numerical examples have been implemented to test
Jia Xu obtained the B.Sc. degree in Statistics from the School of Mathematics and Statistics, Shandong University, Weihai, China. In September 2009 she became a master student in Tongji University, Shanghai, China. She obtained the Ph.D. degree in systems and control from the Delft Center for Systems and Control, Delft University of Technology, Delft, The Netherlands, in 2019. She is currently a postdoc with the Department of Control Science and Engineering, Tongji University, China. Her
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Cited by (0)
Jia Xu obtained the B.Sc. degree in Statistics from the School of Mathematics and Statistics, Shandong University, Weihai, China. In September 2009 she became a master student in Tongji University, Shanghai, China. She obtained the Ph.D. degree in systems and control from the Delft Center for Systems and Control, Delft University of Technology, Delft, The Netherlands, in 2019. She is currently a postdoc with the Department of Control Science and Engineering, Tongji University, China. Her research interests include control of discrete-event and hybrid systems, model predictive control, optimization, distributed control.
Ton van den Boom received his M.Sc. and Ph.D. degrees in Electrical Engineering from the Eindhoven University of Technology, The Netherlands, in 1988 and 1993, respectively. Currently, he is an associate professor at Delft Center for Systems and Control (DCSC) department of Delft University of Technology. His research focus is mainly in modeling and control of discrete event and hybrid systems, in particular max-plus-linear systems, max–min-plus-scaling systems, and switching max-plus-linear systems (both stochastic and deterministic), with applications in manufacturing systems and transportation networks.
Bart De Schutter (IEEE member since 2008, senior member since 2010, fellow since 2019) is a full professor and head of department at the Delft Center for Systems and Control of Delft University of Technology in Delft, The Netherlands. Bart De Schutter is senior editor of the IEEE Transactions on Intelligent Transportation Systems and associate editor of IEEE Transactions on Automatic Control. His current research interests include multi-level and multi-agent control, learning-based control, and control of hybrid systems with applications in intelligent transportation systems and smart energy systems.
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Research supported by the Chinese Scholarship Council grant and the National Science Foundation of China (Grant No. 62003245). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Jan Komenda and Christoforos Hadjicostis under the direction of Editor Christos G. Cassandras.