A Hyper-Matheuristic Methodology for Mixed Integer Linear Optimization Problems

Optimization Online (2020) manuscript 7351, 1–20 A Hyper-Matheuristic Methodology for Mixed Integer Linear Optimization Problems Mart´ın Gonzalez´ a,∗, Jose Juan Lopez-Esp´ ´ına, Juan Aparicioa and El-Ghazali Talbib aCenter of Operations Research, Miguel Hernandez´ University, Elche, Alicante, Spain bUniversity of Lille, France E-mail: [email protected] [Mart´ın Gonzalez]´ Abstract Metaheuristics and exact methods are the most common tools used to solve Mixed Integer Linear Optimization Problems (MILPs). These problems are usually NP-hard problems, being intractable to obtain optimal solutions in reasonable time for large scale instances of the problem. In this work, we propose a MILP-based decomposition in which the main problem is divided in two hierarchical subproblems according to the complexity of the latter. In this paper, we investigate a decomposition separating the discrete and continuous variables, treating each subproblem with different optimization methods. A metaheuristic focuses on the discrete variable search space while the exact method takes into account the continuous variables. Hence, the metaheuristic uses an indirect representation which encode an incomplete solution for the problem, while the exact method is applied to decode the solution and generate a complete solution. The experimental results show that the solutions obtained applying the proposed MILP-based decomposition are better than those obtained solving the problem globally using a metaheuristic method. Moreover, a novel Hyper-Matheuristic scheme is introduced in which the best algorithm selection is found for a set of cooperative metaheuristics and exact optimization algorithms. Keywords: Hyper-Matheuristic, Metaheuristics, Exact Methods, Mixed Integer Problems, MILP Decomposition. 1. Introduction Mixed Integer Linear Optimization (MILP) deals with mathematical optimization problems involving two families of variables: discrete and continuous variables. The objective function and the constraints are linear. This family of optimization problems appears for many real-life applications in various do- 5 mains. Indeed, many real problems can be formulated using MILP models, for example: packing, knap- sack, inventory, production planning, location, resource allocation, routing and scheduling problems, to name but a few (1). This large applicability has led to an increased interest in the development of efficient algorithms for solving this general and popular class of optimization problems. Mixed Integer Linear programming problems are generally NP-hard. In practice, two families of op- 10 timization algorithms are used to solve MILPs: exact algorithms and metaheuristics. The exact methods 1 (e.g. branch and bound, branch and cut, branch and price) are generally applicable for small or easy problems, but are known to be time-consuming for large or more complex ones. When instances become too large or difficult for exact methods, heuristics and particularly metaheuristics are often used. Two categories of metaheuristics can be considered: single solution algorithms (e.g. local search, tabu search) 15 and population based algorithms (e.g. evolutionary algorithms, swarm optimization) (2). However, in general, metaheuristics do not guarantee the optimality of the solutions found. Thus, the combination of metaheuristics and exact optimization algorithms can offer a more efficient and effective resolution method (3). A general classification of this hybridization is discussed in Talbi et al. (4), and some exam- ples can be found in the literature (Pradenas et al. (5), Li et al. (7)). 20 In this paper, a general Matheuristic methodology is first developed to find approximate solutions in a reasonable time for MILP problems. The Matheuristic concept is developed taking into account a MILP-based decomposition, where the main problem is decomposed into two hierarchical subproblems, since it is easier to solve them separately using different families of optimization algorithms. This decomposition is based on the nature of the decision variables: continuous and discrete. Then, a Hyper- 25 Matheuristic methodology is proposed. The hyper-heuristic concept has been applied in the context of hybrid heuristics to find the best combination of heuristics (N.Pillay et al. (8)). In this work we propose a generalization of the hyper-heuristic methodology for Matheuristics combining exact algorithms and metaheuristics, naming the Hyper-Matheuristic concept. The aim of the proposed Hyper-Matheuristic methodology is to find the best combination between 30 metaheuristics and exact methods for a given MILP. For this, some input instances of the problem are evaluated with several iterations of the algorithm, training the Hyper-Matheuristic and obtaining a good solution for any problem. Within the Hyper-Matheuristic, several parameters have been established to generate a large number of different metaheuristics. The functionalities of the generated metaheuristics, which depend on the values of these parameters, where metaheuristics like Evolution Algorithm (EA), 35 Scatter Search (SS), Tabu Search (TS) or Greedy Randomized Adaptive Search Procedure (GRASP) can be generated automatically. All of these parameters have their own upper and lower bound to be tuned. These bounds are studied in the experiments, and they represent inputs for the algorithm. The Matheuristic method has been designed by investigating the different cooperation between metaheuristics and exact methods, to find the best combination to solve a MILP problem. A general classi- 40 fication of existing approaches combining exact methods and metaheuristics for MILP optimization is presented in (10): • Collaborative combinations: self-contained optimization algorithms exchange information extracted from the search. The different algorithms are independent. There is no direct relationship to the internal workings of the algorithms. Exact method and heuristic algorithms can be executed sequentially, 45 intertwined or in parallel. • Integrative combinations: in this type of algorithms, hybridization addresses the functional compo- sition of a single optimization method. A given internal function of an optimization algorithm is replaced by another optimization algorithm. The proposed Matheuristic algorithm in this paper uses an Integrative Combination, in which the 50 metaheuristic provides information to the exact method, which solves the problem returning some new information to the metaheuristic. The main idea is to reduce the problem into much smaller subproblems which can be solved exactly by state-of-the-art mathematical programming algorithms. The variables and the constraints are partitioned into two sets, which decompose the main problem into two hierarchical subproblems: the metaheuristic fixes the decision variables in one set and the exact method optimizes 55 the problem over the other set. Moreover, a hyper-metaheuristic scheme has been included in the proposed methodology for an au- tonomous design of metaheuristics. Some design parameters determine the characteristics of each metaheuristic. Those parameters are framed into different search components: Initialize, Improvement, Se- lection, Combination and Stopping Criteria. In this work, the hyper-metaheuristic methodology (9) has 60 been generalized to Matheuristics, in which exact optimization is combined with a set of metaheuristics. The organisation of the paper is as follows. In section 2, the proposed decomposition of MILPs problems is presented. Section 3 details the Matheuristic strategy combining linear continuous programming and discrete metaheuristics. In section 4, we will focus on the Hyper-Matheuristic methodology in which an automatic design of optimization algorithms is carried out. Finally, section 5 gives some computa- 65 tional experiments on some MILPs problems. 2. MILP-based decomposition Let us consider the following linear problem (LP): n max fcx : Ax ≤ b; x ≥ 0; x 2 R g (1) where A is a m × n matrix, c a n-dimensional row vector, b a m-dimensional column vector, and x a n−dimensional column vector of continuous variables. If we add the restriction that certain variables must take integer values, we have a Mixed Integer Linear Program (MILP), that can be written as: max cx + hy Ax + Gy ≤ b n (2) x ≥ 0; x 2 R p y ≥ 0; y 2 Z where A is again a m × n matrix, G is m × p matrix, h is a p row-vector, and y is a p column-vector of integer variables. A MILP problem is defined as a problem where discrete (y) variables, which are restricted to integer 70 values, and continuous variables (x), which can assume any value on a given continuous interval, are combined with integrality constraints. The integrality constraints allow MILP models to capture the discrete nature of some decisions. For example, a binary variable can be used to decide whether or not any action needs to be taken. Solving large-scale and complex instances using MILP solvers is not efficient in terms of search 75 time. Indeed, large MILPs are often difficult to solve by exact methods, due to the complexity of the combinatorial nature of the discrete part of the problem. One way to solve large MILPs is by dividing them into smaller subproblems, so they can be solved individually. Problem decomposition techniques comprise an approach that is particularly aimed at solving very large and difficult problems. The basic idea is to solve such a large problem by solving a set of smaller problems and aggregating the results 80 obtained appropriately. 2.1. Popular decomposition

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