Matheuristics: using mathematics for heuristic design

Matheuristics are heuristic algorithms based on mathematical tools such as the ones provided by mathematical programming, that are structurally general enough to be applied to different problems with little adaptations to their abstract structure. The result can be metaheuristic hybrids having components derived from the mathematical model of the problems of interest, but the mathematical techniques themselves can define general heuristic solution frameworks. In this paper, we focus our attention on mathematical programming and its contributions to developing effective heuristics. We briefly describe the mathematical tools available and then some matheuristic approaches, reporting some representative examples from the literature. We also take the opportunity to provide some ideas for possible future development

Exact and Heuristic Hybrid Approaches for Scheduling and Clustering Problems

This thesis deals with the design of exact and heuristic algorithms for scheduling and clustering combinatorial optimization problems. All the works are linked by the fact that all the presented methods arebasically hybrid algorithms, that mix techniques used in the world of combinatorial optimization. The algorithms are all efficient in practice, but the one presented in Chapter 4, that has mostly theoretical interest. Chapter 2 presents practical solution algorithms based on an ILP model for an energy scheduling combinatorial problem that arises in a smart building context. Chapter 3 presents a new cutting stock problem and introduce a mathematical formulation and a heuristic solution approach based on a heuristic column generation scheme. Chapter 4 provides an exact exponential algorithm, whose importance is only theoretical so far, for a classical scheduling problem: the Single Machine Total Tardiness Problem. The relevant aspect is that the designed algorithm has the best worst case complexity for the problem, that has been studied for several decades. Furthermore, such result is based on a new technique, called Branch and Merge, that avoids the solution of several equivalent sub-problems in a branching algorithm that requires polynomial space. As a consequence, such technique embeds in a branching algorithm ideas coming from other traditional computer science techniques such as dynamic programming and memorization, but keeping the space requirement polynomial. Chapter 5 provides an exact approach based on semidefinite programming and a matheuristic approach based on a quadratic solver for a fractional clustering combinatorial optimization problem, called Max-Mean Dispersion Problem. The matheuristic approach has the peculiarity of using a non-linear MIP solver. The proposed exact approach uses a general semidefinite programming relaxation and it is likely to be extended to other combinatorial problems with a fractional formulation. Chapter 6 proposes practical solution methods for a real world clustering problem arising in a smart city context. The solution algorithm is based on the solution of a Set Cover model via a commercial ILP solver. As a conclusion, the main contribution of this thesis is given by several approaches of practical or theoretical interest, for two classes of important combinatorial problems: clustering and scheduling. All the practical methods presented in the thesis are validated by extensive computational experiments, that compare the proposed methods with the ones available in the state of the art

Models and Matheuristics for Large-Scale Combinatorial Optimization Problems

Combinatorial optimization deals with efficiently determining an optimal (or at least a good) decision among a finite set of alternatives. In business administration, such combinatorial optimization problems arise in, e.g., portfolio selection, project management, data analysis, and logistics. These optimization problems have in common that the set of alternatives becomes very large as the problem size increases, and therefore an exhaustive search of all alternatives may require a prohibitively long computation time. Moreover, due to their combinatorial nature no closed-form solutions to these problems exist. In practice, a common approach to tackle combinatorial optimization problems is to formulate them as mathematical models and to solve them using a mathematical programming solver (cf., e.g., Bixby et al. 1999, Achterberg et al. 2020). For small-scale problem instances, the mathematical models comprise a manageable number of variables and constraints such that mathematical programming solvers are able to devise optimal solutions within a reasonable computation time. For large-scale problem instances, the number of variables and constraints becomes very large which extends the computation time required to find an optimal solution considerably. Therefore, despite the continuously improving performance of mathematical programming solvers and computing hardware, the availability of mathematical models that are efficient in terms of the number of variables and constraints used is of crucial importance. Another frequently used approach to address combinatorial optimization problems are matheuristics. Matheuristics decompose the considered optimization problem into subproblems, which are then formulated as mathematical models and solved with the help of a mathematical programming solver. Matheuristics are particularly suitable for situations where it is required to find a good, but not necessarily an optimal solution within a short computation time, since the speed of the solution process can be controlled by choosing an appropriate size of the subproblems. This thesis consists of three papers on large-scale combinatorial optimization problems. We consider a portfolio optimization problem in finance, a scheduling problem in project management, and a clustering problem in data analysis. For these problems, we present novel mathematical models that require a relatively small number of variables and constraints, and we develop matheuristics that are based on novel problem-decomposition strategies. In extensive computational experiments, the proposed models and matheuristics performed favorably compared to state-of-the-art models and solution approaches from the literature. In the first paper, we consider the problem of determining a portfolio for an enhanced index-tracking fund. Enhanced index-tracking funds aim to replicate the returns of a particular financial stock-market index as closely as possible while outperforming that index by a small positive excess return. Additionally, we consider various real-life constraints that may be imposed by investors, stock exchanges, or investment guidelines. Since enhanced index-tracking funds are particularly attractive to investors if the index comprises a large number of stocks and thus is well diversified, it is of particular interest to tackle large-scale problem instances. For this problem, we present two matheuristics that consist of a novel construction matheuristic, and two different improvement matheuristics that are based on the concepts of local branching (cf. Fischetti and Lodi 2003) and iterated greedy heuristics (cf., e.g., Ruiz and Stützle 2007). Moreover, both matheuristics are based on a novel mathematical model for which we provide insights that allow to remove numerous redundant variables and constraints. We tested both matheuristics in a computational experiment on problem instances that are based on large stock-market indices with up to 9,427 constituents. It turns out that our matheuristics yield better portfolios than benchmark approaches in terms of out-of-sample risk-return characteristics. In the second paper, we consider the problem of scheduling a set of precedence-related project activities, each of which requiring some time and scarce resources during their execution. For each activity, alternative execution modes are given, which differ in the duration and the resource requirements of the activity. Sought is a start time and an execution mode for each activity, such that all precedence relationships are respected, the required amount of each resource does not exceed its prescribed capacity, and the project makespan is minimized. For this problem, we present two novel mathematical models, in which the number of variables remains constant when the range of the activities' durations and thus also the planning horizon is increased. Moreover, we enhance the performance of the proposed mathematical models by eliminating some symmetric solutions from the search space and by adding some redundant sequencing constraints for activities that cannot be processed in parallel. In a computational experiment based on instances consisting of activities with durations ranging from one up to 260 time units, the proposed models consistently outperformed all reference models from the literature. In the third paper, we consider the problem of grouping similar objects into clusters, where the similarity between a pair of objects is determined by a distance measure based on some features of the objects. In addition, we consider constraints that impose a maximum capacity for the clusters, since the size of the clusters is often restricted in practical clustering applications. Furthermore, practical clustering applications are often characterized by a very large number of objects to be clustered. For this reason, we present a matheuristic based on novel problem-decomposition strategies that are specifically designed for large-scale problem instances. The proposed matheuristic comprises two phases. In the first phase, we decompose the considered problem into a series of generalized assignment problems, and in the second phase, we decompose the problem into subproblems that comprise groups of clusters only. In a computational experiment, we tested the proposed matheuristic on problem instances with up to 498,378 objects. The proposed matheuristic consistently outperformed the state-of-the-art approach on medium- and large-scale instances, while matching the performance for small-scale instances. Although we considered three specific optimization problems in this thesis, the proposed models and matheuristics can be adapted to related optimization problems with only minor modifications. Examples for such related optimization problems are the UCITS-constrained index-tracking problem (cf, e.g., Strub and Trautmann 2019), which consists of determining the portfolio of an investment fund that must comply with regulatory restrictions imposed by the European Union, the multi-site resource-constrained project scheduling problem (cf., e.g., Laurent et al. 2017), which comprises the scheduling of a set of project activities that can be executed at alternative sites, or constrained clustering problems with must-link and cannot-link constraints (cf., e.g., González-Almagro et al. 2020)

Total Tardiness Minimization in a Single-Machine with Periodical Resource Constraints

In this paper we introduce a variant of the single machine considering resource restriction per period. The objective function to be minimized is the total tardiness.  We proposed an integer linear programming modeling based on a bin packing formulation. In view of the NP-hardness of the introduced variant, heuristic algorithms are required to find high-quality solutions within an admissible computation times. In this sense, we present a new hybrid matheuristic called Relax-and-Fix with Variable Fixing Search (RFVFS).  This innovative solution approach combines the relax-and-fix algorithm and a strategy for the fixation of decision variables based on the concept of the variable neighborhood search metaheuristic. As statistical indicators to evaluate the solution procedures under comparison, we employ the Average Relative Deviation Index (ARDI) and the Success Rate (SR). We performed extensive computational experimentation with a testbed composed by 450 proposed test problems. Considering the results for the number of jobs, the RFVFS returned ARDI and SR values of 35.6% and 41.3%, respectively. Our proposal outperformed the best solution approach available for a closely-related problem with statistical significance

A novel solution approach with ML-based pseudo-cuts for the Flight and Maintenance Planning problem

This paper deals with the long-term Military Flight and Maintenance Planning problem. In order to solve this problem efficiently, we propose a new solution approach based on a new Mixed Integer Program and the use of both valid cuts generated on the basis of initial conditions and learned cuts based on the prediction of certain characteristics of optimal or near-optimal solutions. These learned cuts are generated by training a Machine Learning model on the input data and results of 5000 instances. This approach helps to reduce the solution time with little losses in optimality and feasibility in comparison with alternative matheuristic methods. The obtained experimental results show the benefit of a new way of adding learned cuts to problems based on predicting specific characteristics of solutions.French Defense Procurement Agency of the French Ministry of Defense (DGA)

Q-Learnheuristics: towards data-driven balanced metaheuristics

One of the central issues that must be resolved for a metaheuristic optimization process to work well is the dilemma of the balance between exploration and exploitation. The metaheuristics (MH) that achieved this balance can be called balanced MH, where a Q-Learning (QL) integration framework was proposed for the selection of metaheuristic operators conducive to this balance, particularly the selection of binarization schemes when a continuous metaheuristic solves binary combinatorial problems. In this work the use of this framework is extended to other recent metaheuristics, demonstrating that the integration of QL in the selection of operators improves the exploration-exploitation balance. Specifically, the Whale Optimization Algorithm and the Sine-Cosine Algorithm are tested by solving the Set Covering Problem, showing statistical improvements in this balance and in the quality of the solutions

New variants of variable neighbourhood search for 0-1 mixed integer programming and clustering

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Many real-world optimisation problems are discrete in nature. Although recent rapid developments in computer technologies are steadily increasing the speed of computations, the size of an instance of a hard discrete optimisation problem solvable in prescribed time does not increase linearly with the computer speed. This calls for the development of new solution methodologies for solving larger instances in shorter time. Furthermore, large instances of discrete optimisation problems are normally impossible to solve to optimality within a reasonable computational time/space and can only be tackled with a heuristic approach. In this thesis the development of so called matheuristics, the heuristics which are based on the mathematical formulation of the problem, is studied and employed within the variable neighbourhood search framework. Some new variants of the variable neighbourhood searchmetaheuristic itself are suggested, which naturally emerge from exploiting the information from the mathematical programming formulation of the problem. However, those variants may also be applied to problems described by the combinatorial formulation. A unifying perspective on modern advances in local search-based metaheuristics, a so called hyper-reactive approach, is also proposed. Two NP-hard discrete optimisation problems are considered: 0-1 mixed integer programming and clustering with application to colour image quantisation. Several new heuristics for 0-1 mixed integer programming problem are developed, based on the principle of variable neighbourhood search. One set of proposed heuristics consists of improvement heuristics, which attempt to find high-quality near-optimal solutions starting from a given feasible solution. Another set consists of constructive heuristics, which attempt to find initial feasible solutions for 0-1 mixed integer programs. Finally, some variable neighbourhood search based clustering techniques are applied for solving the colour image quantisation problem. All new methods presented are compared to other algorithms recommended in literature and a comprehensive performance analysis is provided. Computational results show that the methods proposed either outperform the existing state-of-the-art methods for the problems observed, or provide comparable results. The theory and algorithms presented in this thesis indicate that hybridisation of the CPLEX MIP solver and the VNS metaheuristic can be very effective for solving large instances of the 0-1 mixed integer programming problem. More generally, the results presented in this thesis suggest that hybridisation of exact (commercial) integer programming solvers and some metaheuristic methods is of high interest and such combinations deserve further practical and theoretical investigation. Results also show that VNS can be successfully applied to solving a colour image quantisation problem.Support from the Mathematical Institute, Serbian Academy of Sciences and Arts, are acknowledged for this research