Matheuristics for robust optimization: application to real-world problems

In the field of optimization, the perspective that the problem data are subject to uncertainty is gaining more and more interest. The uncertainty in an optimization problem represents the measurement errors during the phase of collecting data, or unforeseen changes in the environment while implementing the optimal solution in practice. When the uncertainty is ignored, an optimal solution according to the mathematical model can turn out to be far from optimal, or even infeasible in reality. Robust optimization is an umbrella term for mathematical modelling methodologies focused on finding solutions that are reliable against the data perturbations caused by the uncertainty. Among the relatively more recent robust optimization methodologies, an important concept studied is the degree of conservativeness, which can be explained as the amount of targeted reliability against the uncertainty while looking for a solution. Because the reliability and solution cost usually end up being conflicting objectives, it is important for the decision maker to be able to configure the conservativeness degree, so that the desired balance between the cost and reliability can be obtained, and the most practical solution can be found for the problem at hand. The robust optimization methodologies are typically proposed within the framework of mathematical programming (i.e. linear programming, integer programming). Thanks to the nature of mathematical programming, these methodologies can find the exact optimum, according to the various solution evaluation perspectives they have. However, dependence on mathematical programming might also mean that such methodologies will require too much memory from the computer, and also too much execution time, when large-scale optimization problems are considered. A common strategy to avoid the big memory and execution time requirements of mathematical programming is to use metaheuristic optimization algorithms for solving large problem instances.In this research, we propose an approach for solving medium-to-large-sized robust optimization problem instances. The methodology we propose is a matheuristic (i.e. a hybridization of mathematical programming and metaheuristic). In the matheuristic approach we propose, the mathematical programming part handles the uncertainty, and the metaheuristic part handles the exploration of the solution space. Since the exploration of the solution space is entrusted onto the metaheuristic search, we can obtain practical near-optimal solutions while avoiding the big memory and time requirements that might be brought by pure mathematical programming methods. The mathematical programming part is used for making the metaheuristic favor the solutions which have more protections against the uncertainty. Another important characteristic of the methodology we propose is concurrency with information exchange: we concurrently execute multiple processes of the matheuristic algorithm, each process taking the uncertainty into account with a different degree of conservativeness. During the execution, these processes exchange their best solutions. So, if a process is stuck on a bad solution, it can realize that there is a better solution available thanks to the information exchange, and it can get unstuck. In the end, the solutions of these processes are collected into a solution pool. This solution pool provides the decision maker with alternative solutions with different costs and conservativeness degrees. Having a solution pool available at the end, the decision maker can make the most practical choice according to the problem at hand. In this thesis, we first discuss our studies in the field of robust optimization: a heuristic approach for solving a minimum power multicasting problem in wireless actuator networks under actuator distance uncertainty, and a linear programming approach for solving an aggregate blending problem in the construction industry, where the amounts of components found in aggregates are subject to uncertainty. These studies demonstrate the usage of mathematical programming for handling the uncertainty. We then discuss our studies in the field of matheuristics: a matheuristic approach for solving a large-scale energy management problem, and then a matheuristic approach for solving large instances of minimum power multicasting problem. In these studies, the usage of metaheuristics for handling the large problem instances is emphasized. In our study of solving minimum power multicasting problem, we also incorporate the mechanism of information exchange between different solvers. Later, we discuss the main matheuristic approach that we propose in this thesis. We first apply our matheuristic approach on a well-known combinatorial optimization problem: capacitated vehicle routing problem, by using an ant colony optimization as the metaheuristic part. Finally, we discuss the generality of the methodology that we propose: we suggest that it can be used as a general framework on various combinatorial optimization problems, by choosing the most appropriate metaheuristic algorithm according to the nature of the problem

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

Fixed cardinality stable sets

Given an undirected graph G=(V,E) and a positive integer k in {1, ..., |V|}, we initiate the combinatorial study of stable sets of cardinality exactly k in G. Our aim is to instigate the polyhedral investigation of the convex hull of fixed cardinality stable sets, inspired by the rich theory on the classical structure of stable sets. We introduce a large class of valid inequalities to the natural integer programming formulation of the problem. We also present simple combinatorial relaxations based on computing maximum weighted matchings, which yield dual bounds towards finding minimum-weight fixed cardinality stable sets, and particular cases which are solvable in polynomial time.publishedVersio

Stochastic Service Network Design for Intermodal Freight Transportation

In view of the accelerating climate change, greenhouse gas emissions from freight transportation must be significantly reduced over the next decades. Intermodal transportation can make a significant contribution here. During the transportation process, different modes of transportation are combined, enabling a modal shift to environmentally friendly alternatives such as rail and inland waterway transportation. However, at the same time, the organization of several modes is more complex compared to the unimodal case (where, for example, only trucks are employed). In particular, an efficient management of uncertainties, such as fluctuating transportation demand volumes or delays, is required to realize low costs and transportation times, thereby ensuring the attractiveness of intermodal transportation for a further modal shift. Stochastic service network design can explicitly consider such uncertainities in the planning in order to increase the performance of intermodal transportation. Decisions for the network design as well as for the mode choice are defined by mathematical optimization models, which originate from operations research and include relevant uncertainities by stochastic parameters. As central research gap, this dissertation addresses important operational constraints and decision variables of real-life intermodal networks, which have not been considered in these models so far and, in consequence, strongly limit their application in everyday operations. The resulting research contribution are two new variants of stochastic service network design models: The "stochastic service network design with integrated vehicle routing problem" integrates corresponding routing problems for road vehicles into the planning of intermodal networks. This new variant ensures a cost- and delay-minimal mode choice in the case of uncertain transportation times. The "stochastic service network design with short-term schedule modifications" deals with modifications of intermodal transportation schedules in order to adapt them to fluctuating demand as best as possible. For both new model variants, heuristic solution methods are presented which can efficiently solve even large network instances. Extensive case studies with real-world data demonstrate significant savings potentials compared to deterministic models as well as (simplified) stochastic models that already exist in literature

Multi-constructor CMSA for the maximum disjoint dominating sets problem

We propose the Multi-Constructor CMSA, a Construct, Merge, Solve and Adapt (CMSA) algorithm that employs multiple heuristic procedures, respectively solution constructors, for the Maximum Disjoint Dominating Sets Problem (MDDSP). At every iteration of the search procedure, the solution components built by the constructors are merged into a sub-instance, which is subsequently solved by an exact solver and then adapted to keep only beneficial solution components. In our CMSA the solution constructors are chosen at random according to their relative probabilities, which are adapted during the search, through a mechanism based on reinforcement learning. We test two variants of the new Multi-Constructor CMSA that employ, respectively, two and six solution constructors, on a new set of 3600 problem instances, encompassing random graphs, Watts–Strogatz networks and Barabási-Albert networks, generated through a Hammersley sampling procedure on the instance space. We compare our algorithm against six heuristics from the literature, as well as with the standard version of CMSA. Furthermore, we employ an integer linear programming (ILP) model that is able to achieve a good performance for small, sparse graphs. Overall, the experimental results show that all versions of CMSA outperform by a large margin the previous state of the art and that, among the variants of CMSA, the novel version that combines two constructors provides slightly better results than the other ones, more prominently on larger graphs