# Matheuristics for Robust Optimization: Application to Real-World Problems

Staff - Faculty of Informatics

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**You are cordially invited to attend the PhD Dissertation Defense of Nihat Engin TOKLU on Monday, November 10th 2014 at 13h30 in room SI-003 (Informatics building)**

**Abstract:**

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 demand 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 demands 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.

**Dissertation Committee:**

- Prof. Luca Maria Gambardella, IDSIA-Dalle Molle Institute for Artificial Intelligence, Switzerland (Research Advisor)
- Dr. Roberto Montemanni, IDSIA-Dalle Molle Institute for Artificial Intelligence, Switzerland (Research co-Advisor)
- Prof. Evanthia Papadopoulou, Università della Svizzera italiana, Switzerland (Internal Member)
- Prof. Laura Pozzi, Università della Svizzera italiana, Switzerland (Internal Member)
- Prof. Christian Artigues, LAAS-CNRS, France (External Member)
- Prof. Marino Widmer, University of Fribourg, Switzerland (External Member)