Structural analysis of combinatorial optimization problem characteristics and their resolution using hybrid approaches

Many combinatorial problems coming from the real world may not have a clear and well defined structure, typically being dirtied by side constraints, or being composed of two or more sub-problems, usually not disjoint. Such problems are not suitable to be solved with pure approaches based on a single programming paradigm, because a paradigm that can effectively face a problem characteristic may behave inefficiently when facing other characteristics. In these cases, modelling the problem using different programming techniques, trying to ”take the best” from each technique, can produce solvers that largely dominate pure approaches. We demonstrate the effectiveness of hybridization and we discuss about different hybridization techniques by analyzing two classes of problems with particular structures, exploiting Constraint Programming and Integer Linear Programming solving tools and Algorithm Portfolios and Logic Based Benders Decomposition as integration and hybridization frameworks.

Combinatorial and Robust Optimisation Models and Algorithms for Railway Applications

This thesis deals with an investigation of combinatorial and robust optimisation models to solve railway problems. Railway applications represent a challenging area for operations research. In fact, most problems in this context can be modelled as combinatorial optimisation problems, in which the number of feasible solutions is finite. Yet, despite the astonishing success in the field of combinatorial optimisation, the current state of algorithmic research faces severe difficulties with highly-complex and data-intensive applications such as those dealing with optimisation issues in large-scale transportation networks. One of the main issues concerns imperfect information. The idea of Robust Optimisation, as a way to represent and handle mathematically systems with not precisely known data, dates back to 1970s. Unfortunately, none of those techniques proved to be successfully applicable in one of the most complex and largest in scale (transportation) settings: that of railway systems. Railway optimisation deals with planning and scheduling problems over several time horizons. Disturbances are inevitable and severely affect the planning process. Here we focus on two compelling aspects of planning: robust planning and online (real-time) planning.

Enhanced Mixed Integer Programming Techniques and Routing Problems

Mixed integer programming is up today one of the most widely used techniques for dealing with hard optimization problems. On the one side, many practical optimization problems arising from real-world applications (such as, e.g., scheduling, project planning, transportation, telecommunications, economics and finance, timetabling, etc) can be easily and effectively formulated as Mixed Integer linear Programs (MIPs). On the other hand, 50 and more years of intensive research has dramatically improved on the capability of the current generation of MIP solvers to tackle hard problems in practice. However, many questions are still open and not fully understood, and the mixed integer programming community is still more than active in trying to answer some of these questions. As a consequence, a huge number of papers are continuously developed and new intriguing questions arise every year. When dealing with MIPs, we have to distinguish between two different scenarios. The first one happens when we are asked to handle a general MIP and we cannot assume any special structure for the given problem. In this case, a Linear Programming (LP) relaxation and some integrality requirements are all we have for tackling the problem, and we are ``forced" to use some general purpose techniques. The second one happens when mixed integer programming is used to address a somehow structured problem. In this context, polyhedral analysis and other theoretical and practical considerations are typically exploited to devise some special purpose techniques. This thesis tries to give some insights in both the above mentioned situations. The first part of the work is focused on general purpose cutting planes, which are probably the key ingredient behind the success of the current generation of MIP solvers. Chapter 1 presents a quick overview of the main ingredients of a branch-and-cut algorithm, while Chapter 2 recalls some results from the literature in the context of disjunctive cuts and their connections with Gomory mixed integer cuts. Chapter 3 presents a theoretical and computational investigation of disjunctive cuts. In particular, we analyze the connections between different normalization conditions (i.e., conditions to truncate the cone associated with disjunctive cutting planes) and other crucial aspects as cut rank, cut density and cut strength. We give a theoretical characterization of weak rays of the disjunctive cone that lead to dominated cuts, and propose a practical method to possibly strengthen those cuts arising from such weak extremal solution. Further, we point out how redundant constraints can affect the quality of the generated disjunctive cuts, and discuss possible ways to cope with them. Finally, Chapter 4 presents some preliminary ideas in the context of multiple-row cuts. Very recently, a series of papers have brought the attention to the possibility of generating cuts using more than one row of the simplex tableau at a time. Several interesting theoretical results have been presented in this direction, often revisiting and recalling other important results discovered more than 40 years ago. However, is not clear at all how these results can be exploited in practice. As stated, the chapter is a still work-in-progress and simply presents a possible way for generating two-row cuts from the simplex tableau arising from lattice-free triangles and some preliminary computational results. The second part of the thesis is instead focused on the heuristic and exact exploitation of integer programming techniques for hard combinatorial optimization problems in the context of routing applications. Chapters 5 and 6 present an integer linear programming local search algorithm for Vehicle Routing Problems (VRPs). The overall procedure follows a general destroy-and-repair paradigm (i.e., the current solution is first randomly destroyed and then repaired in the attempt of finding a new improved solution) where a class of exponential neighborhoods are iteratively explored by heuristically solving an integer programming formulation through a general purpose MIP solver. Chapters 7 and 8 deal with exact branch-and-cut methods. Chapter 7 presents an extended formulation for the Traveling Salesman Problem with Time Windows (TSPTW), a generalization of the well known TSP where each node must be visited within a given time window. The polyhedral approaches proposed for this problem in the literature typically follow the one which has been proven to be extremely effective in the classical TSP context. Here we present an overall (quite) general idea which is based on a relaxed discretization of time windows. Such an idea leads to a stronger formulation and to stronger valid inequalities which are then separated within the classical branch-and-cut framework. Finally, Chapter 8 addresses the branch-and-cut in the context of Generalized Minimum Spanning Tree Problems (GMSTPs) (i.e., a class of NP-hard generalizations of the classical minimum spanning tree problem). In this chapter, we show how some basic ideas (and, in particular, the usage of general purpose cutting planes) can be useful to improve on branch-and-cut methods proposed in the literature.

An evolutionary approach to the design of experiments for combinatorial optimization with an application to enzyme engineering

In a large number of problems the high dimensionality of the search space, the vast number of variables and the economical constrains limit the ability of classical techniques to reach the optimum of a function, known or unknown. In this thesis we investigate the possibility to combine approaches from advanced statistics and optimization algorithms in such a way to better explore the combinatorial search space and to increase the performance of the approaches. To this purpose we propose two methods: (i) Model Based Ant Colony Design and (ii) Naïve Bayes Ant Colony Optimization. We test the performance of the two proposed solutions on a simulation study and we apply the novel techniques on an appplication in the field of Enzyme Engineering and Design.

Many-core Algorithms for Combinatorial Optimization

Combinatorial Optimization is becoming ever more crucial, in these days. From natural sciences to economics, passing through urban centers administration and personnel management, methodologies and algorithms with a strong theoretical background and a consolidated real-word effectiveness is more and more requested, in order to find, quickly, good solutions to complex strategical problems. Resource optimization is, nowadays, a fundamental ground for building the basements of successful projects. From the theoretical point of view, Combinatorial Optimization rests on stable and strong foundations, that allow researchers to face ever more challenging problems. However, from the application point of view, it seems that the rate of theoretical developments cannot cope with that enjoyed by modern hardware technologies, especially with reference to the one of processors industry. In this work we propose new parallel algorithms, designed for exploiting the new parallel architectures available on the market. We found that, exposing the inherent parallelism of some resolution techniques (like Dynamic Programming), the computational benefits are remarkable, lowering the execution times by more than an order of magnitude, and allowing to address instances with dimensions not possible before. We approached four Combinatorial Optimization’s notable problems: Packing Problem, Vehicle Routing Problem, Single Source Shortest Path Problem and a Network Design problem. For each of these problems we propose a collection of effective parallel solution algorithms, either for solving the full problem (Guillotine Cuts and SSSPP) or for enhancing a fundamental part of the solution method (VRP and ND). We endorse our claim by presenting computational results for all problems, either on standard benchmarks from the literature or, when possible, on data from real-world applications, where speed-ups of one order of magnitude are usually attained, not uncommonly scaling up to 40 X factors.