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.

Geometric and Combinatorial Aspects of NonEquilibrium Statistical Mechanics

Non-Equilibrium Statistical Mechanics is a broad subject. Grossly speaking, it deals with systems which have not yet relaxed to an equilibrium state, or else with systems which are in a steady non-equilibrium state, or with more general situations. They are characterized by external forcing and internal fluxes, resulting in a net production of entropy which quantifies dissipation and the extent by which, by the Second Law of Thermodynamics, time-reversal invariance is broken. In this thesis we discuss some of the mathematical structures involved with generic discrete-state-space non-equilibrium systems, that we depict with networks in all analogous to electrical networks. We define suitable observables and derive their linear regime relationships, we discuss a duality between external and internal observables that reverses the role of the system and of the environment, we show that network observables serve as constraints for a derivation of the minimum entropy production principle. We dwell on deep combinatorial aspects regarding linear response determinants, which are related to spanning tree polynomials in graph theory, and we give a geometrical interpretation of observables in terms of Wilson loops of a connection and gauge degrees of freedom. We specialize the formalism to continuous-time Markov chains, we give a physical interpretation for observables in terms of locally detailed balanced rates, we prove many variants of the fluctuation theorem, and show that a well-known expression for the entropy production due to Schnakenberg descends from considerations of gauge invariance, where the gauge symmetry is related to the freedom in the choice of a prior probability distribution. As an additional topic of geometrical flavor related to continuous-time Markov chains, we discuss the Fisher-Rao geometry of nonequilibrium decay modes, showing that the Fisher matrix contains information about many aspects of non-equilibrium behavior, including non-equilibrium phase transitions and superposition of modes. We establish a sort of statistical equivalence principle and discuss the behavior of the Fisher matrix under time-reversal. To conclude, we propose that geometry and combinatorics might greatly increase our understanding of nonequilibrium phenomena.

Multiobjective genetic algorithms applied to heat transfer problems

In the present work, the multi-objective optimization by genetic algorithms is investigated and applied to heat transfer problems. Firstly, the work aims to compare different reproduction processes employed by genetic algorithms and two new promising processes are suggested. Secondly, in this work two heat transfer problems are studied under the multi-objective point of view. Specifically, the two cases studied are the wavy fins and the corrugated wall channel. Both these cases have already been studied by a single objective optimizer. Therefore, this work aims to extend the previous works in a more comprehensive study.

Kernel Methods for Tree Structured Data

Machine learning comprises a series of techniques for automatic extraction of meaningful information from large collections of noisy data. In many real world applications, data is naturally represented in structured form. Since traditional methods in machine learning deal with vectorial information, they require an a priori form of preprocessing. Among all the learning techniques for dealing with structured data, kernel methods are recognized to have a strong theoretical background and to be effective approaches. They do not require an explicit vectorial representation of the data in terms of features, but rely on a measure of similarity between any pair of objects of a domain, the kernel function. Designing fast and good kernel functions is a challenging problem. In the case of tree structured data two issues become relevant: kernel for trees should not be sparse and should be fast to compute. The sparsity problem arises when, given a dataset and a kernel function, most structures of the dataset are completely dissimilar to one another. In those cases the classifier has too few information for making correct predictions on unseen data. In fact, it tends to produce a discriminating function behaving as the nearest neighbour rule. Sparsity is likely to arise for some standard tree kernel functions, such as the subtree and subset tree kernel, when they are applied to datasets with node labels belonging to a large domain. A second drawback of using tree kernels is the time complexity required both in learning and classification phases. Such a complexity can sometimes prevents the kernel application in scenarios involving large amount of data. This thesis proposes three contributions for resolving the above issues of kernel for trees. A first contribution aims at creating kernel functions which adapt to the statistical properties of the dataset, thus reducing its sparsity with respect to traditional tree kernel functions. Specifically, we propose to encode the input trees by an algorithm able to project the data onto a lower dimensional space with the property that similar structures are mapped similarly. By building kernel functions on the lower dimensional representation, we are able to perform inexact matchings between different inputs in the original space. A second contribution is the proposal of a novel kernel function based on the convolution kernel framework. Convolution kernel measures the similarity of two objects in terms of the similarities of their subparts. Most convolution kernels are based on counting the number of shared substructures, partially discarding information about their position in the original structure. The kernel function we propose is, instead, especially focused on this aspect. A third contribution is devoted at reducing the computational burden related to the calculation of a kernel function between a tree and a forest of trees, which is a typical operation in the classification phase and, for some algorithms, also in the learning phase. We propose a general methodology applicable to convolution kernels. Moreover, we show an instantiation of our technique when kernels such as the subtree and subset tree kernels are employed. In those cases, Direct Acyclic Graphs can be used to compactly represent shared substructures in different trees, thus reducing the computational burden and storage requirements.

Study of Italian isolates of Alternaria spp.: molecular and morphological characterization and pathogenesis on apple tree

It was decided to carry out a morphological and molecular characterization of the Italian Alternaria isolatescollected from apple , and evaluate their pathogenicity and subsequently combining the data collected. The strain collection (174 isolates) was constructed by collecting material (received from extension service personnel) between June and August of 2007, 2008, and 2009. A Preliminary bioassays were performed on detached plant materials (fruit and leaf wounded and unwounded), belonging to the Golden cultivar, with two different kind of inoculation (conidial suspension and conidial filtrate). Symptoms were monitored daily and a value of pathogenicity score (P.S.) was assigned on the basis of the diameter of the necrotic area that developed. On the basis of the bioassays, the number of isolates to undergo further molecular analysis was restricted to a representative set of single spore strains (44 strains). Morphological characteristics of the colony and sporulation pattern were determined according to previous systematic work on small-spored Alternaria spp. (Pryor and Michaelides, 2002 and Hong et al., 2006). Reference strains (Alternaria alternata, Alternaria tenuissima, Alternaria arborescens and four Japanese strains of Alternaria alternata mali pathotype), used in the study were kindly provided by Prof. Barry Pryor, who allows a open access to his own fungal collection. Molecular characterization was performed combining and comparing different data sets obtained from distinct molecular approach: 1) investigation of specific loci and 2) fingerprinting based on diverse randomly selected polymorphic sites of the genome. As concern the single locus analysis, it was chosen to sequence the EndoPG partial gene and three anonymous region (OPA1-3, OPA2- and OPa10-2). These markers has revealed a powerful tool in the latter systematic works on small-spored Alternaria spp. In fact, as reported in literature small-spored Alternaria taxonomy is complicated due to the inability to resolve evolutionary relationships among the taxa because of the lack of variability in the markers commonly used in fungi systematic. The three data set together provided the necessary variation to establish the phylogenetic relationships among the Italian isolates of Alternaria spp. On Italian strains these markers showed a variable number of informative sites (ranging from 7 for EndoPg to 85 for OPA1-3) and the parsimony analysis produced different tree topologies all concordant to define A. arborescens as a mophyletic clade. Fingerprinting analysis (nine ISSR primers and eight AFLP primers combination) led to the same result: a monophyleic A. arborescens clade and one clade containing both A. tenuissima and the A. alternata strains. This first attempt to characterize Italian Alternaria species recovered from apple produced concordant results with what was already described in a similar phylogenetic study on pistachio (Pryor and Michaelides, 2002), on walnut and hazelnut (Hong et al., 2006), apple (Kang et al., 2002) and citurus (Peever et al., 2004). Together with these studies, this research demonstrates that the three morphological groups are widely distributed and occupy similar ecological niches. Furthermore, this research suggest that these Alternaria species exhibit a similar infection pattern despite the taxonomic and pathogenic differences. The molecular characterization of the pathogens is a fundamental step to understanding the disease that is spreading in the apple orchards of the north Italy. At the beginning the causal agent was considered as Alteraria alternata (Marshall and Bertagnoll, 2006). Their preliminary studies purposed a pathogenic system related to the synthesis of toxins. Experimental data of our bioassays suggest an analogous hypothesis, considering that symptoms could be induced after inoculating plant material with solely the filtrate from pathogenic strains. Moreover, positive PCR reactions using AM-toxin gene specific primers, designed for identification of apple infecting Alternaria pathovar, led to a hypothesis that a host specific toxin (toxins) were involved. It remains an intriguing challenge to discover or not if the agent of the “Italian disease” is the same of the one previously typified as Alternaria mali, casual agent of the apple blotch disease.

Starch distribution in pear tree organs in relation to training systems, rootstocks and fruit quality

Starch is the main form in which plants store carbohydrates reserves, both in terms of amounts and distribution among different plant species. Carbohydrates are direct products of photosynthetic activity, and it is well know that yield efficiency and production are directly correlated to the amount of carbohydrates synthesized and how these are distributed among vegetative and reproductive organs. Nowadays, in pear trees, due to the modernization of orchards, through the introduction of new rootstocks and the development of new training systems, the understanding and the development of new approaches regarding the distribution and storage of carbohydrates, are required. The objective of this research work was to study the behavior of carbohydrate reserves, mainly starch, in different pear tree organs and tissues: i.e., fruits, leaves, woody organs, roots and flower buds, at different physiological stages during the season. Starch in fruit is accumulated at early stages, and reached a maximum concentration during the middle phase of fruit development; after that, its degradation begins with a rise in soluble carbohydrates. Moreover, relationships between fruit starch degradation and different fruit traits, soluble sugars and organic acids were established. In woody organs and roots, an interconversion between starch and soluble carbohydrates was observed during the dormancy period that confirms its main function in supporting the growth and development of new tissues during the following spring. Factors as training systems, rootstocks, types of bearing wood, and their position on the canopy, influenced the concentrations of starch and soluble carbohydrates at different sampling dates. Also, environmental conditions and cultural practices must be considered to better explain these results. Thus, a deeper understanding of the dynamics of carbohydrates reserves within the plant could provide relevant information to improve several management practices to increase crop yield efficiency.