Branching strategies for mixed-integer programs containing logical constraints and decomposable structure

Decision-making optimisation problems can include discrete selections, e.g. selecting a route, arranging non-overlapping items or designing a network of items. Branch-and-bound (B&B), a widely applied divide-and-conquer framework, often solves such problems by considering a continuous approximation, e.g. replacing discrete variable domains by a continuous superset. Such approximations weaken the logical relations, e.g. for discrete variables corresponding to Boolean variables. Branching in B&B reintroduces logical relations by dividing the search space. This thesis studies designing B&B branching strategies, i.e. how to divide the search space, for optimisation problems that contain both a logical and a continuous structure. We begin our study with a large-scale, industrially-relevant optimisation problem where the objective consists of machine-learnt gradient-boosted trees (GBTs) and convex penalty functions. GBT functions contain if-then queries which introduces a logical structure to this problem. We propose decomposition-based rigorous bounding strategies and an iterative heuristic that can be embedded into a B&B algorithm. We approach branching with two strategies: a pseudocost initialisation and strong branching that target the structure of GBT and convex penalty aspects of the optimisation objective, respectively. Computational tests show that our B&B approach outperforms state-of-the-art solvers in deriving rigorous bounds on optimality. Our second project investigates how satisfiability modulo theories (SMT) derived unsatisfiable cores may be utilised in a B&B context. Unsatisfiable cores are subsets of constraints that explain an infeasible result. We study two-dimensional bin packing (2BP) and develop a B&B algorithm that branches on SMT unsatisfiable cores. We use the unsatisfiable cores to derive cuts that break 2BP symmetries. Computational results show that our B&B algorithm solves 20% more instances when compared with commercial solvers on the tested instances. Finally, we study convex generalized disjunctive programming (GDP), a framework that supports logical variables and operators. Convex GDP includes disjunctions of mathematical constraints, which motivate branching by partitioning the disjunctions. We investigate separation by branching, i.e. eliminating solutions that prevent rigorous bound improvement, and propose a greedy algorithm for building the branches. We propose three scoring methods for selecting the next branching disjunction. We also analyse how to leverage infeasibility to expedite the B&B search. Computational results show that our scoring methods can reduce the number of explored B&B nodes by an order of magnitude when compared with scoring methods proposed in literature. Our infeasibility analysis further reduces the number of explored nodes.Open Acces

Tight bounds for some problems in computational geometry: the complete sub-logarithmic parallel time range

There are a number of fundamental problems in computational geometry for which work-optimal algorithms exist which have a parallel running time of $O(\log n)$ in the PRAM model. These include problems like two and three dimensional convex-hulls, trapezoidal decomposition, arrangement construction, dominance among others. Further improvements in running time to sub-logarithmic range were not considered likely because of their close relationship to sorting for which an $\Omega (\log n/\log\log n )$ is known to hold even with a polynomial number of processors. However, with recent progress in padded-sort algorithms, which circumvents the conventional lower-bounds, there arises a natural question about speeding up algorithms for the above-mentioned geometric problems (with appropriate modifications in the output specification). We present randomized parallel algorithms for some fundamental problems like convex-hulls and trapezoidal decomposition which execute in time $O( \log n/\log k)$ in an $nk$ ($k > 1$) processor CRCW PRAM. Our algorithms do not make any assumptions about the input distribution. Our work relies heavily on results on padded-sorting and some earlier results of Reif and Sen [28, 27]. We further prove a matching lower-bound for these problems in the bounded degree decision tree

Small Approximate Pareto Sets with Quality Bounds

We present and empirically characterize a general, parallel, heuristic algorithm for computing small ε-Pareto sets. The algorithm can be used as part of a decision support tool for settings in which computing points in objective space is computationally expensive. We use the graph clearing problem, a formalization of indirect organ exchange markets, as a prototypical example setting. We characterize the performance of the algorithm through ε-Pareto set size, ε value provided, and parallel speedup achieved. Our results show that the algorithm\u27s combination of parallel speedup and small ε-Pareto sets is sufficient to be appealing in settings requiring manual review (i.e., those that have a human in the loop) and real-time solutions

Experimental Study of Compressed Stack Algorithms in Limited Memory Environments

The compressed stack is a data structure designed by Barba et al. (Algorithmica 2015) that allows to reduce the amount of memory needed by a certain class of algorithms at the cost of increasing its runtime. In this paper we introduce the first implementation of this data structure and make its source code publicly available. Together with the implementation we analyse the performance of the compressed stack. In our synthetic experiments, considering different test scenarios and using data sizes ranging up to 2^{30} elements, we compare it with the classic (uncompressed) stack, both in terms of runtime and memory used. Our experiments show that the compressed stack needs significantly less memory than the usual stack (this difference is significant for inputs containing 2000 or more elements). Overall, with a proper choice of parameters, we can save a significant amount of space (from two to four orders of magnitude) with a small increase in the runtime (2.32 times slower on average than the classic stack). These results hold even in test scenarios specifically designed to be challenging for the compressed stack

Unifying a Geometric Framework of Evolutionary Algorithms and Elementary Landscapes Theory

Evolutionary algorithms (EAs) are randomised general-purpose strategies, inspired by natural evolution, often used for finding (near) optimal solutions to problems in combinatorial optimisation. Over the last 50 years, many theoretical approaches in evolutionary computation have been developed to analyse the performance of EAs, design EAs or measure problem difficulty via fitness landscape analysis. An open challenge is to formally explain why a general class of EAs perform better, or worse, than others on a class of combinatorial problems across representations. However, the lack of a general unified theory of EAs and fitness landscapes, across problems and representations, makes it harder to characterise pairs of general classes of EAs and combinatorial problems where good performance can be guaranteed provably. This thesis explores a unification between a geometric framework of EAs and elementary landscapes theory, not tied to a specific representation nor problem, with complementary strengths in the analysis of population-based EAs and combinatorial landscapes. This unification organises around three essential aspects: search space structure induced by crossovers, search behaviour of population-based EAs and structure of fitness landscapes. First, this thesis builds a crossover classification to systematically compare crossovers in the geometric framework and elementary landscapes theory, revealing a shared general subclass of crossovers: geometric recombination P-structures, which covers well-known crossovers. The crossover classification is then extended to a general framework for axiomatically analysing the population behaviour induced by crossover classes on associated EAs. This shows the shared general class of all EAs using geometric recombination P-structures, but no mutation, always do the same abstract form of convex evolutionary search. Finally, this thesis characterises a class of globally convex combinatorial landscapes shared by the geometric framework and elementary landscapes theory: abstract convex elementary landscapes. It is formally explained why geometric recombination P-structure EAs expectedly can outperform random search on abstract convex elementary landscapes related to low-order graph Laplacian eigenvalues. Altogether, this thesis paves a way towards a general unified theory of EAs and combinatorial fitness landscapes

The Convex Hull Problem in Practice : Improving the Running Time of the Double Description Method

The double description method is a simple but widely used algorithm for computation of extreme points in polyhedral sets. One key aspect of its implementation is the question of how to efficiently test extreme points for adjacency. In this dissertation, two significant contributions related to adjacency testing are presented. First, the currently used data structures are revisited and various optimizations are proposed. Empirical evidence is provided to demonstrate their competitiveness. Second, a new adjacency test is introduced. It is a refinement of the well known algebraic test featuring a technique for avoiding redundant computations. Its correctness is formally proven. Its superiority in multiple degenerate scenarios is demonstrated through experimental results. Parallel computation is one further aspect of the double description method covered in this work. A recently introduced divide-and-conquer technique is revisited and considerable practical limitations are demonstrated

Convex Hulls: Complexity and Applications (a Survey)

Computational geometry is, in brief, the study of algorithms for geometric problems. Classical study of geometry and geometric objects, however, is not well-suited to efficient algorithms techniques. Thus, for the given geometric problems, it becomes necessary to identify properties and concepts that lend themselves to efficient computation. The primary focus of this paper will be on one such geometric problems, the Convex Hull problem

A framework of face recognition with set of testing images

We propose a novel framework to solve the face recognition problem base on set of testing images. Our framework can handle the case that no pose overlap between training set and query set. The main techniques used in this framework are manifold alignment, face normalization and discriminant learning. Experiments on different databases show our system outperforms some state of the art methods