Curvature as a Complexity Bound in Interior-Point Methods

In this thesis, we investigate the curvature of interior paths as a component of complexity bounds for interior-point methods (IPMs) in Linear Optimization (LO). LO is an optimization paradigm, where both the objective and the constraints of the model are represented by linear relationships of the decision variables. Among the class ofalgorithms for LO, our focus is on IPMs which have been an extremely active research area in the last three decades. IPMs in optimization are unique in the sense that they enjoy the best iteration-complexity bounds which are polynomial in the size of the LO problem. The main objects of our interest in this thesis are two distinct curvature measures in the literature, the geometric and the Sonnevend curvature of the central path. The central path is a fundamental tool for the design and the study of IPMs and we will see both that the geometric and Sonnevend\u27s curvature of the central path are proven to be useful in approaching the iteration-complexity questions in IPMs. While the Sonnevend curvature of the central path has been rigorously shown to determine the iteration-complexity of certain IPMs, the role of the geometric curvature in the literature to explain the iteration-complexity is still not well-understood. The novel approach in this thesis is to explore whether or not there is a relationship between these two curvature concepts aiming to bring the geometric curvature into the picture. The structure of the thesis is as follows. In the first three chapters, we present the basic knowledge of path-following IPMs algorithms and review the literature on Sonnevend\u27s curvature and the geometric curvature of the central path. In Chapter 4, we analyze a certain class ofLO problems and show that the geometric and Sonnevend\u27s curvature for these problems display analogous behavior. In particular, the main result of this chapter states that in order to establish an upper bound for the total Sonnevend curvature of the central path, it is sufficient to consider only the case when the number of inequalities is twice as big as the dimension. In Chapter 5, we study the redundant Klee-Minty (KM) construction and prove that the classical polynomial upper bound for IPMs is essentially tight for the Mizuno-Todd-Ye predictor-corrector algorithm. This chapter also provides a negative answer to an open problem about the Sonnevend curvature posed by Stoer et al. in 1993. Chapter 6 investigates a condition number relevant to the Sonnevend curvature and yields a strongly polynomial bound for that curvature in some special cases. Chapter 7 deals with another self-concordant barrier function, the volumetric barrier, and the volumetric path. That chapter investigates some of the basic properties of the volumetric path and shows that certain fundamental properties of the central path failto hold for the volumetric path. Chapter 8 concludes the thesis by providing some final remarks and pointing out future research directions

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

A simpler and tighter redundant Klee-Minty construction

By introducing redundant Klee-Minty examples, we have previously shown that the central path can be bent along the edges of the Klee-Minty cubes, thus having 2 n − 2 sharp turns in dimension n. In those constructions the redundant hyperplanes were placed parallel with the facets active at the optimal solution. In this paper we present a simpler and more powerful construction, where the redundant constraints are parallel with the coordinate-planes. An important consequence of this new construction is that one of the sets of redundant hyperplanes is touching the feasible region, and N, the total number of the redundant hyperplanes is reduced by a factor of n 2, further tightening the gap between iteration-complexity upper and lower bounds

A Comprehensive Analysis of Lift-and-Project Methods for Combinatorial Optimization

In both mathematical research and real-life, we often encounter problems that can be framed as finding the best solution among a collection of discrete choices. Many of these problems, on which an exhaustive search in the solution space is impractical or even infeasible, belong to the area of combinatorial optimization, a lively branch of discrete mathematics that has seen tremendous development over the last half century. It uses tools in areas such as combinatorics, mathematical modelling and graph theory to tackle these problems, and has deep connections with related subjects such as theoretical computer science, operations research, and industrial engineering. While elegant and efficient algorithms have been found for many problems in combinatorial optimization, the area is also filled with difficult problems that are unlikely to be solvable in polynomial time (assuming the widely believed conjecture $\mathcal{P} \neq \mathcal{NP}$). A common approach of tackling these hard problems is to formulate them as integer programs (which themselves are hard to solve), and then approximate their feasible regions using sets that are easier to describe and optimize over. Two of the most prominent mathematical models that are used to obtain these approximations are linear programs (LPs) and semidefinite programs (SDPs). The study of these relaxations started to gain popularity during the 1960's for LPs and mid-1990's for SDPs, and in many cases have led to the invention of strong approximation algorithms for the underlying hard problems. On the other hand, sometimes the analysis of these relaxations can lead to the conclusion that a certain problem cannot be well approximated by a wide class of LPs or SDPs. These negative results can also be valuable, as they might provide insights into what makes the problem difficult, which can guide our future attempts of attacking the problem. One mathematical framework that generates strong LP and SDP relaxations for integer programs is lift-and-project methods. Among many attractive features, an important advantage of this approach is that tighter relaxations can often be obtained without sacrificing polynomial-time solvability. Also, these procedures are able to generate relaxations systematically, without relying on problem-specific observations. Thus, they can be applied to improve any given relaxation. In the past two decades, lift-and-project methods have garnered a lot of research attention. Many operators under this approach have been proposed, most notably those by Sherali and Adams; Lov{\'a}sz and Schrijver; Balas, Ceria and Cornu{\'e}jols; Lasserre; and Bienstock and Zuckerberg. These operators vary greatly both in strength and complexity, and their performances and limitations on many optimization problems have been extensively studied, with the exception of the Bienstock--Zuckerberg operator (and to a lesser degree, the Lasserre operator) in terms of limitations. In this thesis, we aim to provide a comprehensive analysis of the existing lift and project operators, as well as many new variants of these operators that we propose in our work. Our new operators fill the spectrum of lift-and-project operators in a way which makes all of them more transparent, easier to relate to each other, and easier to analyze. We provide new techniques to analyze the worst-case performances as well as relative strengths of these operators in a unified way. In particular, using the new techniques and a recent result of Mathieu and Sinclair, we prove that the polyhedral Bienstock--Zuckerberg operator requires at least $\sqrt{2n}- \frac{3}{2}$ iterations to compute the matching polytope of the $(2n+1)$-clique. We further prove that the operator requires approximately $\frac{n}{2}$ iterations to reach the stable set polytope of the $n$-clique, if we start with the fractional stable set polytope. Moreover, we obtained an example in which the Bienstock--Zuckerberg operator with positive semidefiniteness requires $\Omega(n^{1/4})$ iterations to compute the integer hull of a set contained in $[0,1]^n$. These examples provide the first known instances where the Bienstock--Zuckerberg operators require more than a constant number of iterations to return the integer hull of a given relaxation. In addition to relating the performances of various lift-and-project methods and providing results for specific operators and problems, we provide some general techniques that can be useful in producing and verifying certificates for lift-and-project relaxations. These tools can significantly simply the task of obtaining hardness results for relaxations that have certain desirable properties. Finally, we characterize some sets on which one of the strongest variants of the Sherali--Adams operator with positive semidefinite strengthenings does not perform better than Lov\'{a}sz and Schrijver's weakest polyhedral operator, providing examples where even imposing a very strong positive semidefiniteness constraint does not generate any additional cuts. We then prove that some of the worst-case instances for many known lift-and-project operators are also bad instances for this significantly strengthened version of the Sherali--Adams operator, as well as the Lasserre operator. We also discuss how the techniques we presented in our analysis can be applied to obtain the integrality gaps of convex relaxations

Implicit Loss of Surjectivity and Facial Reduction: Theory and Applications

Facial reduction, pioneered by Borwein and Wolkowicz, is a preprocessing method that is commonly used to obtain strict feasibility in the reformulated, reduced constraint system. The importance of strict feasibility is often addressed in the context of the convergence results for interior point methods. Beyond the theoretical properties that the facial reduction conveys, we show that facial reduction, not only limited to interior point methods, leads to strong numerical performances in different classes of algorithms. In this thesis we study various consequences and the broad applicability of facial reduction. The thesis is organized in two parts. In the first part, we show the instabilities accompanied by the absence of strict feasibility through the lens of facially reduced systems. In particular, we exploit the implicit redundancies, revealed by each nontrivial facial reduction step, resulting in the implicit loss of surjectivity. This leads to the two-step facial reduction and two novel related notions of singularity. For the area of semidefinite programming, we use these singularities to strengthen a known bound on the solution rank, the Barvinok-Pataki bound. For the area of linear programming, we reveal degeneracies caused by the implicit redundancies. Furthermore, we propose a preprocessing tool that uses the simplex method. In the second part of this thesis, we continue with the semidefinite programs that do not have strictly feasible points. We focus on the doubly-nonnegative relaxation of the binary quadratic program and a semidefinite program with a nonlinear objective function. We closely work with two classes of algorithms, the splitting method and the Gauss-Newton interior point method. We elaborate on the advantages in building models from facial reduction. Moreover, we develop algorithms for real-world problems including the quadratic assignment problem, the protein side-chain positioning problem, and the key rate computation for quantum key distribution. Facial reduction continues to play an important role for providing robust reformulated models in both the theoretical and the practical aspects, resulting in successful numerical performances

Strategic Biopharmaceutical Production Planning for Batch and Perfusion Processes

Capacity planning for multiple biopharmaceutical therapeutics across a large network of manufacturing facilities, including contract manufacturers, is a complex task. Production planning is further complicated by portfolios of products requiring different modes of manufacture: batch and continuous. Capacity planning decisions each have their own costs and risks which must be carefully considered when determining manufacturing schedules. Hence, this work describes a framework which can assimilate various input data and provide intelligent capacity planning solutions. First of all, a mathematical model was created with the objective of minimising total cost. Various challenges surrounding the biomanufacturing of both perfusion and fed-batch products were solved. Sequence-dependent changeover times and full decoupling between upstream and downstream production suites were incorporated into the mixed integer linear program, which was used on an industrial case study to determine optimal manufacturing schedules and capital expenditure requirements. The effect of varying demands and fermentation titres was investigated via scenario analysis. To improve computational performance of the model, a rolling time horizon was introduced, and was shown to not only improve performance but also solution quality. The performance of the model was then improved via appropriate reformulations which consider the state task network (STN) topology of the problem domain. Two industrial case studies were used to demonstrate the merits of using the new formulation, and results showed that the STN improved performance in all test cases, and even performed better than the rolling time horizon approach from the previous model in one test case. Various strategic options regarding capacity expansion were analysed, in addition to an illustration of how the framework could be used to de-bottleneck existing capacity issues. Finally, a multi-objective component is added to the model, enabling the consideration of strategic multi-criteria decision making. The ε-constraint method was shown to be the superior multi-objective technique, and was used to demonstrate how uncertain input parameters could affect the different objectives and capacity plans in question

Thioesterases in fruit and the generation of high impact aroma chemicals

Volatile organosulphur compounds (VOSCs) are key ingredients in the aroma of tropical fruit where they are active as both free thiols and the respective thioesters. Present as trace high-impact flavourings, VOSCs are however problematic to extract and have therefore become targets for bioproduction. This work focuses on the endogenous thioesterases in tropical fruit believed to catalyze the liberation of thiol VOSCs from thioester precursors. Using a simple and sensitive colourimetric assay, a thioesterase activity toward VOSCs was identified in purple passion fruit (Passiflora edulis Sims). The enzyme was identified as a cell wall-bound protein in the mesocarp of the fruit. Following extraction with salt solutions, the thioesterase was purified 150-fold and shown to be associated with a 43 kDa polypeptide. Affinity labelling with a biotinylated fluorophosphonate suicide probe showed the enzyme to be a serine hydrolase, with MS-MS sequencing of tryptic digests identifying it as a pectin acetylesterase (PAE). Putative thioesterase PAEs were subsequently cloned from passion fruit and Arabidopsis thaliana. The observation that an esterase involved in cell wall modification had a secondary role in hydrolysing esterified VOSCs led to the consideration of further fruit species as a source of the enzyme. Orange (Citrus sinensis) was particularly abundant in thioesterase activity. The enzyme was purified 85-fold and identified as a homologous 43 kDa basic (pi: 9) PAE. The enzyme was stable (t(_1/2): 7 days 22 hours) and demonstrated a high turnover toward VOSCs (k(_cat): 7.85 sec(^-1)). Freeze-dried orange peel was found to retain activity (>90% activity, 3 months 4 C) and demonstrated comparable productivities to those of immobilized microbial enzymes. Here we have initiated a programme for developing processes for the bioproduction of VOSCs, in which the potential of plant glycohydrolases has been demonstrated