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

Mixed-Integer Convex Nonlinear Optimization with Gradient-Boosted Trees Embedded

Decision trees usefully represent sparse, high dimensional and noisy data. Having learned a function from this data, we may want to thereafter integrate the function into a larger decision-making problem, e.g., for picking the best chemical process catalyst. We study a large-scale, industrially-relevant mixed-integer nonlinear nonconvex optimization problem involving both gradient-boosted trees and penalty functions mitigating risk. This mixed-integer optimization problem with convex penalty terms broadly applies to optimizing pre-trained regression tree models. Decision makers may wish to optimize discrete models to repurpose legacy predictive models, or they may wish to optimize a discrete model that particularly well-represents a data set. We develop several heuristic methods to find feasible solutions, and an exact, branch-and-bound algorithm leveraging structural properties of the gradient-boosted trees and penalty functions. We computationally test our methods on concrete mixture design instance and a chemical catalysis industrial instance

Designing Energy-Efficient Heat Recovery Networks using Mixed-Integer Nonlinear Optimisation

Many industrial processes involve heating and cooling liquids: a quarter of the EU 2012 energy consumption came from industry and industry uses 73% of this energy on heating and cooling. We discuss mixed-integer nonlinear optimisation and its applications to energy efficiency. Our particular emphasis is on algorithms and solution techniques enabling optimisation for large-scale industrial networks. As a first application, optimising heat exchangers networks may increase efficiency in industrial plants. We develop deterministic global optimisation algorithms for a mixed-integer nonlinear optimisation model that simultaneously incorporates utility cost, equipment area, and hot/cold stream matches. We automatically recognise and exploit special mathematical structures common in heat recovery. We also computationally demonstrate the impact on the global optimisation solver ANTIGONE and benchmark large-scale test cases against heuristic approaches. As a second application, we discuss special structure in nonconvex quadratically-constrained optimisation problems, particularly through the lens of stream mixing and intermediate blending on process systems engineering networks. We take a parametric approach to uncovering topological structure and sparsity of the standard pooling problem in its p-formulation. We show that the sparse patterns of active topological structure are associated with a piecewise objective function. Finally, the presentation explains the conditions under which sparsity vanishes and where the combinatorial complexity emerges to cross over the P/NP boundary. We formally present the results obtained and their derivations for various specialised instances

Exact lexicographic scheduling and approximate rescheduling

In industrial resource allocation problems, an initial planning stage may solve a nominal problem instance and a subsequent recovery stage may intervene to repair inefficiencies and infeasibilities due to uncertainty, e.g.\ machine failures and job processing time variations. In this context, we investigate the minimum makespan scheduling problem, a.k.a.\ $P||C_{\max}$, under uncertainty. We propose a two-stage robust scheduling approach where first-stage decisions are computed with exact lexicographic scheduling and second-stage decisions are derived using approximate rescheduling. We explore recovery strategies accounting for planning decisions and constrained by limited permitted deviations from the original schedule. Our approach is substantiated analytically, with a price of robustness characterization parameterized by the degree of uncertainty, and numerically. This analysis is based on optimal substructure imposed by lexicographic optimality. Thus, lexicographic optimization enables more efficient rescheduling. Further, we revisit state-of-the-art exact lexicographic optimization methods and propose a lexicographic branch-and-bound algorithm whose performance is validated computationally

Approximation algorithms for process systems engineering

Designing and analyzing algorithms with provable performance guarantees enables efficient optimization problem solving in different application domains, e.g.\ communication networks, transportation, economics, and manufacturing. Despite the significant contributions of approximation algorithms in engineering, only limited and isolated works contribute from this perspective in process systems engineering. The current paper discusses three representative, NP-hard problems in process systems engineering: (i) pooling, (ii) process scheduling, and (iii) heat exchanger network synthesis. We survey relevant results and raise major open questions. Further, we present approximation algorithms applications which are relevant to process systems engineering: (i) better mathematical modeling, (ii) problem classification, (iii) designing solution methods, and (iv) dealing with uncertainty. This paper aims to motivate further research at the intersection of approximation algorithms and process systems engineering