977 research outputs found
A Convex-Analysis Perspective on Disjunctive Cuts
An updated version of this paper has appeared in Math. Program., Ser. A 106, pp 567-586 (2006), DOI 10.1007/s10107-005-0670-8We treat the general problem of cutting planes with tools from convex analysis. We emphasize the case of disjunctive polyhedra and the generation of facets. We conclude with some considerations on the design of disjunctive cut generators
Convex Relaxations for Gas Expansion Planning
Expansion of natural gas networks is a critical process involving substantial
capital expenditures with complex decision-support requirements. Given the
non-convex nature of gas transmission constraints, global optimality and
infeasibility guarantees can only be offered by global optimisation approaches.
Unfortunately, state-of-the-art global optimisation solvers are unable to scale
up to real-world size instances. In this study, we present a convex
mixed-integer second-order cone relaxation for the gas expansion planning
problem under steady-state conditions. The underlying model offers tight lower
bounds with high computational efficiency. In addition, the optimal solution of
the relaxation can often be used to derive high-quality solutions to the
original problem, leading to provably tight optimality gaps and, in some cases,
global optimal soluutions. The convex relaxation is based on a few key ideas,
including the introduction of flux direction variables, exact McCormick
relaxations, on/off constraints, and integer cuts. Numerical experiments are
conducted on the traditional Belgian gas network, as well as other real larger
networks. The results demonstrate both the accuracy and computational speed of
the relaxation and its ability to produce high-quality solutions
When Lift-and-Project Cuts are Different
In this paper, we present a method to determine if a lift-and-project cut for
a mixed-integer linear program is irregular, in which case the cut is not
equivalent to any intersection cut from the bases of the linear relaxation.
This is an important question due to the intense research activity for the past
decade on cuts from multiple rows of simplex tableau as well as on
lift-and-project cuts from non-split disjunctions. While it is known since
Balas and Perregaard (2003) that lift-and-project cuts from split disjunctions
are always equivalent to intersection cuts and consequently to such multi-row
cuts, Balas and Kis (2016) have recently shown that there is a necessary and
sufficient condition in the case of arbitrary disjunctions: a lift-and-project
cut is regular if, and only if, it corresponds to a regular basic solution of
the Cut Generating Linear Program (CGLP). This paper has four contributions.
First, we state a result that simplifies the verification of regularity for
basic CGLP solutions from Balas and Kis (2016). Second, we provide a
mixed-integer formulation that checks whether there is a regular CGLP solution
for a given cut that is regular in a broader sense, which also encompasses
irregular cuts that are implied by the regular cut closure. Third, we describe
a numerical procedure based on such formulation that identifies irregular
lift-and-project cuts. Finally, we use this method to evaluate how often
lift-and-project cuts from simple -branch split disjunctions are irregular,
and thus not equivalent to multi-row cuts, on 74 instances of the MIPLIB
benchmarks.Comment: INFORMS Journal on Computing (to appear
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
Nonlinear Integer Programming
Research efforts of the past fifty years have led to a development of linear
integer programming as a mature discipline of mathematical optimization. Such a
level of maturity has not been reached when one considers nonlinear systems
subject to integrality requirements for the variables. This chapter is
dedicated to this topic.
The primary goal is a study of a simple version of general nonlinear integer
problems, where all constraints are still linear. Our focus is on the
computational complexity of the problem, which varies significantly with the
type of nonlinear objective function in combination with the underlying
combinatorial structure. Numerous boundary cases of complexity emerge, which
sometimes surprisingly lead even to polynomial time algorithms.
We also cover recent successful approaches for more general classes of
problems. Though no positive theoretical efficiency results are available, nor
are they likely to ever be available, these seem to be the currently most
successful and interesting approaches for solving practical problems.
It is our belief that the study of algorithms motivated by theoretical
considerations and those motivated by our desire to solve practical instances
should and do inform one another. So it is with this viewpoint that we present
the subject, and it is in this direction that we hope to spark further
research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G.
Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50
Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art
Surveys, Springer-Verlag, 2009, ISBN 354068274
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