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Developments in linear and integer programming
In this review we describe recent developments in linear and integer (linear) programming. For over 50 years Operational Research practitioners have made use of linear optimisation models to aid decision making and over this period the size of problems that can be solved has increased dramatically, the time required to solve problems has decreased substantially and the flexibility of modelling and solving systems has increased steadily. Large models are no longer confined to large computers, and the flexibility of optimisation systems embedded in other decision support tools has made on-line decision making using linear programming a reality (and using integer programming a possibility). The review focuses on recent developments in algorithms, software and applications and investigates some connections between linear optimisation and other technologies
An investigation of pricing strategies within simplex
The PRICE step within the revised simplex method for the LP problems is considered in this report. Established strategies which have proven to be computationally efficient are first reviewed. A method based on the internal rate of return is then described. The implementation of this method and the results obtained by experimental investigation are discussed
On Augmentation Algorithms for Linear and Integer-Linear Programming: From Edmonds-Karp to Bland and Beyond
Motivated by Bland's linear-programming generalization of the renowned
Edmonds-Karp efficient refinement of the Ford-Fulkerson maximum-flow algorithm,
we discuss three closely-related natural augmentation rules for linear and
integer-linear optimization. In several nice situations, we show that
polynomially-many augmentation steps suffice to reach an optimum. In
particular, when using "discrete steepest-descent augmentations" (i.e.,
directions with the best ratio of cost improvement per unit 1-norm length), we
show that the number of augmentation steps is bounded by the number of elements
in the Graver basis of the problem matrix, giving the first ever strongly
polynomial-time algorithm for -fold integer-linear optimization. Our results
also improve on what is known for such algorithms in the context of linear
optimization (e.g., generalizing the bounds of Kitahara and Mizuno for the
number of steps in the simplex method) and are closely related to research on
the diameters of polytopes and the search for a strongly polynomial-time
simplex or augmentation algorithm
Learning to Pivot as a Smart Expert
Linear programming has been practically solved mainly by simplex and interior
point methods. Compared with the weakly polynomial complexity obtained by the
interior point methods, the existence of strongly polynomial bounds for the
length of the pivot path generated by the simplex methods remains a mystery. In
this paper, we propose two novel pivot experts that leverage both global and
local information of the linear programming instances for the primal simplex
method and show their excellent performance numerically. The experts can be
regarded as a benchmark to evaluate the performance of classical pivot rules,
although they are hard to directly implement. To tackle this challenge, we
employ a graph convolutional neural network model, trained via imitation
learning, to mimic the behavior of the pivot expert. Our pivot rule, learned
empirically, displays a significant advantage over conventional methods in
various linear programming problems, as demonstrated through a series of
rigorous experiments
Discrete Convex Functions on Graphs and Their Algorithmic Applications
The present article is an exposition of a theory of discrete convex functions
on certain graph structures, developed by the author in recent years. This
theory is a spin-off of discrete convex analysis by Murota, and is motivated by
combinatorial dualities in multiflow problems and the complexity classification
of facility location problems on graphs. We outline the theory and algorithmic
applications in combinatorial optimization problems
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