An implementation of steepest-descent augmentation for linear programs

Borgwardt, Steffen; Viss, Charles

doi:10.1016/j.orl.2020.04.004

Cited by 10 publications

(5 citation statements)

References 16 publications

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“…The steepest-descent direction can be formulated by an LP; but without any restrictions on the input problem, this may not be simpler than the original one. However, it could be easier to solve in practice; Borgwardt and Viss [BV20] exhibits an implementation of a steepest-descent circuit augmentation algorithm with encouraging computational results.…”

Section: Circuit Augmentation Algorithmsmentioning

confidence: 99%

Circuit imbalance measures and linear programming

Ekbatani¹,

Natura²,

Végh³

2021

Preprint

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We study properties and applications of various circuit imbalance measures associated with linear spaces. These measures describe possible ratios between nonzero entries of support-minimal nonzero vectors of the space. The fractional circuit imbalance measure turns out to be a crucial parameter in the context of linear programming, and two integer variants can be used to describe integrality properties of associated polyhedra.We give an overview of the properties of these measures, and survey classical and recent applications, in particular, for linear programming algorithms with running time dependence on the constraint matrix only, and for circuit augmentation algorithms. We also present new bounds on the diameter and circuit diameter of polyhedra in terms of the fractional circuit imbalance measure.

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Section: Circuit Augmentation Algorithmsmentioning

confidence: 99%

Circuit imbalance measures and linear programming

Ekbatani¹,

Natura²,

Végh³

2021

Preprint

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“…To determine the solution of an ill-posed matrix, equation ( 4) is sufficient. Generally, the preferred method is to describe a point x 0 and then find the solution as the function approaches zero [38]. e next iteration is set to be at the next stage, x i+1 , that function reached x i .…”

Section: Methodsmentioning

confidence: 99%

Iterative Algorithms for Deblurring of Images in Case of Electrical Capacitance Tomography

Jaffri

Almogren

Ahmad

et al. 2021

Mathematical Problems in Engineering

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Electrical capacitance tomography (ECT) has been used to measure flow by applying gas-solid flow in coal gasification, pharmaceutical, and other industries. ECT is also used for creating images of physically confined objects. The data collected by the acquisition system to produce images undergo blurring because of ambient conditions and the electronic circuitry used. This research includes the principle of ECT techniques for deblurring images that were created during measurement. The data recorded by the said acquisition system ascends a large number of linear equations. This system of equations is sparse and ill-conditioned and hence is ill-posed in nature. A variety of reconstruction algorithms with many pros and cons are available to deal with ill-posed problems. Large-scale systems of linear equations resulting during image deblurring problems are solved using iterative regularization algorithms. The conjugate gradient algorithm for least-squares problems (CGLS), least-squares QR factorization (LSQR), and the modified residual norm steepest descent (MRNSD) algorithm are the famous variations of iterative algorithms. These algorithms exhibit a semiconvergence behavior; that is, the computed solution quality first improves and then reduces as the error norm decreases and later increases with each iteration. In this work, soft thresholding has been used for image deblurring problems to tackle the semiconvergence issues. Numerical test problems were executed to indicate the efficacy of the suggested algorithms with criteria for optimal stopping iterations. Results show marginal improvement compared to the traditional iterative algorithms (CGLS, LSQR, and MRNSD) for resolving semiconvergence behavior and image restoration.

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“…On the other hand, if not even the circuit diameter satisfies the Hirsch bound, the reason for this would be the maximality of steps in Definition 2(iii): if the step lengths are not required to be maximal, the so-called conformal sum property [1,28,35] guarantees the existence of a walk of at most f − d circuit steps between any pair of vertices (see also Section 2). Finally, circuit diameters are intimately related to the efficiency of circuit augmentation schemes to solve linear programs [6,8,12,16,17,18].…”

Section: Circuit Diameters and The Circuit Diameter Conjecturementioning

confidence: 99%

Short Circuit Walks on Hirsch Counterexamples

E.¹,

Borgwardt²,

Brugger³

2023

Preprint

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Circuit diameters of polyhedra are a fundamental tool for studying the complexity of circuit augmentation schemes for linear programming and for lower bounding the combinatorial diameters of polyhedra. The main open problem in this area is the circuit diameter conjecture, the analogue of the Hirsch conjecture in the circuit setting. A natural approach to this question is to see whether counterexamples to the Hirsch conjecture carry over. Previously, Stephen and Yusun showed that the Klee-Walkup counterexample to the unbounded Hirsch conjecture does not transfer to the circuit setting. Our main contribution is to show that the remaining counterexamples for monotone walks and polytopes also do not transfer. The key ingredients for our results are sign-compatible circuit walks on spindles, a class of polytopes underlying all known Hirsch counterexamples. To resolve the monotone case, we enumerate all linear programs with feasible region given by Todd's monotone Hirsch counterexample and find four new orientations that contradict the monotone Hirsch conjecture, while the remaining 7107 satisfy the bound.

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An implementation of steepest-descent augmentation for linear programs

Cited by 10 publications

References 16 publications

Circuit imbalance measures and linear programming

Circuit imbalance measures and linear programming

Iterative Algorithms for Deblurring of Images in Case of Electrical Capacitance Tomography

Short Circuit Walks on Hirsch Counterexamples

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