The Relaxation Method for Solving Systems of Linear Inequalities

Goffin, Jean-Louis

doi:10.1287/moor.5.3.388

Cited by 130 publications

(102 citation statements)

References 22 publications

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“…e T x = 1 in the von Neumann algorithm. An important quantity in the convergence analysis of the algorithms we will describe is the condition measure introduced by Goffin [10]:…”

Section: Denote By Supp(l+) ⊆ [N] the Maximum Support Of A Point In L+mentioning

confidence: 99%

Rescaled Coordinate Descent Methods for Linear Programming

Dadush

Végh

Zambelli

2016

Integer Programming and Combinatorial Optimization

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Abstract. We propose two simple polynomial-time algorithms to find a positive solution to Ax = 0. Both algorithms iterate between coordinate descent steps similar to von Neumann's algorithm, and rescaling steps. In both cases, either the updating step leads to a substantial decrease in the norm, or we can infer that the condition measure is small and rescale in order to improve the geometry. We also show how the algorithms can be extended to find a solution of maximum support for the system Ax = 0, x ≥ 0.

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“…e T x = 1 in the von Neumann algorithm. An important quantity in the convergence analysis of the algorithms we will describe is the condition measure introduced by Goffin [10]:…”

Section: Denote By Supp(l+) ⊆ [N] the Maximum Support Of A Point In L+mentioning

confidence: 99%

Rescaled Coordinate Descent Methods for Linear Programming

Dadush

Végh

Zambelli

2016

Integer Programming and Combinatorial Optimization

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“…In the 1980's the relaxation method was revisited with interest because of its similarities to the ellipsoid method (see [AH05,Bet04,Gof80,Tel82] and references therein). One can show that the relaxation method is finite in all cases when using rational data, in that it can be modified to detect infeasible systems.…”

Section: Introductionmentioning

confidence: 99%

“…In some special cases the method gives a polynomial time algorithm (e.g. for totally unimodular matrices [MTA81]), but there are also examples of exponential running times (see [Gof82,Tel82]). In late 2010, Chubanov [Chu12], announced a modification of the traditional relaxation style method, which gives a strongly polynomial-time algorithm in some situations [BDJ14,VZ14].…”

Section: Introductionmentioning

confidence: 99%

A Sampling Kaczmarz--Motzkin Algorithm for Linear Feasibility

Loera¹,

Haddock²,

Needell³

2017

SIAM J. Sci. Comput.

108

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“…τ F is a variation on the notion of the "inner measure" of Goffin [11] when the norm is Euclidean, and has also been used in similar format in [9,7].…”

Section: Measuring the Behavior Of F : Geometry And Complexitymentioning

confidence: 99%

Projective Pre-Conditioners for Improving the Behavior of a Homogeneous Conic Linear System

Belloni

Freund

2006

SSRN Journal

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Abstract. In this paper we present a general theory for transforming a normalized homogeneous conic system F : Ax = 0,s T x = 1, x ∈ C to an equivalent system via projective transformation induced by the choice of a point v in the setSuch a projective transformation serves to pre-condition the conic system into a system that has both geometric and computational properties with certain guarantees. We characterize both the geometric behavior and the computational behavior of the transformed system as a function of the symmetry ofv in H • s as well as the complexity parameter ϑ of the barrier for C. Under the assumption that F has an interior solution, H • s must contain a point v whose symmetry is at least 1/m; if we can find a point whose symmetry is Ω(1/m) then we can projectively transform the conic system to one whose geometric properties and computational complexity will be strongly-polynomial-time in m and ϑ. We present a method for generating such a pointv based on sampling and on a geometric random walk on H • s with associated complexity and probabilistic analysis. Finally, we implement this methodology on randomly generated homogeneous linear programming feasibility problems, constructed to be poorly behaved. Our computational results indicate that the projective pre-conditioning methodology holds the promise to markedly reduce the overall computation time for conic feasibility problems; for instance we observe a 46% decrease in average IPM iterations for 100 randomly generated poorly-behaved problem instances of dimension 1000 × 5000.

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The Relaxation Method for Solving Systems of Linear Inequalities

Cited by 130 publications

References 22 publications

Rescaled Coordinate Descent Methods for Linear Programming

Rescaled Coordinate Descent Methods for Linear Programming

A Sampling Kaczmarz--Motzkin Algorithm for Linear Feasibility

Projective Pre-Conditioners for Improving the Behavior of a Homogeneous Conic Linear System

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