On the Curvature of the Central Path of Linear Programming Theory

Dedieu, Jean-Pierre; Malajovich, Gregorio; Shub, Michael

doi:10.1007/s10208-003-0116-8

Cited by 22 publications

(39 citation statements)

References 36 publications

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“…The later follows from the finiteness of the total curvature of P established in [2], or intuitively from the fact that P is asymptotically straight as ν → 0 and ν → ∞.…”

Section: The Continuous Analogue Of the Results Of Klee And Walkupmentioning

confidence: 99%

A Continuous d-Step Conjecture for Polytopes

Deza

Terlaky

Zinchenko

2008

Discrete Comput Geom

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The curvature of a polytope, defined as the largest possible total curvature of the associated central path, can be regarded as the continuous analogue of its diameter. We prove the analogue of the result of Klee and Walkup. Namely, we show that if the order of the curvature is less than the dimension d for all polytope defined by 2d inequalities and for all d, then the order of the curvature is less that the number of inequalities for all polytopes.

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“…The later follows from the finiteness of the total curvature of P established in [2], or intuitively from the fact that P is asymptotically straight as ν → 0 and ν → ∞.…”

Section: The Continuous Analogue Of the Results Of Klee And Walkupmentioning

confidence: 99%

A Continuous d-Step Conjecture for Polytopes

Deza

Terlaky

Zinchenko

2008

Discrete Comput Geom

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“…The relationship between the number of approximately straight segments of the central path introduced by Vavasis and Ye [13] and a certain curvature measure of the central path introduced by Sonnevend, Stoer and Zhao [11] and further analyzed in [14], was further studied by Monteiro and Tsuchiya in [8]. Dedieu, Malajovich and Shub [1] investigated a properly averaged total curvature of the central path. Nesterov and Todd [9] studied the Riemannian curvature of the central path in particular relevant to the so-called short-step methods.…”

Section: Introductionmentioning

confidence: 99%

Central Path Curvature and Iteration-Complexity for Redundant Klee—Minty Cubes

Deza

Terlaky

Zinchenko

2009

Advances in Mechanics and Mathematics

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Summary. We consider a family of linear optimization problems over the ndimensional Klee-Minty cube and show that the central path may visit all of its vertices in the same order as simplex methods do. This is achieved by carefully adding an exponential number of redundant constraints that forces the central path to take at least 2 n −2 sharp turns. This fact suggests that any feasible path-following interior-point method will take at least O(2 n ) iterations to solve this problem, while in practice typically only a few iterations, e.g., 50, suffices to obtain a high quality solution. Thus, the construction potentially exhibits the worst-case iterationcomplexity known to date which almost matches the theoretical iteration-complexity bound for this type of methods. In addition, this construction gives a counterexample to a conjecture that the total central path curvature is O(n).

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“…Noteworthy are the Riemannian geometry of the central path in linear programming [17,45], and an intriguing continuous-time system on the Grassmann manifold associated with linear programs [63,1].…”

Section: Resultsmentioning

confidence: 99%

Optimization On Manifolds: Methods and Applications

Absil

Mahony

Sepulchre

2010

Recent Advances in Optimization and Its Applications in Engineering

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Summary. This paper provides an introduction to the topic of optimization on manifolds. The approach taken uses the language of differential geometry, however, we choose to emphasise the intuition of the concepts and the structures that are important in generating practical numerical algorithms rather than the technical details of the formulation. There are a number of algorithms that can be applied to solve such problems and we discuss the steepest descent and Newton's method in some detail as well as referencing the more important of the other approaches. There are a wide range of potential applications that we are aware of, and we briefly discuss these applications, as well as explaining one or two in more detail.

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On the Curvature of the Central Path of Linear Programming Theory

Cited by 22 publications

References 36 publications

A Continuous d-Step Conjecture for Polytopes

A Continuous d-Step Conjecture for Polytopes

Central Path Curvature and Iteration-Complexity for Redundant Klee—Minty Cubes

Optimization On Manifolds: Methods and Applications

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