Path-Following Methods for Linear Programming

Gonzaga, Clóvis C.

doi:10.1137/1034048

Cited by 299 publications

(153 citation statements)

References 68 publications

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“…Much of the modern work in numerical algorithms has focused on interior-point methods [166,37,1631. Initially such work was limited to LPs [88,133,73,89,109,105,621, but was soon extended to encompass other CPs as well 1117, 118,7,87,119,162,15,1201. Now a number of excellent solvers are readily available, both commercial and freely distributed.…”

Section: Numerical Algorithmsmentioning

confidence: 99%

Disciplined Convex Programming

Grant

Boyd

Nonconvex Optimization and Its Applications

3,684

4,391

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Section: Numerical Algorithmsmentioning

confidence: 99%

Disciplined Convex Programming

Grant

Boyd

Nonconvex Optimization and Its Applications

3,684

4,391

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“…The positive definite matrix S (k) 2 (y, x (k) ) can be factored into the product of three matrices: a unit lower triangular matrix L, a positive definite diagonal matrix M , and the transpose of L, such that [22]. So (10) can be rewritten in the form of (13). Therefore, by Theorem 1, local minimizers of (12) exist for each y ∈ F 0 and k = 1, 2, .…”

Section: +∞ Let S Be a Scalar-valued Function Of The Single Variablementioning

confidence: 99%

“…The purpose of this paper is to derive a class of decomposition algorithms for SSDPs based on a volumetric barrier, and to prove polynomial complexity of short step [3,13] and long step [3,13] members of the class.…”

Section: Introductionmentioning

confidence: 99%

A class of polynomial volumetric barrier decomposition algorithms for stochastic semidefinite programming

Ariyawansa

Zhu

2010

Math. Comp.

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Abstract. Ariyawansa and Zhu have recently proposed a new class of optimization problems termed stochastic semidefinite programs (SSDPs). SSDPs may be viewed as an extension of two-stage stochastic (linear) programs with recourse (SLPs). Zhao has derived a decomposition algorithm for SLPs based on a logarithmic barrier and proved its polynomial complexity. Mehrotra and Ozevin have extended the work of Zhao to the case of SSDPs to derive a polynomial logarithmic barrier decomposition algorithm for SSDPs. An alternative to the logarithmic barrier is the volumetric barrier of Vaidya. There is no work based on the volumetric barrier analogous to that of Zhao for SLPs or to the work of Mehrotra andÖzevin for SSDPs. The purpose of this paper is to derive a class of volumetric barrier decomposition algorithms for SSDPs, and to prove polynomial complexity of certain members of the class.

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“…A signature of interior methods is the existence of continuously parameterized families of approximate solutions that asymptotically converge to the exact solution; see, for example, [20]. As the parameter approaches its limit, these paths trace smooth trajectories with geometric properties (such as being "centered" in a precisely defined sense) that can be analyzed and exploited algorithmically.…”

Section: Complexitymentioning

confidence: 99%

The interior-point revolution in optimization: History, recent developments, and lasting consequences

Wright

2004

Bull. Amer. Math. Soc.

282

131

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Abstract. Interior methods are a pervasive feature of the optimization landscape today, but it was not always so. Although interior-point techniques, primarily in the form of barrier methods, were widely used during the 1960s for problems with nonlinear constraints, their use for the fundamental problem of linear programming was unthinkable because of the total dominance of the simplex method. During the 1970s, barrier methods were superseded, nearly to the point of oblivion, by newly emerging and seemingly more efficient alternatives such as augmented Lagrangian and sequential quadratic programming methods. By the early 1980s, barrier methods were almost universally regarded as a closed chapter in the history of optimization.This picture changed dramatically in 1984, when Narendra Karmarkar announced a fast polynomial-time interior method for linear programming; in 1985, a formal connection was established between his method and classical barrier methods. Since then, interior methods have continued to transform both the theory and practice of constrained optimization. We present a condensed, unavoidably incomplete look at classical material and recent research about interior methods.

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Path-Following Methods for Linear Programming

Cited by 299 publications

References 68 publications

Disciplined Convex Programming

Disciplined Convex Programming

A class of polynomial volumetric barrier decomposition algorithms for stochastic semidefinite programming

The interior-point revolution in optimization: History, recent developments, and lasting consequences

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