Local convergence of predictor—corrector infeasible-interior-point algorithms for SDPs and SDLCPs

Kojima, M.; Shida, Masayuki; Shindoh, Susumu

doi:10.1007/bf01581723

Cited by 71 publications

(78 citation statements)

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“…Since our paths are defined by fields of search directions which decide the convergence behavior of an IPM, the bad behavior (not being analytic and having unbounded derivative if y 2 = −y 1 in the example) of the underlying paths strongly indicates slow convergence of IPM near a solution to a SDLCP. Kojima, et al, [9] observed certain evidence that IPMs do not converge superlinearly without "shrinking" the neighborhood using the HKM direction. Through the analysis of underlying paths here, we provide further evidence which anticipates the slow local convergence of IPMs for SDP and SDLCP using the HKM direction.…”

Section: Discussionmentioning

confidence: 99%

“…What we discussed below using this example, however, is not directly related to its discussion in [9].…”

Section: Qedmentioning

confidence: 99%

“…This example is taken from [9]. Note that the example satisfies the standard assumptions for SDP that appear in the literature.…”

Section: Qedmentioning

confidence: 99%

“…(The concept of nondegeneracy is discussed for example in [9] and is widely used in the literature.) In this sense, the example is a nice, typical SDLCP example.…”

Section: Qedmentioning

confidence: 99%

“…We choose this example from [9] mainly because it is simple and its nice properties. What we discussed below using this example, however, is not directly related to its discussion in [9].…”

Section: Qedmentioning

confidence: 99%

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Underlying paths in interior point methods for the monotone semidefinite linear complementarity problem

Sim¹,

Zhao

2006

Math. Program.

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An interior point method (IPM) defines a search direction at each interior point of the feasible region. The search directions at all interior points together form a direction field, which gives rise to a system of ordinary differential equations (ODEs). Given an initial point in the interior of the feasible region, the unique solution of the ODE system is a curve passing through the point, with tangents parallel to the search directions along the curve. We call such curves off-central paths. We study off-central paths for the monotone semidefinite linear complementarity problem (SDLCP). We show that each off-central path is a well-defined analytic curve with parameter µ ranging over (0, ∞) and any accumulation point of the off-central path is a solution to SDLCP. Through a simple example we show that the off-central paths are not analytic as a function of √ µ and have first derivatives which are unbounded as a function of µ at µ = 0 in general. On the other hand, for the same example, we can find a subset of off-central paths which are analytic at µ = 0. These "nice" paths are characterized by some algebraic equations.

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Section: Discussionmentioning

confidence: 99%

“…What we discussed below using this example, however, is not directly related to its discussion in [9].…”

Section: Qedmentioning

confidence: 99%

“…This example is taken from [9]. Note that the example satisfies the standard assumptions for SDP that appear in the literature.…”

Section: Qedmentioning

confidence: 99%

“…(The concept of nondegeneracy is discussed for example in [9] and is widely used in the literature.) In this sense, the example is a nice, typical SDLCP example.…”

Section: Qedmentioning

confidence: 99%

“…We choose this example from [9] mainly because it is simple and its nice properties. What we discussed below using this example, however, is not directly related to its discussion in [9].…”

Section: Qedmentioning

confidence: 99%

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Underlying paths in interior point methods for the monotone semidefinite linear complementarity problem

Sim¹,

Zhao

2006

Math. Program.

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Complementarity Problems Over Symmetric Cones: A Survey of Recent Developments in Several Aspects

Yoshise

2011

International Series in Operations Research &Amp; Management Science

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The complementarity problem over a symmetric cone (that we call the Symmetric Cone Complementarity Problem, or the SCCP) has received much attention of researchers in the last decade. Most of studies done on the SCCP can be categorized into the three research themes, interior point methods for the SCCP, merit or smoothing function methods for the SCCP, and various properties of the SCCP. In this paper, we will provide a brief survey on the recent developments on these three themes.

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Exploiting sparsity in primal-dual interior-point methods for semidefinite programming

Fujisawa

Kojima

Nakata

1997

Mathematical Programming

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Local convergence of predictor—corrector infeasible-interior-point algorithms for SDPs and SDLCPs

Cited by 71 publications

References 20 publications

Underlying paths in interior point methods for the monotone semidefinite linear complementarity problem

Underlying paths in interior point methods for the monotone semidefinite linear complementarity problem

Complementarity Problems Over Symmetric Cones: A Survey of Recent Developments in Several Aspects

Exploiting sparsity in primal-dual interior-point methods for semidefinite programming

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