NDLI: On Interior Logarithmic Smoothing and Strongly Stable Stationary Points

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Rckmann, Jan-J. Jongen, Hubertus Th.
Copyright Year	2010
Abstract	The paper deals with a nonlinear programming problem (P) and, by using a logarithmic barrier function, a parametric family of interior point approximations $M^\gamma$ of its feasible set M, where $M^\gamma$ is described by a single smooth inequality constraint. Assuming that a stationary point $\bar{x}$ of (P) under consideration is strongly stable, it is shown that for all sufficiently small $\gamma>0$ there exists locally around $\bar{x}$ a uniquely determined stationary point $x(\gamma)$ of (P$^\gamma$), where (P$^\gamma$) is obtained from (P) by substituting M by $M^\gamma$. In particular, $x(\gamma)$ is strongly stable, even nondegenerate, and it has the same stationary index as $\bar{x}$. Furthermore, it turns out that $x(\gamma)$ and its uniquely determined Lagrange multiplier $\mu(\gamma)$ form a solution pair of a corresponding interior-point problem, where $(x(\gamma),\mu(\gamma))$ depends continuously differentiable (under linear independence constraint qualification (LICQ)) or continuous (under MangasarianFromovitz constraint qualification (MFCQ)) on the parameter $\gamma$ and $x(\gamma)$ converges to $\bar{x}$ as $\gamma\rightarrow0$. The stationary point $\bar{x}$ might be degenerate and, a priori, no strict complementarity is assumed. Finally, a globalization of this one-to-one correspondence between the stationary points of (P) and (P$^\gamma$) as well as some further topological properties of M and $M^\gamma$ are discussed.
Starting Page	2137
Ending Page	2156
Page Count	20
File Format	PDF
ISSN	10526234
DOI	10.1137/070711566
e-ISSN	10957189
Journal	SIAM Journal on Optimization (SJOPE8)
Issue Number	5
Volume Number	20
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2010-04-14
Access Restriction	Subscribed
Subject Keyword	interior point approximation degenerate solution of interior-point methods strongly stable stationary point Mathematical programming methods stationary index logarithmic barrier function smoothing Nonlinear programming Methods of nonlinear programming type Sensitivity, stability, parametric optimization
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Software

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