NDLI: Smoothing SQP Methods for Solving Degenerate Nonsmooth Constrained Optimization Problems with Applications to Bilevel Programs

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Zhang, Liwei Ye, Jane J. Xu, Mengwei
Copyright Year	2015
Abstract	We consider a degenerate nonsmooth and nonconvex optimization problem for which the standard constraint qualification such as the generalized Mangasarian--Fromovitz constraint qualification (GMFCQ) may not hold. We use smoothing functions with the gradient consistency property to approximate the nonsmooth functions and introduce a smoothing sequential quadratic programming (SQP) algorithm under the $l_\infty$ penalty framework. We show that any accumulation point of a selected subsequence of the iteration sequence generated by the smoothing SQP algorithm is a Clarke stationary point, provided that the sequence of multipliers and the sequence of penalty parameters are bounded. Furthermore, we propose a new condition called the weakly generalized Mangasarian--Fromovitz constraint qualification (WGMFCQ) that is weaker than the GMFCQ. We show that the extended version of the WGMFCQ guarantees the boundedness of the sequence of multipliers and the sequence of penalty parameters and thus guarantees the global convergence of the smoothing SQP algorithm. We demonstrate that the WGMFCQ can be satisfied by bilevel programs for which the GMFCQ never holds. Preliminary numerical experiments show that the algorithm is efficient for solving degenerate nonsmooth optimization problems such as the simple bilevel program.
Starting Page	1388
Ending Page	1410
Page Count	23
File Format	PDF
ISSN	10526234
DOI	10.1137/140971580
e-ISSN	10957189
Journal	SIAM Journal on Optimization (SJOPE8)
Issue Number	3
Volume Number	25
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2015-07-14
Access Restriction	Subscribed
Subject Keyword	Optimization and variational techniques smoothing function Nonconvex programming, global optimization bilevel program sequential quadratic programming algorithm Nonlinear programming constraint qualification nonsmooth optimization constrained optimization
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Software

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