Smoothing SQP methods for solving degenerate nonsmooth constrained optimization problems with applications to bilevel programs

摘要

We consider a degenerate nonsmooth and nonconvex optimization problem forwhich the standard constraint qualification such as the generalized MangasarianFromovitz constraint qualification (GMFCQ) may not hold. We use smoothingfunctions with the gradient consistency property to approximate the nonsmoothfunctions and introduce a smoothing sequential quadratic programming (SQP)algorithm under the exact penalty framework. We show that any accumulationpoint of a selected subsequence of the iteration sequence generated by thesmoothing SQP algorithm is a Clarke stationary point, provided that thesequence of multipliers and the sequence of exact penalty parameters arebounded. Furthermore, we propose a new condition called the weakly generalizedMangasarian Fromovitz constraint qualification (WGMFCQ) that is weaker than theGMFCQ. We show that the extended version of the WGMFCQ guarantees theboundedness of the sequence of multipliers and the sequence of exact penaltyparameters and thus guarantees the global convergence of the smoothing SQPalgorithm. We demonstrate that the WGMFCQ can be satisfied by bilevel programsfor which the GMFCQ never holds. Preliminary numerical experiments show thatthe algorithm is efficient for solving degenerate nonsmooth optimizationproblem such as the simple bilevel program.