2014/03/07 by Mengwei Xu, Xu, Mengwei, Jane J. Ye +3
Computer Science · Decision Sciences · Mathematics · #Advanced Optimization Algorithms Research #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis #Risk and Portfolio Optimization
paper · pdf · doi:10.48550/arxiv.1403.1636
openalex publication_date 2014/03/07 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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 exact 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 exact 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 exact 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 problem such as the simple bilevel program.