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A nonmonotone trust region algorithm for unconstrained nonsmooth optimization. (English) Zbl 0846.90099

Summary: The following unconstrained nonsmooth optimization problem (P) is considered: \(\min_{x\in \mathbb{R}^n} f(x)\), where \(f(x): \mathbb{R}^n\to \mathbb{R}\) is only a locally Lipschitzian function. Many papers appear on the convergence properties of the trust region algorithm to solve several different particular nonsmooth problems. Dennis, Li and Tapia proposed a general trust region model by using regular functions. They proved the global convergence of the general trust region model under some mild conditions which are shown to be satisfied by many trust region algorithms including smooth one. Qi and Sun provided another trust region model which can be applied to the minimax problem and the nonlinear complementarity problem. Under some weak conditions based on the subhomogeneity assumption, they also proved global convergence of the corresponding trust region algorithm. These algorithms are all monotone of which the values objective functions are monotone decrease.
Some nonmonotone algorithms for smooth nonlinear programs have been provided. In this note, we propose a general nonmonotone trust region algorithm to solve the nonsmooth optimization problem (P) and prove its global convergence under suitable conditions.

MSC:

90C30 Nonlinear programming
49J52 Nonsmooth analysis
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