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A modified BFGS method and its global convergence in nonconvex minimization. (English) Zbl 0984.65055
A modification of the BFGS method for unconstrained optimization is proposed. The authors study the following unconstrained optimization problem: $$\min f(x)$$, $$x\in\mathbb{R}^n$$, where $$f: \mathbb{R}^n\to \mathbb{R}$$ is continuously differentiable function. The objective function $$f$$ has Lischitz continuous gradients.
Main result: The authors show (the precise proofs are given) a global convergence property even without convexity assumption on the objective function. Under certain conditions superlinear convergence of the proposed method is presented.

##### MSC:
 65K05 Numerical mathematical programming methods 90C26 Nonconvex programming, global optimization
ve08
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##### References:
 [1] Broyden, C.G.; Dennis, J.E.; Moré, J.J., On the local and superlinear convergence of quasi-Newton methods, J. inst. math. appl., 12, 223-246, (1973) · Zbl 0282.65041 [2] Byrd, R.; Nocedal, J., A tool for the analysis of quasi-Newton methods with application to unconstrained minimization, SIAM J. numer. anal., 26, 727-739, (1989) · Zbl 0676.65061 [3] Byrd, R.; Nocedal, J.; Yuan, Y., Global convergence of a class of quasi-Newton methods on convex problems, SIAM J. numer. anal., 24, 1171-1189, (1987) · Zbl 0657.65083 [4] Dennis, J.E.; Moré, J.J., A characterization of superlinear convergence and its application to quasi-Newton methods, Math. comput., 28, 549-560, (1974) · Zbl 0282.65042 [5] Dennis, J.E.; Moré, J.J., Quasi – newton methods, motivation and theory, SIAM rev., 19, 46-89, (1977) · Zbl 0356.65041 [6] Dixon, L.C.W., Variable metric algorithms: necessary and sufficient conditions for identical behavior on nonquadratic functions, J. optim. theory appl., 10, 34-40, (1972) · Zbl 0226.49014 [7] Fletcher, R., Practical methods of optimization, (1987), Wiley Chichester · Zbl 0905.65002 [8] Fletcher, R., An overview of unconstrained optimization, (), 109-143 · Zbl 0828.90123 [9] Griewank, A.; Toint, Ph.L., Local convergence analysis for partitioned quasi-Newton updates, Numer. math., 39, 429-448, (1982) · Zbl 0505.65018 [10] Griewank, A., The global convergence of partitioned BFGS on problems with convex decompositions and Lipschitzian gradients, Math. programming, 50, 141-175, (1991) · Zbl 0736.90068 [11] Li, D.-H., On the global convergence of DFP method, J. hunan univ. (natural sciences), 20, 16-20, (1993) · Zbl 0774.90064 [12] Powell, M.J.D., On the convergence of the variable metric algorithm, J. inst. math. appl., 7, 21-36, (1971) · Zbl 0217.52804 [13] Powell, M.J.D., Some global convergence properties of a variable metric algorithm for minimization without exact line searches, (), 53-72 · Zbl 0338.65038 [14] Toint, Ph.L., Global convergence of the partitioned BFGS algorithm for convex partially separable optimization, Math. programming, 36, 290-306, (1986) · Zbl 0626.90076 [15] Zhang, Y.; Tewarson, R.P., Quasi-Newton algorithms with updates from the preconvex part of Broyden’s family, IMA J. numer. anal., 8, 487-509, (1988) · Zbl 0661.65061
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