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Nonmonotone spectral gradient-type methods for large-scale unconstrained optimization and nonlinear systems of equations. (English) Zbl 1228.49038
Summary: This paper presents some spectral gradient-type methods under a new non-monotone line search, which possesses a favorite property that each iteration is well defined even when the search direction is not a descent direction. That may be useful when the function values are inaccurate (e.g., in real-life problems, the values are sometimes noised or derivatives are computed by differences). Global convergence is established under some suitable conditions and a nonmonotone Spectral Conjugate Gradient (SCG) method is presented as a special case. Then this SCG method is extended to solve nonlinear systems of equations, which results in a new derivative-free method for solving large-scale nonlinear equations. Preliminary numerical results are reported on a set of large-scale problems to show the convergence and efficiency of this method.
MSC:
49M37Methods of nonlinear programming type in calculus of variations
65H10Systems of nonlinear equations (numerical methods)
65K05Mathematical programming (numerical methods)
90C26Nonconvex programming, global optimization
90C30Nonlinear programming
93A15Large scale systems
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