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New sequential and parallel derivative-free algorithms for unconstrained minimization. (English) Zbl 1049.90133

Summary: This paper presents sequential and parallel derivative-free algorithms for finding a local minimum of smooth and nonsmooth functions of practical interest. It is proved that, under mild assumptions, a sufficient decrease condition holds for a nonsmooth function. Based on this property, the algorithms explore a set of search directions and move to a point with a sufficiently lower functional value. If the function is strictly differentiable at its limit points, a (sub)sequence of points generated by the algorithm converges to a first-order stationary point (\(\nabla f(x) = 0\)). If the function is convex around its limit points, convergence (of a subsequence) to a point with nonnegative directional derivatives on a set of search directions is ensured. Preliminary numerical results on sequential algorithms show that they compare favorably with the recently introduced pattern search methods.

MSC:

90C56 Derivative-free methods and methods using generalized derivatives
65K10 Numerical optimization and variational techniques

Software:

KELLEY; DFO; fminsearch
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