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A nonsmooth version of the univariate optimization algorithm for locating the nearest extremum (locating extremum in nonsmooth univariate optimization)

100%
Smietanski M.
Open Mathematics
|
2008
|
tom 6
|
nr 3
469-481
EN
An algorithm for univariate optimization using a linear lower bounding function is extended to a nonsmooth case by using the generalized gradient instead of the derivative. A convergence theorem is proved under the condition of semismoothness. This approach gives a globally superlinear convergence of algorithm, which is a generalized Newton-type method.
2
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On the convergence of the secant method under the gamma condition

64%
Argyros I.
Open Mathematics
|
2007
|
tom 5
|
nr 2
205-214
EN
We provide sufficient convergence conditions for the Secant method of approximating a locally unique solution of an operator equation in a Banach space. The main hypothesis is the gamma condition first introduced in [10] for the study of Newton’s method. Our sufficient convergence condition reduces to the one obtained in [10] for Newton’s method. A numerical example is also provided.
3
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On a quadratically convergent method using divided differences of order one under the gamma condition

52%
Argyros I., Ren H.
Open Mathematics
|
2008
|
tom 6
|
nr 2
262-271
EN
We re-examine a quadratically convergent method using divided differences of order one in order to approximate a locally unique solution of an equation in a Banach space setting [4, 5, 7]. Recently in [4, 5, 7], using Lipschitz conditions, and a Newton-Kantorovich type approach, we provided a local as well as a semilocal convergence analysis for this method which compares favorably to other methods using two function evaluations such as the Steffensen’s method [1, 3, 13]. Here, we provide an analysis of this method under the gamma condition [6, 7, 19, 20]. In particular, we also show the quadratic convergence of this method. Numerical examples further validating the theoretical results are also provided.
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