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Some recent developments in nonlinear optimization algorithms. (English) Zbl 1035.65061

Summary: This article provides a condensed overview of some of the major today’s features (both classical or recently developed), used in the design and development of algorithms to solve nonlinear continuous optimization problems.
We first consider the unconstrained optimization case to introduce the line-search and trust-region approaches as globalization techniques to force an algorithm to converge from any starting point. We then focus on constrained optimization and give the main ideas of two classes of methods, the sequential quadratic programming methods and the interior-point methods. We briefly discuss why interior-point methods are now so popular, in their primal-dual version, while they have been abandoned about twenty years ago.
We also introduce a newly emerging alternative, called filter method, to the use of a merit function as a tool to measure progress from one iteration to the next in constrained optimization. We relate some of the most widely used nonlinear optimization solvers to the algorithmic features presented, and we finally give some useful tools for an easy and comprehensive access to recent developments in nonlinear optimization algorithms and to practical solvers and their performance.

MSC:

65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
90C51 Interior-point methods
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming
90C55 Methods of successive quadratic programming type

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