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Derivative-free optimization methods. (English) Zbl 07099161
The paper gives a “survey of methods for derivative-free optimization and key results for their analysis” to determine a local minimum of mathematical programming problems. Global optimization and multi-objective optimization, beside some special exceptions in section 8, are not under extensive consideration. Hints to associated reviews are given. References to applications, beside a link to some book, are not mentioned. “At the persistent page https://archive.org/services/purl/dfomethods we intend to link all works that cite the entries in our bibliography and those that cite this survey.” The link contains all bibtex sources and in much cases an available pdf-source of papers via doi. The most references are from the ninetieth till now and over half of them are from the past ten years. There are about 340 references. In order to apply a mentioned method it is ever necessary to consult suitable given references. The contents is as follows:
1 Introduction p. 288
2 Deterministic methods for deterministic objectives p. 293
3 Randomized methods for deterministic objectives p. 318
4 Methods for convex objectives p. 324
5 Methods for structured objectives p. 336
6 Methods for stochastic optimization p. 344
7 Methods for constrained optimization p. 354
8 Other extensions and practical considerations p. 369
Appendix: Collection of Worst Case Complexity analysis (WCC) results p. 376
References pp. 380-404
In the Introduction the main purpose and the restrictions of the paper are explained. In Section 2 the main search methods are discussed by presenting the main principal algorithms (Simplex of Nelder Mead, direct directional search (DDS), Descent test etc., generalized patter search (GPS), mesh adaptive direct search (MADS), polynomial models, quadratic interpolation models, radial basis function interpolation models (RBF), model based methods like Trust Region (TR), Filtering methods). The section is essential for understanding all further sections.
In all the following sections, new notions are defined by formulas with corresponding explanations. From Section 2 till Section 6, presented methods are mainly without formulas discussed w.r.t. the performance, the convergence, existing counterexamples and the complexity (WCC), ever highlighted by cited references. In the appendix the formulas of some of the WCC are given explicitly. Section 3 deals with random search, randomized direct-search and randomized TR methods. In Section 4 deterministic objectives (DSS with better WCC) and stochastic objectives (bandit feedback, convex expected value as objective) are considered. In Section 5 nonlinear least squares, composite non-smooth optimization, bilevel and general minimax problems are covered. The stochastic problems in Section 6 possess again an expected value as objective but now nonconvex.
In Section 7 the former unconstrained methods are extended to constrained optimization using penalty, augmented Lagrangian, SQP with TR models. The kind of the constraints is relaxable or unrelaxable algebraic, simulation based, modelled or unrelaxable discret. The discussion is more general again with a lot of references but mainly without convergence results and without WCC. Modelling is the main task. In Section 8 some extensions and practical considerations are given for special cases of multi-objective optimization (Pareto optimality), global optimization (Lipschitz under estimators, multistart methods) and multi fidelity in the focus of extensions of former presented methods and in the focus of future developments.

MSC:
65K05 Numerical mathematical programming methods
90C56 Derivative-free methods and methods using generalized derivatives
90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming
90-08 Computational methods for problems pertaining to operations research and mathematical programming
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