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A derivative-free trust-funnel method for equality-constrained nonlinear optimization

Journal paper
Ph. R. Sampaio, Ph. L. Toint
Computational Optimization and Applications, Volume 61, Issue 1, Pages 25-49, 2015

Abstract

A new derivative-free method is proposed for solving equality-constrained nonlinear optimization problems. This method is of the trust-funnel variety and is also based on the use of polynomial interpolation models. In addition, it uses a self-correcting geometry procedure in order to ensure that the interpolation problem is well de ned in the sense that the geometry of the set of interpolation points does not di er too much from the ideal one. The algorithm is described in detail and some encouraging numerical results are presented.

Worst-case evaluation complexity of non-monotone gradient-related algorithms for unconstrained optimization

Journal paper
C. Cartis, Ph. R. Sampaio, Ph. L. Toint
Optimization, Volume 64, Issue 5, Pages 1349-1361, 2015

Abstract

The worst-case evaluation complexity of finding an approximate first-order critical point using gradient-related non-monotone methods for smooth nonconvex and unconstrained problems is investigated. The analysis covers a practical linesearch implementation of these popular methods, allowing for an unknown number of evaluations of the objective function (and its gradient) per iteration. It is shown that this class of methods shares the known complexity properties of a simple steepest-descent scheme and that an approximate first-order critical point can be computed in at most O(ε^{-2}) function and gradient evaluations, where ε > 0 is the user-defined accuracy threshold on the gradient norm.

Numerical experience with a derivative-free trust-funnel method for nonlinear optimization problems with general nonlinear constraints

Journal paper
Ph. R. Sampaio and Ph. L. Toint
Optimization Methods and Software, Volume 31, Issue 3, Pages 511-534, 2016

Abstract

A trust-funnel method is proposed for solving nonlinear optimization problems with general nonlinear constraints. It extends the one presented by Gould and Toint [Nonlinear programming without a penalty function or a filter. Math. Prog. 122(1):155–196, 2010], originally proposed for equality-constrained optimization problems only, to problems with both equality and inequality constraints and where simple bounds are also considered. As the original one, our method makes use of neither filter nor penalty functions and considers the objective function and the constraints as independently as possible. To handle the bounds, an active-set approach is employed. We then exploit techniques developed for derivative-free optimization (DFO) to obtain a method that can also be used to solve problems where the derivatives are unavailable or are available at a prohibitive cost. The resulting approach extends the DEFT-FUNNEL algorithm presented by Sampaio and Toint [A derivative-free trust-funnel method for equality-constrained nonlinear optimization. Comput. Optim. Appl. 61(1):25–49, 2015], which implements a derivative-free trust-funnel method for equality-constrained problems. Numerical experiments with the extended algorithm show that our approach compares favourably to other well-known model-based algorithms for DFO.