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Delaunay-based derivative-free optimization via global surrogates. III: nonconvex constraints. (English) Zbl 1447.90035
Summary: This paper introduces a Delaunay-based derivative-free optimization algorithm, dubbed $$\varDelta$$-DOGS $$(\varOmega)$$, for problems with both (a) a nonconvex, computationally expensive objective function $$f(x)$$, and (b) nonlinear, computationally expensive constraint functions $$c_\ell (x)$$ which, taken together, define a nonconvex, possibly even disconnected feasible domain $$\varOmega$$, which is assumed to lie within a known rectangular search domain $$\varOmega_s$$, everywhere within which the $$f(x)$$ and $$c_\ell (x)$$ may be evaluated. Approximations of both the objective function $$f(x)$$ as well as the feasible domain $$\varOmega$$ are developed and refined as the iterations proceed. The approach is practically limited to the problems with less than about ten adjustable parameters. The work is an extension of our original Delaunay-based optimization algorithm (see [the second author et al.,ibid. 66, No. 3, 331–382 (2016; Zbl 1383.90045)]), and inherits many of the constructions and strengths of that algorithm, including: (1) a surrogate function $$p(x)$$ interpolating all existing function evaluations and summarizing their trends, (2) a synthetic, piecewise-quadratic uncertainty function $$e(x)$$ built on the framework of a Delaunay triangulation amongst existing datapoints, (3) a tunable balance between a global exploration (large $$K$$) and a local refinement (small $$K$$), (4) provable global convergence for a sufficiently large $$K$$, under the assumption that the objective and constraint functions are twice differentiable with bounded Hessians, (5) an adaptive-$$K$$ variant of the algorithm that efficiently tunes $$K$$ automatically based on a target value of the objective function, and (6) remarkably fast global convergence on a variety of benchmark problems.

##### MSC:
 90C26 Nonconvex programming, global optimization 90C56 Derivative-free methods and methods using generalized derivatives
##### Keywords:
response surface methods; nonconvex constraints