Delaunay-based derivative-free optimization via global surrogates, part I: linear constraints

A derivative-free optimization algorithm for the efficient minimization of functions obtained via statistical averaging

“…A comparison between the minima of these two search functions is made in order to decide between further sampling (and, therefore, refining) an existing measurement, or sampling at a new point in parameter space. The method developed builds closely on the Delaunay-based Derivative-free Optimization via Global Surrogates algorithm, dubbed ∆-DOGS, proposed in [12][13][14]. Convergence of the algorithm is established in problems for which a.…”

“…This section presents a simplified version of the ∆-DOGS(Z) algorithm, the full version of which is given as Algorithm 2 of [12], where it is analyzed in detail. The ∆-DOGS(Z) algorithm is a grid-based acceleration of the ∆-DOGS algorithm originally developed in [14], and is designed to minimize problems in which precise function evaluations are available, while avoiding an accumulation of unnecessary function evaluations on the boundary of the feasible domain. The optimization problem considered in this section is the minimization of an objective function f (x), approximations of which are assumed to be available, in the feasible domain L = {x|x ∈ R n , a ≤ x ≤ b}.…”

“…The interpolation and remoteness functions at iteration k are denoted here by p k (x) and e k (x), the latter of which is defined below. Note that the function e k (x) is called an "uncertainty" function in [12][13][14]; we use the name "remoteness" function for the same construction in the present paper, as it plays a slightly different role in the sections that follow. Definition 1.…”

“…The best technique for computing the regression is problem dependent. As with [12][13][14], a key advantage of our Delaunay-based approach in the present work is that it facilitates the use of any suitable regression technique, subject to it satisfying the "strict" regression property given in Definition 4. Since our numerical tests all implement the polyharmonic spline regression technique, the derivation of this regression technique is briefly explained in this appendix; additional details may be found in [46].…”

“…In this paper, the Delaunay-based derivative-free optimization algorithm developed in [1,2,[12][13][14]47], dubbed ∆-DOGS, is modified to minimize an objective function, f (x) : R n → R, which cannot be evaluated directly, but can be approximated to a tuneable degree of precision. As the precision of any function evaluation is increased, the computational (or experimental) cost of that function evaluation is increased accordingly.…”