Delaunay-based derivative-free optimization via global surrogates, part II: convex 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.…”

“…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.…”

“…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.…”

A parameter-free unconstrained reformulation for nonsmooth problems with convex constraints

“…Other works stem from research presented in [11,21] whose line-search approach has been extended for box and linearly constrained problems in [20] and [22] while more general constraints are covered in [19]. An interesting work in the field of global optimization is [8]. We refer the reader interested in derivative-based methods, which are not considered in this work, to [7].…”