Constrained derivative-free optimization on thin domains

Abstract

Many derivative-free methods for constrained problems are not efficient for minimizing functions on "thin" domains. Other algorithms, like those based on Augmented Lagrangians, deal with thin constraints using penalty-like strategies. When the constraints are computationally inexpensive but highly nonlinear, these methods spend many potentially expensive objective function evaluations motivated by the difficulties of improving feasibility. An algorithm that handles efficiently this case is proposed in this paper. The main iteration is splitted into two steps: restoration and minimization. In the restoration step the aim is to decrease infeasibility without evaluating the objective function. In the minimization step the objective function f is minimized on a relaxed feasible set. A global minimization result will be proved and computational experiments showing the advantages of this approach will be presented.

Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO

“…Algorithmic developments and convergence have been studied for box-constrained problems (BCDFO) (Lewis and Torczon, 1999;Lucidi and Sciandrone, 2002;Audet and Dennis, 2002;Garcia-Palomares et al, 2013); for known linear constrained problems (Lewis and Torczon, 2000;Lewis et al, 2007;Audet and Dennis, 2002;Kolda et al, 2006) (Liuzzi et al, 2010); smoothed exact l −∞ penalty function (Liuzzi and Lucidi, 2009) and exact penalty merit functions (Fasano et al, 2014;Gratton and Vicente, 2014), or restoration steps which are independent from objective function (Martinez and Sobral, 2013;Arouxét et al, 2015). Allowing pattern-search methods to handle a dense set of polling directions instead of a finite set of polling directions was a significant development for studying the convergence of generally constrained derivative-free problems which are Lipschitz even near a limit point (Audet and Dennis, 2006;Audet et al, 2008b).…”

Penalty-free method for nonsmooth constrained optimization via radial basis functions

“…Table 3 shows the comparison of the best solution of our method in terms of the value of design variables and function value. The comparison of the OSRB algorithm against the Skinny method [31] and HOPSPACK [38] shows that our algorithm is more robust, because the number function evaluations are less than in most problems; also in HOPSPACK [38] some problems have no feasible solution, whereas our algorithm is always able to find a feasible point. These overall results suggest that the proposed OSRB can be considered an effective optimization technique for solving nonsmooth constrained optimization problems.…”

Trust-Region-Based Methods for Nonlinear Programming: Recent Advances and Perspectives

“…For problems with thin domains, defined by computationally inexpensive but highly nonlinear functions, Martínez & Sobral [51] have proposed the algorithm SKINNY, that splits the main iteration into a restoration step, where infeasibility is decreased without evaluating the objective function, followed by the derivative-free minimization on a relaxed feasible set. In the presented comparative numerical experiments, SKINNY were able to solve more problems than DFO [15].…”