Implementing Generating Set Search Methods for Linearly Constrained Minimization

Abstract

Abstract. We discuss an implementation of a derivative-free generating set search method for linearly constrained minimization with no assumption of nondegeneracy placed on the constraints. The convergence guarantees for generating set search methods require that the set of search directions possesses certain geometrical properties that allow it to approximate the feasible region near the current iterate. In the hard case, the calculation of the search directions corresponds to finding the extreme rays of a cone with a degenerate vertex at the origin, a difficult problem. We discuss here how state-of-the-art computational geometry methods make it tractable to solve this problem in connection with generating set search. We also discuss a number of other practical issues of implementation, such as the careful treatment of equality constraints and the desirability of augmenting the set of search directions beyond the theoretically minimal set. We illustrate the behavior of the implementation on several problems from the CUTEr test suite. We have found it to be successful on problems with several hundred variables and linear constraints.

Weight and power optimization of steam bottoming cycle for offshore oil and gas installations

“…This algorithm combines the positive features of the Constrained Particle Swarm Optimizer (CPSO) [28], Generating Set Search (GSS) [29], and the Complex [30]. Each iteration of the algorithm consists of three steps: (i) a search step corresponding to a population update of a revised CPSO, (ii) an optional (skipped if the CPSO improves the best solution found so far)…”

A generating set direct search augmented Lagrangian algorithm for optimization with a combination of general and linear constraints.

“…Observe that the the initialization of 60 in Algorithm 4.1 and the definition of 0 in (13), together with the update rule (15), ensure that for all k 2 0, As a practical matter, in the implementation of Algorithm 3.1 discussed in [15] the directions in G are normalized, so Pmax = 1, which simplifies both (15) and (16).…”

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).…”