A global optimization algorithm for simulation-based problems via the extended DIRECT scheme

Abstract

This article presents a global optimization algorithm via the extension of the DIviding RECTangles (DIRECT) scheme to handle problems with computationally expensive simulations efficiently. The new optimization strategy improves the regular partition scheme of DIRECT to a flexible irregular partition scheme in order to utilize information from irregular points. The metamodelling technique is introduced to work with the flexible partition scheme to speed up the convergence, which is meaningful for simulation-based problems. Comparative results on eight representative benchmark problems and an engineering application with some existing global optimization algorithms indicate that the proposed global optimization strategy is promising for simulation-based problems in terms of efficiency and accuracy.

Constrained global optimization via a DIRECT-type constraint-handling technique and an adaptive metamodeling strategy

“…The sensitivity analysis of eDIRECT to different metamodel types by Liu et al (2015b) has also confirmed this finding.…”

“…As the basis of the unconstrained eDIRECT global optimization algorithm which was proposed by the authors earlier (Liu et al 2015b), the original DIRECT algorithm (Jones et al 1993) has been studied and developed intensively for boxconstrained problems (Sergeyev and Kvasov 2006;Paulavičius and Žilinskas 2013;Liu and Cheng 2014;Paulavičius et al 2014). However, to the best of our knowledge, there is little work of the DIRECT-type algorithm for constrained problems (Jones 2001;Finkel 2005;Paulavičius and Žilinskas 2016).…”

“…According to the recommendations by Liu et al (2015b), the optimization parameters of eDIRECT-C are the same as that of eDIRECT: (1) in line 2 we generate 5n initial points by the translational propagation latin hypercube design (TPLHD) algorithm with a one-point seed (Viana et al 2010); (2) in lines 9 and 26 we use 50mn random points to describe the irregular Voronoi cells; (3) in lines 10 and 27 we set the tolerance parameter ε as 0.0001; and (4) in line 23 the stopping criterion can be defined by users, it could be an acceptable error measurement or limited computing resource. In the numerical experiments below, this article employs the relative error (see (28)) and the maximal number of evaluations as stopping criteria.…”

“…For any point x ∈ D and any δ > 0, there exists an iteration number k(δ) ≥ 1 and a point x′ Proof. Similar to Liu et al (2015b), we define a parameter s k in the kth iteration as…”

“…The DIRECT algorithm however has some disadvantages: (1) for rectangle-based partition the points should be regularly sampled, i.e., the sampling procedure is exclusive and can not handle irregular points; and (2) as a direct algorithm it converges slowly. The novel eDIRECT algorithm (Liu et al 2015b) overcomes these disadvantages of the DIRECT algorithm, i.e., exclusive sampling procedure and slow convergence, by using an inclusive partition scheme and the metamodeling techniques.…”