Deterministic approaches for solving practical black-box global optimization problems

Abstract

In many important design problems, some decisions should be made by finding the global optimum of a multiextremal objective function subject to a set of constrains. Frequently, especially in engineering applications, the functions involved in optimization process are black-box with unknown analytical representations and hard to evaluate. Such computationally challenging decision-making problems often cannot be solved by traditional optimization techniques based on strong suppositions about the problem (convexity, differentiability, etc.). Nature and evolutionary inspired metaheuristics are also not always successful in finding global solutions to these problems due to their multiextremal character. In this paper, some innovative and powerful deterministic approaches developed by the authors to construct numerical methods for solving the mentioned problems are surveyed. Their efficiency is shown on solving both the classes of random test problems and some practical engineering tasks.

Global optimization for structured low rank approximation

“…Figure 1 contains some plots of (2) with (3), q = 2 and N = 10. Although the objective functions are often Lipschitz-continuous (see, e. g., [7,10,16,17,19]), it has very high Lipschitz constants which increase with N, the number of observations. Adding noise to the observed data increases the complexity of the objective function (see, e. g., [3,20]) and moves the global minimizer away from the vector of true parameters.…”

“…Objective functions are typically highly multiextremal with the objective functions possessing many local minima (see also the related discussion in [10]). Figure 1 contains some plots of (2) with (3), q = 2 and N = 10.…”

Truss Structure Optimization with Subset Simulation and Augmented Lagrangian Multiplier Method

“…The minimum cross-sectional area of all design variables is 0.1 inch 2 . The structure is designed against three independent loading conditions: (1) 1.0 kip acting in the positive x-direction at nodes 1,6,15,20,29,34,43,48,57,62, and 71; (2) 10.0 kips acting in the negative y-direction at nodes 1,2,3,4,5,6,8,10,12,14,15,16,17,18,19,20,22,24,26,28,29,30,31,32,33,34,36,38,40,42,43,44,45,46,47,48,50,52,54,56,57,…”

“…An alternative could be the deterministic global optimization methods, e.g., the widely used DIRECT method [2,3], which is a branch of gradient-free methods. Recently, Kvasov and his colleagues provided a good guide on the deterministic global optimization methods [4] and carried out a comprehensive comparison study between the deterministic and stochastic global optimization methods for one-dimensional problems [5]. One of the main disadvantages of the deterministic global optimization methods is the high-dimensionality issue caused by the larger number of design variables [3] or constraints.…”

Internal volumetric heat generation and heat capacity prediction during a material electromagnetic treatment process using hybrid algorithms

“…To overcome this situation, those parameters can be estimated by solving the corresponding inverse problem. Besides, it is now possible to work out these type of problems using diverse global optimization algorithms (Hào et al, 2017;Wang et al, 2017;Cebo-Rudnicka et al, 2016;Nedin et al, 2016;Chen et al, 2016;Strongin & Sergeyev, 2000;Zhigljavsky & Žilinskas, 2008;Kvasov & Sergeyev, 2014). Previous works related to heat generation proposed estimating heat generation in the case of a rotatory friction welding system (Yang et al, 2011).…”