Objective and violation upper bounds on a DIRECT-filter method for global optimization

This paper addresses the problem of solving a constrained global optimization problem using a modification of the DIRECT method that incorporates the filter methodology to simultaneously minimize the objective function and the constraints violation. Thus, in the “Selection” step of the herein proposed DIRECT-filter algorithm, the hyperrectangles are classified in four categories and subsequently handled separately. The new algorithm also imposes upper bounds on the objective function and constraints violation aiming to discard some hyperrectangles from the process of identifying the potentially optimal ones. A heuristic to avoid the exploration of the hyperrectangles that have been mostly divided is also implemented. Preliminary numerical experiments are carried out to show the effectiveness of the imposed upper bounds on the objective and violation as well as the goodness of the heuristic.The authors wish to thank two anonymous referees for theircomments and suggestions to improve the paper. This work has been supported by FCT{ Fundação para a Ciência e Tecnologia within the Projects Scope: UID/CEC/00319/2019 and UID/MAT/00013/2013

A two-phase heuristic coupled DIRECT method for bound constrained global optimization

In this paper, we investigate the use of a simple heuristic in the DIRECT method context, aiming to select a set of the hyperrectangles that have the lowest function values in each size group. For solving bound constrained global optimization problems, the proposed heuristic divides the region where the hyperrectangles with the lowest function values in each size group lie into three subregions. From each subregion, different numbers of hyperrectangles are selected depending on the subregion they lie. Subsequently, from those selected hyperrectangles, the potentially optimal ones are identified for further division. Furthermore, the two-phase strategy aims to firstly encourage the global search and secondly enhance the local search. Global and local phases differ on the number of selected hyperrectangles from each subregion. The process is repeated until convergence. Numerical experiments carried out until now show that the proposed two-phase heuristic coupled DIRECT method is effective in converging to the optimal solution.H2020 - Horizon 2020 Framework Programme(UIDB/00013/2020)This work has been supported by European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 734205 - H2020-MSCA-RISE-2017 and by national funds through FCT Fundação para a Ciência e Tecnologia within the Projects Scope: UID/CEC/00319/2019 and UID/MAT/00013/2013

A two-phase heuristic coupled DIRECT method for bound constrained global optimization

In this paper, we investigate the use of a simple heuristic in the DIRECT method context, aiming to select a set of the hyperrectangles that have the lowest function values in each size group. For solving bound constrained global optimization problems, the proposed heuristic divides the region where the hyperrectangles with the lowest function values in each size group lie into three subregions. From each subregion, different numbers of hyperrectangles are selected depending on the subregion they lie. Subsequently, from those selected hyperrectangles, the potentially optimal ones are identified for further division. Furthermore, the two-phase strategy aims to firstly encourage the global search and secondly enhance the local search. Global and local phases differ on the number of selected hyperrectangles from each subregion. The process is repeated until convergence. Preliminary numerical experiments show that the proposed two-phase heuristic coupled DIRECT method is effective in converging to the optimal solution.FCT – Fundação para a Ciência e Tecnologia within the Projects Scope: UID/CEC/00319/2019 and UID/MAT/ 00013/2013