On parallel Branch and Bound frameworks for Global Optimization

Branch and Bound (B&B) algorithms are known to exhibit an irregularity of the search tree. Therefore, developing a parallel approach for this kind of algorithms is a challenge. The efficiency of a B&B algorithm depends on the chosen Branching, Bounding, Selection, Rejection, and Termination rules. The question we investigate is how the chosen platform consisting of programming language, used libraries, or skeletons influences programming effort and algorithm performance. Selection rule and data management structures are usually hidden to programmers for frameworks with a high level of abstraction, as well as the load balancing strategy, when the algorithm is run in parallel. We investigate the question by implementing a multidimensional Global Optimization B&B algorithm with the help of three frameworks with a different level of abstraction (from more to less): Bobpp, Threading Building Blocks (TBB), and a customized Pthread implementation. The following has been found. The Bobpp implementation is easy to code, but exhibits the poorest scalability. On the contrast, the TBB and Pthread implementations scale almost linearly on the used platform. The TBB approach shows a slightly better productivity

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

BASBL: Branch-And-Sandwich BiLevel solver. Implementation and computational study with the BASBLib test set

We describe BASBL, our implementation of the deterministic global optimization algorithm Branch-and-Sandwich for a general class of nonconvex/nonlinear bilevel problems, within the open-source MINOTAUR framework. The solver incorporates the original Branch-and-Sandwich algorithm and modifications proposed in (Paulavičius and Adjiman, J. Glob. Opt., 2019, Submitted). We also introduce BASBLib, an extensive online library of bilevel benchmark problems collected from the literature and designed to enable contributions from the bilevel optimization community. We use the problems in the current release of BASBLib to analyze the performance of BASBL using different algorithmic options and we identify a set of default options that provide good overall performance. Finally, we demonstrate the application of BASBL to a set of flexibility index problems including linear and nonlinear constraints

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