Continuous relaxation of MINLP problems by penalty functions: a practical comparison

A practical comparison of penalty functions for globally solving mixed-integer nonlinear programming (MINLP) problems is presented. The penalty approach relies on the continuous relaxation of the MINLP problem by adding a specific penalty term to the objective function. A new penalty algorithm that addresses simultaneously the reduction of the error tolerances for optimality and feasibility, as well as the reduction of the penalty parameter, is designed. Several penalty terms are tested and different penalty parameter update schemes are analyzed. The continuous nonlinear optimization problem is solved by the deterministic DIRECT optimizer. The numerical experiments show that the quality of the produced solutions are satisfactory and that the selected penalties have different performances in terms of efficiency and robustness.This work has been supported by COMPETE: POCI-01-0145-FEDER-007043 and FCT - Fundação para a Ciência e Tecnologia, within the projects UID/CEC/00319/2013 and UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

Penalty-based heuristic direct method for constrained global optimization

This paper is concerned with an extension of the heuristic DIRECT method, presented in[8], to solve nonlinear constrained global optimization (CGO) problems. Using a penalty strategy based on a penalty auxiliary function, the CGO problem is transformed into a bound constrained problem. We have analyzed the performance of the proposed algorithm using fixed values of the penalty parameter, and we may conclude that the algorithm competes favourably with other DIRECT-type algorithms in the literature.The authors wish to thank two anonymous referees for their comments and suggestions to improve the paper. This work has been supported by FCT – Fundação para a Ciência e Tecnologia within the R&D Units Project Scope: UIDB/00319/2020, UIDB/00013/2020 and UIDP/00013/2020 of CMAT-UM

Theoretical and practical convergence of a self-adaptive penalty algorithm for constrained global optimization

This paper proposes a self-adaptive penalty function and presents a penalty-based algorithm for solving nonsmooth and nonconvex constrained optimization problems. We prove that the general constrained optimization problem is equivalent to a bound constrained problem in the sense that they have the same global solutions. The global minimizer of the penalty function subject to a set of bound constraints may be obtained by a population-based meta-heuristic. Further, a hybrid self-adaptive penalty firefly algorithm, with a local intensification search, is designed, and its convergence analysis is established. The numerical experiments and a comparison with other penalty-based approaches show the effectiveness of the new self-adaptive penalty algorithm in solving constrained global optimization problems.The authors would like to thank the referees, the Associate Editor and the Editor-in-Chief for their valuable comments and suggestions to improve the paper. This work has been supported by COMPETE: POCI-01-0145-FEDER-007043 and FCT - Funda¸c˜ao para a Ciˆencia e Tecnologia within the projects UID/CEC/00319/2013 and UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

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

Filter-based stochastic algorithm for global optimization

We propose the general Filter-based Stochastic Algorithm (FbSA) for the global optimization of nonconvex and nonsmooth constrained problems. Under certain conditions on the probability distributions that generate the sample points, almost sure convergence is proved. In order to optimize problems with computationally expensive black-box objective functions, we develop the FbSA-RBF algorithm based on the general FbSA and assisted by Radial Basis Function (RBF) surrogate models to approximate the objective function. At each iteration, the resulting algorithm constructs/updates a surrogate model of the objective function and generates trial points using a dynamic coordinate search strategy similar to the one used in the Dynamically Dimensioned Search method. To identify a promising best trial point, a non-dominance concept based on the values of the surrogate model and the constraint violation at the trial points is used. Theoretical results concerning the sufficient conditions for the almost surely convergence of the algorithm are presented. Preliminary numerical experiments show that the FbSA-RBF is competitive when compared with other known methods in the literature.The authors are grateful to the anonymous referees for their fruitful comments and suggestions.The first and second authors were partially supported by Brazilian Funds through CAPES andCNPq by Grants PDSE 99999.009400/2014-01 and 309303/2017-6. The research of the thirdand fourth authors were partially financed by Portuguese Funds through FCT (Fundação para Ciência e Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020 of CMAT-UM and UIDB/00319/2020

On a smoothed penalty-based algorithm for global optimization

This paper presents a coercive smoothed penalty framework for nonsmooth and nonconvex constrained global optimization problems. The properties of the smoothed penalty function are derived. Convergence to an ε -global minimizer is proved. At each iteration k, the framework requires the ε(k) -global minimizer of a subproblem, where ε(k)→ε . We show that the subproblem may be solved by well-known stochastic metaheuristics, as well as by the artificial fish swarm (AFS) algorithm. In the limit, the AFS algorithm convergence to an ε(k) -global minimum of the real-valued smoothed penalty function is guaranteed with probability one, using the limiting behavior of Markov chains. In this context, we show that the transition probability of the Markov chain produced by the AFS algorithm, when generating a population where the best fitness is in the ε(k)-neighborhood of the global minimum, is one when this property holds in the current population, and is strictly bounded from zero when the property does not hold. Preliminary numerical experiments show that the presented penalty algorithm based on the coercive smoothed penalty gives very competitive results when compared with other penalty-based methods.The authors would like to thank two anonymous referees for their valuable comments and suggestions to improve the paper. This work has been supported by COMPETE: POCI-01-0145-FEDER-007043 and FCT - Fundac¸ao para a Ci ˜ encia e Tecnologia within the projects UID/CEC/00319/2013 and ˆ UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

Solving large 0–1 multidimensional knapsack problems by a new simplified binary artificial fish swarm algorithm

The artificial fish swarm algorithm has recently been emerged in continuous global optimization. It uses points of a population in space to identify the position of fish in the school. Many real-world optimization problems are described by 0-1 multidimensional knapsack problems that are NP-hard. In the last decades several exact as well as heuristic methods have been proposed for solving these problems. In this paper, a new simpli ed binary version of the artificial fish swarm algorithm is presented, where a point/ fish is represented by a binary string of 0/1 bits. Trial points are created by using crossover and mutation in the different fi sh behavior that are randomly selected by using two user de ned probability values. In order to make the points feasible the presented algorithm uses a random heuristic drop item procedure followed by an add item procedure aiming to increase the profit throughout the adding of more items in the knapsack. A cyclic reinitialization of 50% of the population, and a simple local search that allows the progress of a small percentage of points towards optimality and after that refines the best point in the population greatly improve the quality of the solutions. The presented method is tested on a set of benchmark instances and a comparison with other methods available in literature is shown. The comparison shows that the proposed method can be an alternative method for solving these problems.The authors wish to thank three anonymous referees for their comments and valuable suggestions to improve the paper. The first author acknowledges Ciˆencia 2007 of FCT (Foundation for Science and Technology) Portugal for the fellowship grant C2007-UMINHO-ALGORITMI-04. Financial support from FEDER COMPETE (Operational Programme Thematic Factors of Competitiveness) and FCT under project FCOMP-01-0124-FEDER-022674 is also acknowledged

Filter-based DIRECT method for constrained global optimization

This paper presents a DIRECT-type method that uses a filter methodology to assure convergence to a feasible and optimal solution of nonsmooth and nonconvex constrained global optimization problems. The filter methodology aims to give priority to the selection of hyperrectangles with feasible center points, followed by those with infeasible and non-dominated center points and finally by those that have infeasible and dominated center points. The convergence properties of the algorithm are analyzed. Preliminary numerical experiments show that the proposed filter-based DIRECT algorithm gives competitive results when compared with other DIRECT-type methods.The authors would like to thank two anonymous referees and the Associate Editor for their valuable comments and suggestions to improve the paper. This work has been supported by COMPETE: POCI-01-0145-FEDER-007043 and FCT - Fundac¸ao para a Ciência e Tecnologia within the projects UID/CEC/00319/2013 and ˆ UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

Myofascial trigger points in cluster headache patients: a case series

Active myofascial trigger points (MTrPs) have been found to contribute to chronic tension-type headache and migraine. The purpose of this case series was to examine if active trigger points (TrPs) provoking cluster-type referred pain could be found in cluster headache patients and, if so, to evaluate the effectiveness of active TrPs anaesthetic injections both in the acute and preventive headache's treatment. Twelve patients, 4 experiencing episodic and 8 chronic cluster headache, were studied. TrPs were found in all of them. Abortive infiltrations could be done in 2 episodic and 4 chronic patients, and preemptive infiltrations could be done in 2 episodic and 5 chronic patients, both kind of interventions being successful in 5 (83.3%) and in 6 (85.7%) of the cases respectively. When combined with prophylactic drug therapy, injections were associated with significant improvement in 7 of the 8 chronic cluster patients. Our data suggest that peripheral sensitization may play a role in cluster headache pathophysiology and that first neuron afferent blockade can be useful in cluster headache management