Abstract
<inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> BB is an elegant deterministic branch and bound global optimisation that guarantees global optimum convergence with minimal parameter tuning. However, the method suffers from a slow convergence speed calling for computational improvements in several areas. The current paper proposes hybridising the branch and bound process with particle swarm optimisation to improve its global convergence speed when solving twice differentiable (<inline-formula> <tex-math notation="LaTeX">$C^{2}$ </tex-math></inline-formula>) box-constrained multimodal functions. This hybridisation complemented with interval analysis leads to an early discovery of the global optimum, quicker pruning of suboptimal regions in the problem space, thus improving global convergence. Also, when used as a heuristic search algorithm, the hybrid algorithm yields superior solution accuracy owing to the combined search capabilities of PSO and the branch and bound framework. Computational experiments have been conducted on CEC 2017/2019 test sets and on n-dimensional classical test sets yielding improved convergence speed in the complete search configuration and superior solution accuracy in the heuristic search configuration.
Highlights
Deterministic global optimisation techniques are rigorous and complete search algorithms [1] that aims to find an −accurate global optimum solution in finite time unlike its stochastic counterpart
The study proposes an efficient use of particle swarm optimisation (PSO) as an upper bound solver in the BB-procedure to allow early discovery of the true optimum, which could lead to early pruning of suboptimal regions and speed up global convergence provided the availability of tight bounds supplemented by interval analysis
The current study extends from the work of [35] and proceeds with the hybridisation of αBB, a problem-agnostic generic branch and bound framework for twice differentiable problems with guaranteed −global optimal verification where PSO is used as a substitute for the upper bound solver of classical αBB inclusive of additional optimisation in a bid to improve the computation time performance of the αBB as a unit as well as guarantee global convergence of PSO
Summary
Deterministic global optimisation techniques are rigorous and complete search algorithms [1] that aims to find an −accurate global optimum solution in finite time unlike its stochastic counterpart (particle swarm optimisation [2], genetic algorithm [3], differential evolution [4], simulated annealing [5], ant colony optimisation [6] and other competing techniques [7]). The study proposes an efficient use of particle swarm optimisation (PSO) as an upper bound solver in the BB-procedure to allow early discovery of the true optimum, which could lead to early pruning of suboptimal regions and speed up global convergence provided the availability of tight bounds supplemented by interval analysis. Deterministic global optimisation typically proceeds by an exhaustive partitioning of the solution space in which upper and lower bounding of the problem inner regions anticipates pruning of sub-optimal areas in the search for the −global optimum. This divide and conquer approach is performed within a branch and bound framework [8].
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