Citation: Rosso, M.M.; Cucuzza, R.;
Aloisio, A.; Marano, G.C. Enhanced
Multi-Strategy Particle Swarm
Optimization for Constrained
Problems with an Evolutionary-
Strategies-Based Unfeasible Local
Search Operator. Appl. Sci. 2022, 12,
2285. https://doi.org/10.3390/
app12052285
Academic Editors: Nikos D. Lagaros,
Vagelis Plevris and Jong Wan Hu
Received: 13 January 2022
Accepted: 15 February 2022
Published: 22 February 2022
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Article
Enhanced Multi-Strategy Particle Swarm Optimization for
Constrained Problems with an Evolutionary-Strategies-Based
Unfeasible Local Search Operator
Marco Martino Rosso
1
, Raffaele Cucuzza
1,
* , Angelo Aloisio
2
and Giuseppe Carlo Marano
1
1
DISEG, Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino,
Corso Duca degli Abruzzi, 24, 10128 Turin, Italy; marco.rosso@polito.it (M.M.R.);
giuseppe.marano@polito.it (G.C.M.)
2
Civil Environmental and Architectural Engineering Department, Università degli Studi dell’Aquila,
Via Giovanni Gronchi n.18, 67100 L’Aquila, Italy; angelo.aloisio1@univaq.it
* Correspondence: raffaele.cucuzza@polito.it
Abstract:
Nowadays, optimization problems are solved through meta-heuristic algorithms based
on stochastic search approaches borrowed from mimicking natural phenomena. Notwithstanding
their successful capability to handle complex problems, the No-Free Lunch Theorem by Wolpert and
Macready (1997) states that there is no ideal algorithm to deal with any kind of problem. This issue
arises because of the nature of these algorithms that are not properly mathematics-based, and the
convergence is not ensured. In the present study, a variant of the well-known swarm-based algorithm,
the Particle Swarm Optimization (PSO), is developed to solve constrained problems with a different
approach to the classical penalty function technique. State-of-art improvements and suggestions
are also adopted in the current implementation (inertia weight, neighbourhood). Furthermore, a
new local search operator has been implemented to help localize the feasible region in challenging
optimization problems. This operator is based on hybridization with another milestone meta-heuristic
algorithm, the Evolutionary Strategy (ES). The self-adaptive variant has been adopted because of its
advantage of not requiring any other arbitrary parameter to be tuned. This approach automatically
determines the parameters’ values that govern the Evolutionary Strategy simultaneously during the
optimization process. This enhanced multi-strategy PSO is eventually tested on some benchmark
constrained numerical problems from the literature. The obtained results are compared in terms of
the optimal solutions with two other PSO implementations, which rely on a classic penalty function
approach as a constraint-handling method.
Keywords:
particle swarm optimization (PSO); multi-strategy PSO; self-adaptive evolutionary
strategies (ES); local search operator; constraints handling
1. Introduction
In optimization problems, the aim is optimizing certain mathematical functions, called
Objective Functions (OF)
f (x)
. These problems can be divided into single-objective or
multi-objective problems, depending on the number of OFs, and a further subdivision for
single-objective problems is based on the presence of constraints. Unconstrained problems
are defined as:
min
x∈Ω
{f (x)} (1)
meanwhile, constrained problems are defined as:
min
x∈Ω
{f ( x)}
s.t. g
q
(x) ≤ 0 ∀q = 1, . . . , n
q
h
r
(x) = 0 ∀r = 1, . . . , n
r
(2)
Appl. Sci. 2022, 12, 2285. https://doi.org/10.3390/app12052285 https://www.mdpi.com/journal/applsci
