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A multiple local search algorithm for continuous dynamic optimization

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Abstract

Many real-world optimization problems are dynamic (time dependent) and require an algorithm that is able to track continuously a changing optimum over time. In this paper, we propose a new algorithm for dynamic continuous optimization. The proposed algorithm is based on several coordinated local searches and on the archiving of the optima found by these local searches. This archive is used when the environment changes. The performance of the algorithm is analyzed on the Moving Peaks Benchmark and the Generalized Dynamic Benchmark Generator. Then, a comparison of its performance to the performance of competing dynamic optimization algorithms available in the literature is done. The obtained results show the efficiency of the proposed algorithm.

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Notes

  1. The strategy of DE consists in generating a new position for an individual, according to the differences calculated between other randomly selected individuals.

Abbreviations

\(\alpha \) :

Number of evaluations that makes a time span in the benchmarks

\(A_{i}\) :

Archive of the last \(n_{m}\) initial positions of agents that are created or relocated, in MLSDO

\(A_{m}\) :

Archive of the local optima found by the agents, in MLSDO

\(\Delta _{i}\) :

Size of the interval that defines the search space on the \(i{\mathrm{th}}\) axis in the “non-normalized” basis

\(\delta _{ph}\) :

Parameter of MLSDO that defines the highest precision parameter of the stagnation criterion of the agents local searches

\(\delta _{pl}\) :

Parameter of MLSDO that defines the lowest precision parameter of the stagnation criterion of the agents local searches

\(d\) :

Dimension of the search space

\(\mathbf{D}\) :

Direction vector of the preceding displacement of an agent, in MLSDO

\(\mathbf{D}^{\prime }\) :

Direction vector of the current displacement of an agent, in MLSDO

\(\mathbf{D}_{n}\) :

Direction vector of the displacement of an agent at the \(n{\mathrm{th}}\) step of its local search, in MLSDO

\(\mathbf{e}_{i}\) :

\(i{\mathrm{th}}\) unit vector of the “non-normalized” basis of the search space

\(f(\mathbf{x})\) :

The objective function of a static optimization problem

\(f(\mathbf{x}, t)\) :

The objective function of a dynamic optimization problem

\(f^{*}(t)\) :

Value of the best solution found at the \(t{\mathrm{th}}\) evaluation since the last change in the objective function

\(f_{i}(t)\) :

Value of the best solution found at the \(t{\mathrm{th}}\) evaluation of GDBG

\(f^{*}_{i}(t)\) :

Value of the global optimum at the \(t{\mathrm{th}}\) evaluation of GDBG

\(f^{*}_{j}\) :

Value of the global optimum for the \(j{\mathrm{th}}\) time span in the benchmarks

\(f^{*}_{ji}\) :

Value of the best solution found at the \(i{\mathrm{th}}\) evaluation of the \(j{\mathrm{th}}\) time span of MPB

\(\mathrm{F}_{k}\) :

\(k{\mathrm{th}}\) problem of GDBG

\(\bar{f}^{*}_{x}\) :

Average relative error of the best fitness found at the \(x{\mathrm{th}}\) evaluation of a time span of MPB

\(fitness\) :

Function that returns the value of the objective function of a given solution, in MLSDO

\(g_{k}(\mathbf{x}, t)\) :

The \(k{\mathrm{th}}\) inequality constraint of a dynamic optimization problem

\(h_{j}(\mathbf{x}, t)\) :

The \(j{\mathrm{th}}\) equality constraint of a dynamic optimization problem

\(isNotUpToDate\) :

Flag that indicates if a change in the objective function occurred since the detection of a given stored optimum

\(m\) :

Number of optima currently stored in the archive \(A_{m}\)

\(mark_{max}\) :

Maximal mark that can be obtained on the considered test case of GDBG

\(mark_{pct}\) :

Mark obtained on the considered test case of GDBG

\(max\) :

Function that returns the maximum value among several given values

\(min\) :

Function that returns the minimum value among several given values

\(N\) :

Number of agents currently existing during the execution of MLSDO

\(n_{a}\) :

Parameter of MLSDO that defines the maximum number of “exploring” agents

\(n_{c}\) :

Parameter of MLSDO that defines the maximum number of “tracking” agents created after the detection of a change

\(N_{c}\) :

Number of changes in the benchmarks

\(N_{e}(j)\) :

Evaluations performed on the \(j{\mathrm{th}}\) time span of MPB

\(n_{m}\) :

Capacity of the archives \(A_{i}\) and \(A_{m}\)

\(\mathbf{O}_\mathbf{c}\) :

A newly found optimum

\(oe\) :

Offline error used in MPB

\(op\) :

Overall performance used in GDBG

\(R\) :

Step size of an agent of MLSDO

\(r_{e}\) :

Parameter of MLSDO that defines the exclusion radius of the agents, and the initial step size of “exploring” agents

\(r_{i}(t)\) :

Relative error of the best fitness found at the \(t{\mathrm{th}}\) evaluation of the \(i{\mathrm{th}}\) run of GDBG

\(r_{l}\) :

Parameter of MLSDO that defines the initial step size of “tracking” agents

\(r_{new}\) :

Initial step size of an agent, in MLSDO (can be equal to either \(r_{e}\) or \(r_{l}\))

\(round\) :

Function that rounds a given number to the nearest integer

\(s\) :

Change severity used in MPB

\(\mathbf{S}_\mathbf{c}\) :

Current solution of an agent, in MLSDO

\(\mathbf{S^{\prime }}_\mathbf{c}\) :

Best candidate solution of an agent at the current step of its local search, in MLSDO

\(\mathbf{S}_\mathbf{prev}\) :

A candidate solution generated with \(\mathbf{S}_\mathbf{next}\) in the local search of an agent, in MLSDO

\(\mathbf{S}_\mathbf{new}\) :

The initial solution of the local search of an agent, in MLSDO

\(\mathbf{S}_\mathbf{next}\) :

A candidate solution generated with \(\mathbf{S}_\mathbf{prev}\) in the local search of an agent, in MLSDO

\(\mathbf{S}_\mathbf{w}\) :

Worst candidate solution of an agent at the current step of its local search, in MLSDO

\(t\) :

Number of evaluations performed since the beginning of the tested algorithm

\(\mathrm{T}_{k}\) :

\(k{\mathrm{th}}\) change scenario of GDBG

\(u\) :

Number of equality constraints

\(U\) :

Value of the cumulative dot product used to adapt the step size of an agent, in MLSDO

\(\mathbf{u}_{i}\) :

\(i{\mathrm{th}}\) vector of the “normalized” basis of the search space

\(U_{n}\) :

Value of the cumulative dot product of an agent at the \(n{\mathrm{th}}\) step of its local search, in MLSDO

\(v\) :

Number of inequality constraints

\(\mathbf{x}\) :

A solution in the search space of an optimization problem

\(x_{i}\) :

\(i{\mathrm{th}}\) coordinate of the solution vector \(\mathbf{x}\)

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  1. Laboratoire Images, Signaux et Systèmes Intelligents, LISSI, E.A. 3956, Université Paris-Est Créteil, 61 Avenue du Général de Gaulle, 94010 , Créteil, France

    Julien Lepagnot, Amir Nakib, Hamouche Oulhadj & Patrick Siarry

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  1. Julien Lepagnot
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Lepagnot, J., Nakib, A., Oulhadj, H. et al. A multiple local search algorithm for continuous dynamic optimization. J Heuristics 19, 35–76 (2013). https://doi.org/10.1007/s10732-013-9215-0

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