# On the Behaviour of Differential Evolution for Problems with Dynamic Linear Constraints

###### Abstract

Evolutionary algorithms have been widely applied for solving dynamic constrained optimization problems (DCOPs) as a common area of research in evolutionary optimization. Current benchmarks proposed for testing these problems in the continuous spaces are either not scalable in problem dimension or the settings for the environmental changes are not flexible. Moreover, they mainly focus on non-linear environmental changes on the objective function. While the dynamism in some real-world problems exists in the constraints and can be emulated with linear constraint changes. The purpose of this paper is to introduce a framework which produces benchmarks in which a dynamic environment is created with simple changes in linear constraints (rotation and translation of constraint’s hyperplane). Our proposed framework creates dynamic benchmarks that are flexible in terms of number of changes, dimension of the problem and can be applied to test any objective function. Different constraint handling techniques will then be used to compare with our benchmark. The results reveal that with these changes set, there was an observable effect on the performance of the constraint handling techniques.

Key words— Benchmarking, benchmark generator, constraint handling techniques, dynamic constrained optimization, differential evolution, evolutionary computation

## 1 Introduction

Dynamic constrained optimization problems (DCOPs) in which the objective function or/and the constraints change over time is observed in a variety of real world problems. Examples include hydro-thermal power scheduling [de2008plant], source identification [liu2008adaptive], and parameter estimation [prata2006simultaneous] in which the dynamism arise because the available resources or demand vary over time, the information about the problem is gradually revealed, or parameter tuning is needed as time passes.

Multiple evolutionary algorithms have been designed so far to solve these problems [ECDCOPs, Nguyen20121, Yaneli2016]. The focus of these papers are either on dealing with dynamism in the environment including introducing [Goh_2009] or maintaining diversity [Bui2005], memory-based approaches [Richter2013] and multi-population approaches [branke2000multi], or mechanisms to deal with constraints including penalty [CEC09], repair methods [Das, bu2017continuous, nguyen2012continuous] and feasibility rules [Yaneli2016]. In addition, some papers enhanced both constraint handling and dynamic handling mechanisms [Nguyen20121]. Among the many evolutionary algorithms, DE has showed competitive results in dynamic and constrained optimization problems so far [ameca2018comparison].

However, besides to developing algorithms there should be a comprehensive benchmark suit that can test algorithms considering a range of characteristics. Although there are a range of benchmarks proposed to test the relevant algorithms for discrete spaces [roostapour2018performance], and/or multi-objective optimization in dynamic environments [jiang2017evolutionary], for continuous spaces in single objective optimization so far, the most used benchmark is the proposed benchmark in [Nguyen20121]. In this benchmark, the dynamic changes are applied by adding time-dependent terms to the objective function and the constraints of one of the functions () of the static benchmark proposed in CEC 2006 [liang2006problem]. However, there are parameters defined to alter the severity of changes in the environment, this benchmark is based on one objective function and the transformation of this function and is not applicable to test different functions to consider a range of characteristics. Moreover, the proposed problem is a two-dimensional in size and is not flexible to be applied for larger dimension of the problem. In addition, the feasible regions of the dynamic constraint function in this benchmark are very large, which might not be sufficiently complicated, Bu et all in [bu2017continuous], introduces two variants of this benchmark suit that have a parameter to control the size and the number of the feasible regions.

A similar benchmark is proposed in [zhang2014danger] that is based on dynamic transformations introduced by Nguyen in [Nguyen20121]. However, the problem information, including the number of feasible regions, the global optimum, and the dynamics of each feasible region, is lacking. The lack of such information makes it difficult to measure and analyze the performance of an algorithm and probably this is the reason this benchmark have become less popular than Nguyen benchmark [Nguyen20121].

In terms of having a scalable and flexible benchmark, in the literature there are some benchmark generators proposed. Like in [li2008benchmark] a dynamic benchmark generator is proposed that is designed with the idea of constructing dynamic environments across binary, real, and combinatorial solution spaces. The dynamism is obtained by tuning some system control parameters, creating six change types: small step, large step, random, chaotic, recurrent, and recurrent change with noise.

While the aforementioned benchmark generator’s main focus is on creating dynamic objective functions, in this paper we put our focus on creating dynamism in the constraints. Our motivation come from characteristic of some real-world problems like scheduling power system problem having dynamic linear constraints (due to the variable demand and available resources over-time). For a better insight about the effects of constraint changes we keep the objective function static. Indeed, this is the case in some real world problems in which only constraints will change like the problem of hydro-thermal power scheduling in continuous spaces [deb2007dynamic] or the ship scheduling problem in discrete spaces [mertens2006dyncoaa].

Dynamic changes are imposed by the translation and rotation of the constraint’s hyperplane. The examples of these two operations on the constraint in a real-world dynamic environment are: the reduction and increment of demand that happens regularly at power system (hyperplane translation) or changes on the share of each plant power production (hyperplane rotation).

Our proposed benchmark generator is flexible (frequency and severity of changes, number of environmental changes, and dimension of the problem), simple to implement (with any objective function), analyze, or evaluate and computationally efficient and finally allows conjectures to real-world problems.

In the experiments we apply differential evolution (DE) algorithm with different constraint handling techniques and observe how they deal with these changes depending on the magnitude and frequency of changes.

Our experiments are repeated across some well-known functions including sphere, Rastrigin, Ackley and Rosenbrock. For the analysis on the performance of the tested algorithms, a ranking procedure is introduced that uses the values of the objective function and the constraint violations to rank the performance of the algorithms. In addition, the common measure modified offline error is also evaluated for the experiments and the results are investigated. The results reveal that the changes on frequency and hyperplane rotation and translation have a direct correlation with the performance of the constraint handling techniques. Therefore with imposing simple linear changes we could effectively put the algorithms to struggle and test their performance.

The outline of the paper is as follows. Section 2 introduces a short overview of the problem statement. In Section LABEL:sec:change-setup, our proposed dynamic changes’ framework is described. Experimental investigations will be presented in Section LABEL:sec:Experiment and finally in Section LABEL:sec:conclusion conclusions and future work are summarized.

## 2 Problem statement

In this section a general overview of the problem statement is presented.

### 2.1 Dynamic constraint optimization problems

A dynamic constrained optimization problem (DCOP) is an optimization problem where the objective function and/or the constraints can change over time [Nguyen20121]. Such an optimization problem ideally must be solved at every time instant or whenever there is a change in any of the objective function and/or the constraints with . In such optimization problems, the time parameter can be mapped with the iteration counter of the optimization algorithm. Such problems often arise in real-world problem solving, particularly in optimal control problems or problems requiring an on-line optimization [de2008plant].

Generally, the problem statement in DCOPs can be defined as follows.

Find , at each time , which:

(1) |

where is a single objective function, is a solution vector and is the current time,

(2) |

is called the search space (), where and are the lower and upper boundaries of the th variable,

subject to: