Handling time-varying constraints and objectives in dynamic evolutionary multi-objective optimization

doi:10.1016/j.swevo.2017.10.005

Swarm and Evolutionary Computation

Volume 39, April 2018, Pages 222-248

https://doi.org/10.1016/j.swevo.2017.10.005 Get rights and content

Abstract

Recently, several researchers within the evolutionary and swarm computing community have been interested in solving dynamic multi-objective problems where the objective functions, the problem's parameters, and/or the constraints may change over time. According to the related literature, most works have focused on the dynamicity of objective functions, which is insufficient since also constraints may change over time along with the objectives. For instance, a feasible solution could become infeasible after a change occurrence, and vice versa. Besides, a non-dominated solution may become dominated, and vice versa. Motivated by these observations, we devote this paper to focus on the dynamicity of both: (1) problem's constraints and (2) objective functions. To achieve our goal, we propose a new self-adaptive penalty function and a new feasibility driven strategy that are embedded within the NSGA-II and that are applied whenever a change is detected. The feasibility driven strategy is able to guide the search towards the new feasible directions according to the environment changes. The empirical results have shown that our proposal is able to handle various challenges raised by the problematic of dynamic constrained multi-objective optimization. Moreover, we have compared our new dynamic constrained NSGA-II version, denoted as DC-MOEA, against two existent dynamic constrained evolutionary algorithms. The obtained results have demonstrated the competitiveness and the superiority of our algorithm on both aspects of convergence and diversity.

Introduction

Dynamic Multi-objective Optimization Problems (DMOPs) involve the simultaneous optimization of several objectives subject to a number of constraints where the objective functions, constraints and/or problem parameters may vary over the course of time. Dealing with such problems impose several challenges. In fact, the optimal Pareto Front (PF) and the optimal Pareto Set (PS) may change over time. Moreover, as the constraints are also time-dependent, the feasibility area would not remain unchanged. Such complex dynamic behaviors make the task of optimization more difficult. Therefore, solving constrained DMOPs needs to handle essentially the following issues: (1) evolving a near-optimal and diverse PF, (2) preserving the required diversity to ensure a high level of adaptability, and (3) ensuring a good trade-off between feasible and infeasible solutions to converge towards simultaneously, feasibility and optimality.

Evolutionary Algorithms (EAs) have been used to solve DMOPs, where their earliest state of the art application to dynamic environments dates from 1966 [1]. However, this subject has not gained a significant attention until the 1980s. A number of approaches have been suggested such as diversity introduction-based approaches [2], change predictive approaches [3], [4], [5], memory-based approaches [6], [7], parallel approaches [8], and approaches that convert DMOP into multiple static Multi-objective Optimization Problems (MOPs) [9]. The majority of these works focused only on unconstrained problems. Nevertheless, in real world, we usually confront problems that not only need to optimize a number of conflictual goals but also have a set of constraint conditions to be satisfied. Referring to the literature, despite the increasing interest for the use of EAs to solve DMOPs, their application to the constrained ones is not yet largely explored [10]. Encouraged by the significant importance of dynamic constrained optimization problems in practice [11] and by this lack of attention on solving constrained DMOPs using EAs, in this paper:

•
we develop a new self-adaptive penalty function that, in addition to feasible solutions, it makes use of infeasible solutions having high fitness values in an effective and efficient manner. This helps increasing population diversity and avoiding to be trapped on local optimal PFs;
•
as constraints are also time-dependent, we propose a feasibility driven strategy that we apply at each change detection. This strategy is able to guide the search towards the new feasible regions according to environment changes; and
•
we integrate the proposed penalty function and feasibility driven strategy within the NSGA-II. The performance of the resulting algorithm, called DC-MOEA, is evaluated on various constrained dynamic benchmark problems characterized by different challenging difficulties. Our proposal is also compared against two algorithms. The first one is a well known dynamic EA [2] that has demonstrated the ability to handle constrained as well as unconstrained problems. The second one is among the most recent dynamic constrained EAs [10].

The rest of this paper is structured as follows. Section 2 presents the background definitions while Section 3 introduces the motivations behind our work. Section 4 gives an overview of the related work. Section 5 presents the proposed algorithm and describes its building-blocks. The experimental study is provided in Section 6. In Section 7 we discuss the possible threats to validity of the empirical results. Finally, Section 8 concludes the paper and presents future perspectives.

Section snippets

Background definitions

Definition 1

Dynamic Multi-objective Optimization Problem. Considering a minimization problem, a DMOP can be formally defined as follows: $\begin{align} Min F (x, t) = {f_{1} (x, t), f_{2} (x, t), \dots, f_{M} (x, t)} \\ s . t . g_{i} (x, t) \leq 0, i = 1, 2, \dots, p \\ a n d h_{j} (x, t) = 0, j = 1, 2, \dots, q \end{align}$ where $t$ represents the time variable; $x = (x_{1}, x_{2}, \dots, x_{n}) {\in R}^{n}$ is $n$ dimension decision vector; $S$ is the set of all feasible solutions and $f_{k}$ are objectives to be minimized for $k = 1, 2, \dots, M$ depending on $t$ . $g_{i} (x, t) \leq 0$ are inequality constraints for $i = 1, 2, \dots, p$ while $h_{j} (x, t)$ are equality constraints for $j = 1,$

Overview and motivations

In daily life we often encounter dynamic problems that have a set of constraints to be satisfied and so belong to the category of constrained DMOPs. For example, we can cite dynamic scheduling problems [12], transportation problems, chemical engineering design like the dynamic hydro-thermal power scheduling problem, optimal control problems, portfolio investment and so forth. Due to the significant and growing importance of this kind of problems in practice, we tackle in this paper the issue of

Literature review

Dynamic constrained multi-objective optimization is found at the intersection of dynamic multi-objective optimization and constrained optimization research fields. In this section, we present the literature review related to both research fields and we discuss the lack of focus on constrained DMOPs.

Basic idea

In this paper, we proposed a dynamic EA based on the framework of NSGA-II to conduct crossover, mutation and selection operators. On one hand, to address the constraints issue, we choose to use a penalty-based approach. In fact, we propose a new dynamic and self-adaptive penalty function which, in addition to feasible solutions, makes use of infeasible ones having high fitness values in an effective and efficient manner. This amount of penalty added to the original objective function values

Dynamic constrained benchmark functions

We opted in this paper to the use of a set of dynamic constrained benchmark functions that we have proposed in [10] and denoted as DCTPs. These test problems are characterized by simultaneously time dependent PF, PS and constraints. These characteristics challenge the optimization algorithm much more than dynamic unconstrained test problems. Moreover, as presented in [10], DCTPs present two kinds of tunable difficulties: (1) difficulty in the proximity of the optimal PF where constraints make

Threats to validity discussion

This section is devoted to discuss the possible threats to validity of the empirical results:

•
The conclusion validity: It is concerned with the statistically significant relationship between the treatment and the outcome. Since all compared algorithms used randomly generated initial populations and they are based on a set of stochastic operators, it is obvious that it is not fair enough to draw conclusions from a single run of each algorithm. Therefore, all experiments were executed several

Conclusion and future works

Encouraged by the growing and significant importance of constrained DMOPs in practice, we have developed in this work a dynamic EA based on a new self-adaptive penalty function that effectively and efficiently makes use of infeasible solutions having high fitness values. The proposed penalty function gives attention to infeasible solutions with the goal of evolving them towards feasibility while evolving feasible solutions towards optimality. Moreover, we proposed a feasibility driven strategy

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View full text

Handling time-varying constraints and objectives in dynamic evolutionary multi-objective optimization

Abstract

Introduction

Section snippets

Background definitions

Overview and motivations

Literature review

Basic idea

Dynamic constrained benchmark functions

Threats to validity discussion

Conclusion and future works

Artificial Intelligence through Simulated Evolution

Dynamic multi-objective optimization and decision-making using modified nsga-ii: a case study on hydro-thermal power scheduling

Dynamic multi-objective optimization with evolutionary algorithms: a forward-looking approach

Topology of anticipatory populations for evolutionary dynamic multi-objective optimization

A predictive gradient strategy for multi-objective evolutionary algorithms in a fast changing environment

Memet. Comput.

A competitive-cooperative coevolutionary paradigm for dynamic multi-objective optimization

IEEE Trans. Evol. Comput.

Dynamic multi-objective optimization based on membrane computing for control of time-varying unstable plants

Inf. Sci.

Parallel processing for multi-objective optimization in dynamic environments

An evolutionary algorithm for dynamic multi-objective optimization

Appl. Math. Comput.

Multi-objective optimization with dynamic constraints and objectives: new challenges for evolutionary algorithms

Dynamic constrained optimization with offspring repair based gravitational search algorithm

A prediction-based online soft scheduling algorithm for the real-world steelmaking-continuous casting production

Knowledge-Based Syst.

Evolutionary algorithms for constrained parameter optimization problems

Evol. Comput.

Constraint handling improvements for multiobjective genetic algorithms

Struct. Multidiscip. Optim.

An Evolutionary Algorithm for Constrained Multiobjective for Optimization Problems

Mobes: a multiobjective evolution strategy for constrained optimization problems

Constrained multi-objective optimization using steady state genetic algorithms

Constraint handling in multi-objective evolutionary optimization

IEEE Trans. Evol. Comput.

Moea/d for constrained multiobjective optimization: some preliminary experimental results

A modified objective function method with feasible-guiding strategy to solve constrained multi-objective optimization problems

Appl. Soft Comput.

Constraint-handling method for multi-objective function optimization: Pareto descent repair operator

Infeasibility driven evolutionary algorithm for constrained optimization

Multiobjective optimization and multiple constraint handling with evolutionary algorithms: parti: a unified formulation

IEEE Trans. Syst. Cybern. Part A Syst. Humans

Treating constraints as objectives in multiobjective optimization problems using niched Pareto genetic algorithm

IEEE Trans. Magn.

A decomposition-based constraint-handling technique for constrained multi-objective optimization

Using genetic algorithms in engineering design optimization with non-linear constraints

Multiobjective design optimization by an evolutionary algorithm

Eng. Optim.

A fast and elitist multiobjective genetic algorithm: nsga-ii

IEEE Trans. Evol. Comput.

An adaptive stochastic ranking mechanism in moea/d for constrained multi-objective optimization

Constrained multi-objective optimization algorithm with an ensemble of constraint handling methods

Eng. Optim.

Bstbga: a hybrid genetic algorithm for constrained multi-objective optimization problems

Comput. Oper. Res.