Stopping criteria for MAPLS-AW, a hybrid multi-objective evolutionary algorithm

Abstract

Evolutionary algorithms are widely used to solve multi-objective optimization problems effectively by performing global search over the solution space to find better solutions. Hybrid evolutionary algorithms have been introduced to enhance the quality of solutions obtained. One such hybrid algorithm is memetic algorithm with preferential local search using adaptive weights (MAPLS-AW) (Bhuvana and Aravindan in Soft Comput, doi:10.1007/s00500-015-1593-9, 2015). MAPLS-AW, a variant of NSGA-II algorithm, recognizes the elite solutions of the population and preferences are given to them for local search during the evolution. This paper proposes a termination scheme derived from the features of MAPLS-AW. The objective of the proposed scheme is to detect convergence of population without compromising quality of solutions generated by MAPLS-AW. The proposed termination scheme consists of five stopping measures, among which two are newly proposed in this paper to predict the convergence of the population. Experimental study has been carried out to analyze the performance of the proposed termination scheme and to compare with existing termination schemes. Several constrained and unconstrained multi-objective benchmark test problems are used for this comparison. Additionally, a real-time application economic emission and load dispatch has also been used to check the performance of the proposed scheme. The results show that the proposed scheme identifies convergence of population much earlier than the existing stopping schemes without compromising the quality of solutions.

Appendix

Here, we present memetic algorithm with preferential local search using adaptive weights for multi-objective optimization problems, which is reported in citation (Bhuvana and Aravindan 2015). It is included here to make this paper self-contained.

1.1 Design of memetic algorithm

The goal of memetic algorithm is to generate quality solutions by combining global and local search together. The purpose of combining two different search processes is to perform exploration of the search space and exploitation of the neighborhood locality. Two major proposals in combining two search heuristics are adaptive weight assignment scheme for performing local search and preferential local search. These two key ideas can be incorporated into any of the global search algorithms to arrive at a new memetic algorithm. We have integrated the above two approaches into NSGA-II arriving at a new algorithm, MAPLS-AW. Balance should be maintained, when a global search is integrated with a local search process. Our proposed algorithm maintains that balance between exploration and exploitation through preferential local search and by using adaptive weight assignment scheme. Due to this, the explicit exploration parameter, $\eta _c,$ in simulated binary crossover used by NSGA-II is not needed. We have eliminated the need of that parameter in our proposed algorithm and kept $\eta _c$ as a constant. In the following subsections we have introduced the working of adaptive weight assignment and preferential local search.

1.2 Adaptive weights

Using the weighted sum method, multi-objectives are combined together into single objective before the local optimization process is performed. This requires the weights to be assigned to the functional objectives. An adaptive weights mechanism has been introduced in this work that dynamically adapts weights for the objectives.

We assume that the objectives are minimizing ones and this is not a limitation, since any minimization problem can be converted into maximization problem.

Uniformly distributed optimal solutions are the solutions expected out of the evolutionary process, where providing equal weights will affect the diversity of solutions obtained. Equal weights never explore the extreme regions of the optimal front. Hence, we decided to provide weights for the multiple objectives in an adaptive manner. Classical aggregation techniques aggregate the objectives before the evolution begins and also does it only once. We are proposing a new aggregation method that aggregates during evolution and it is prudent to use information available in the objective space.

Adaptive weights are assigned by collecting information about a solution from its multidimensional objective space. Weights computed from these functional values will keep on changing during the course of the evolutionary process.

The aim is to provide lesser preference for any objective which has larger functional value in a minimization problem. Higher preference can be given to sustain a lesser functional objective and vice versa. While associating weight in this manner to one objective function, we need to consider other objectives at the same time. If we do not consider other objectives, then the evolution may take the solutions toward one particular region and make them crowded. This will affect the diversity of optimal solutions. Instead, we need to move an objective functional value toward its optimal minimum with respect to every other functional objectives in the search space. By this, a solution is shifted proportionately toward the Pareto optimal. Proportionate movement in the objective space is achieved with the help of the Euclidean norm.

If $f_i^{(x)}$ is the $i$th functional objective of a solution x, then the proportionate movement is given by $\omega _i$.

$$\begin{aligned} \omega _i= \frac{f_i^{(x)}}{\Vert f^{(x)}\Vert } \end{aligned}$$

(22)

where $\Vert f^{(x)}\Vert$, the euclidean norm is given by

$$\begin{aligned} \Vert f^{(x)}\Vert ~=~\sqrt{f_1^{2^{(x)}}+f_2^{2^{(x)}}+f_3^{2^{(x)}}+\cdots +f_M^{2^{(x)}}}. \end{aligned}$$

(23)

Since the sum of weights in the weighted sum aggregation approach should be equal to 1, the following scaling down of $\omega _i$ is done to become the adaptive individual weight of the objectives of a solution:

$$\begin{aligned} \alpha _i= \frac{\omega _i}{\sum _{i=1}^M \omega _i}. \end{aligned}$$

(24)

Once the individual weights are determined for all the objectives, they are combined together into a single objective F and is given by

$$\begin{aligned} F=\alpha _1f_1+\alpha _2f_2+\alpha _3f_3+\cdots +\alpha _Mf_M, \end{aligned}$$

(25)

where the sum of the weights, $\alpha _1+\alpha _2+\alpha _3+\cdots +\alpha _M=1$. Local search applied after such dynamic weight adaptation will overcome the drawbacks, such as convergence time, taking optimization in the wrong direction and diversity of obtained optimal solutions.

1.3 Preferential local search (PLS)

The objective of integrating a local search in a global search process is to enhance the quality of solutions by fine-tuning them. To establish a balance between exploration and exploitation, we are proposing PLS. PLS addresses the issues related to combining global and local searches together. Issues addressed by PLS are choosing individuals for local search, deciding the depth of local search and determining the frequency of local search.

1.3.1 Choosing individuals for LS

Time incurred in allowing all the individuals for local search adds to the complexity of the memetic algorithm. Decision should be made to selectively allow a few for the local search. An individual in the population can be passed on to the next generation only when it has potential enough to survive and compete with other peer solutions and offspring. In any $i$th generation where the population is a mix of varied set of solutions, PLS identifies the elite solutions. Preference can be given to such elite solutions to undergo local search. This kind of preferences to good solutions strengthen them to counter new offspring.

The offspring that are generated after the genetic operations may lose their chance in the evolution when they compete with the potential parents. PLS selects the new offspring for depth-limited local search. That is, each solution will undergo at least one depth-limited local search once they are newly generated. If they survive to the next generation, they will be identified as elites and local search will be deepened further.

1.3.2 Depth of LS

PLS is designed in such a away as to limit the depth of local search, that is, potential solutions will undergo depth-limited LS. The depth of LS is determined by a predetermined number of steps. If potential solutions survive the next generation, local search will be deepened further. This way the local search is continued and iteratively deepened on good solutions across generations. Lesser the potential of one candidate solution, the lesser is the depth of local searches applied on it. Greater the potential, the deeper will be the local search applied on that solution across generations. The potential of a candidate depends on the fitness of that individual, which is associated with both exploration of the search space and exploitation of its neighborhood. We decided to use the steepest descent as local search method, since it suits the decision of limiting the depth of applying local search. Thus PLS chooses individuals for depth-limited local search and fine-tunes them. This fine-tuning spreads across generations and iteratively deepens. These solutions are collectively referred to as preferential local search. Procedure followed by PLS is given in Algorithm 5. These two ideas of PLS and AW are incorporated into NSGA-II and a new hybrid algorithm MAPLS-AW was developed.