A hybrid multi-objective evolutionary algorithm with feedback mechanism

Abstract

Exploration and exploitation are two cornerstones for multi-objective evolutionary algorithms (MOEAs). To balance exploration and exploitation, we propose an efficient hybrid MOEA (i.e., MOHGD) by integrating multiple techniques and feedback mechanism. Multiple techniques include harmony search, genetic operator and differential evolution, which can improve the search diversity. Whereas hybrid selection mechanism contributes to the search efficiency by integrating the advantages of the static and adaptive selection scheme. Therefore, multiple techniques based on the hybrid selection strategy can effectively enhance the exploration ability of the MOHGD. Besides, we propose a feedback strategy to transfer some non-dominated solutions from the external archive to the parent population. This feedback strategy can strengthen convergence toward Pareto optimal solutions and improve the exploitation ability of the MOHGD. The proposed MOHGD has been evaluated on benchmarks against other state of the art MOEAs in terms of convergence, spread, coverage, and convergence speed. Computational results show that the proposed MOHGD is competitive or superior to other MOEAs considered in this paper.

Appendix

The three operators used in this paper are described as follows, respectively.

• 1) HS technique

HS is inspired by the improvisation process where music players improvise their instruments’ pitches or notes to search a beautiful harmony. The procedure of the HS algorithm is described as follows [25, 47, 48].

• Step 1. Initialize the parameters. The parameters used in HS are as follows. Harmony memory size (HMS); harmony memory considering rate (HMCR); pith adjusting rate (PAR); bandwidth (BW ); and the number of improvisations (NI).

• Step 2. Initialize the harmony memory. Generate random solution vectors $\left \{ \mathbf {x}^{1},\mathbf {x}^{2},\mathellipsis ,\mathbf {x}^{HMS} \right \}$ with size HMS. The harmony memory (HM) is a memory archive that is used to store all the solutions. The HM matrix is defined as follow.

  $$ HM\mathrm{=}\left[ {\begin{array}{*{20}c} {x_{1}^{1}} & \mathrm{\cdots} & {x_{n}^{1}}\\ \mathrm{\vdots} & \mathrm{\ddots} & \mathrm{\vdots} \\ x_{1}^{HMS} & \mathrm{\cdots} & x_{n}^{HMS} \end{array}} \right] $$

  (6)

• Step 3. Improvisation. A new harmony vector $\mathbf {x}^{t}=\left ({x_{1}^{t}},{x_{2}^{t}},\mathellipsis ,{x_{n}^{t}} \right )$ is generated based on three rules. (a) harmony memory consideration, (b) pitch adjustment and (c) random selection. Creating a new harmony is also called improvisation.

In the memory consideration, the value of variable ${x_{i}^{t}}$ is selected from $\left \{ {x_{i}^{1}},{x_{i}^{2}},\mathellipsis ,x_{i}^{HMS} \right \}$ with the probability of HMCR (HMCR∈ [0, 1]). In random selection, ${x_{i}^{t}}$ can be any feasible value not limited to those stored in HM with the probability of (1-HMCR). The formula can be written as follow.

$$ {x_{i}^{t}}=\left\{ {\begin{array}{lllllll} {x_{i}^{t}}\in \{{x_{1}^{t}},{x_{2}^{t}},\mathellipsis ,{x_{n}^{t}}\} & if\, rand\, \le HMCR \\ {x_{i}^{t}}\in X_{i} &otherwise \end{array}} \right. $$

(7)

where rand is a uniform random number in the range between 0 and 1.

Thereafter, each pitch obtained by the memory consideration is adjusted by the pitch adjustment rule with the rate of PAR. The adjustment rule works as follow.

$$ {x_{i}^{t}}=\left\{ {\begin{array}{lllll} {x_{i}^{t}}\pm rand\times BW &if \,rand\,\le PAR \\ {x_{i}^{t}} &otherwise \end{array}} \right. $$

(8)

where BW is the arbitrary distance bandwidth.

HS has many advantages. (1) HS can handle discrete problems as well as continuous problems. (2) It is easy for HS to combine with other algorithms, constructing a new algorithm with better performance. (3) HS overcomes the drawback of GA’s building block theory. However, premature phenomenon is likely to occur in HS.

• 2) GA technique

The GA is one of the most well-known and commonly used evolutionary algorithms (EAs). It was developed in the early 1970s by John Holland [26]. The basic GA is very generic, and it mainly includes the following steps: selection, crossover and mutation. In general GA works as follows. First, GA begins by creating a random initial population, and evaluate each individual of the current population by calculating its fitness value; then select two parents from the current population based on their fitness, that is to say, individual with higher fitness is often selected as a parent; child is from parents by executing recombination and mutation operation on parents; finally replace the current population with child according to fitness in every iteration. GA runs until stopping condition is met. The pseudocode of GA is given in Algorithm 1.

The merits of GA are as follows. (1) GA has good global search ability. (2) It is easy for GA to be extended and merged with other algorithms. However, it also has some limitations. For example, it easily falls into local optimum [49, 50].

• 3) DE technique

DE, proposed by Storn Price, is one of the most popular EAs in recent years [27]. The four main steps in DE are initialization, mutation, recombination and selection. In DE, a new solution is created by adding the weighted difference vector between two parents to a third parent. In the current generation G, a candidate solution x_i,G, $i =$ 1,…, N (N is population size) can be used to generate a new trial solution u by using this update formula as in Algorithm 2. This paper uses DE/rand/1/bin strategy to update population, even though different strategies have been proposed. In this pseudocode CR controls the crossover operation and F is the scaling factor. Both CR and F are constant number. After the applications of these two operators, the new solution $\mathbf {u}_{i,G}$ is compared with the old vectorx_i,G; the latter will be replaced by the former if this one has a better objective value.

The main advantages of DE contain the following aspects. (1) DE has fewer control parameters, namely N, F, and CR. (2) It is also competitive to the other EAs. (3) It can effectively solve high-dimensional complex optimization problems. However, DE has several drawbacks including unstable convergence in the last period [51].