A Steep Thermodynamical Selection Rule for Evolutionary Algorithms

Abstract

The genetic algorithm (GA) often suffers from the premature convergence because of the loss of population diversity at an early stage of searching. This paper proposes a steep thermodynamical evolutionary algorithm (STEA), which utilizes a steep thermodynamical selection (STS) rule. STEA simulates the competitive mechanism between energy and entropy in annealing to systematically resolve the conflicts between selective pressure and population diversity in GA. This paper also proves that the rule STS has the approximate steepest descent ability of the free energy. Experimental results show that STEA is both far more efficient and much stabler than the thermodynamical genetic algorithm (TDGA).

A Thermodynamical Selection-Based Discrete Differential Evolution for the 0-1 Knapsack Problem

“…In order to enhance the selection operator of the traditional DE, in this section, we propose an improved selection operator, MFES, which is inspired by the principle of the minimal free energy in thermodynamics, trying to palliate the conflict between the selective pressure and the population diversity to some degree. The principle of the minimal free energy refers to [39][40][41][42], in the annealing process, a metal, beginning with high temperature and chaotic state, is gradually cooled, so that the system at any temperature approximately reaches thermodynamic equilibrium. This cooling process can be regarded as an adaptation procedure to reach the stability of the final crystalline solid.…”

“…In addition, any change from non-equilibrium to equilibrium of the system at each temperature abides by the principle of the minimal free energy. This means the system will change spontaneously to reach a lower total free energy and the system achieves equilibrium when its free energy seeks a minimum [40][41][42]. The free energy F is defined by:…”

“…According to the principle of the minimal free energy, we can realize that any change from non-equilibrium to equilibrium of the system can be regarded as a consequence of the competition between the mean energy and the entropy, and the temperature T determines their relative weights in the competition [39,41,42]. Accordingly, the competition between the mean energy and the entropy in the annealing process is the same as the competition between the selective pressure and the population diversity of DE during the evolution process.…”

“…Therefore, we can alleviate the conflict between the selective pressure and the diversity of population according to the principle of the minimal free energy. Inspired by [41][42][43], in the improved selection scheme, we firstly map the selective pressure and the population diversity of DE into the mean energy E and the entropy H in the principle of the minimal free energy, respectively. Then, the criterion of the proposed selection scheme is converted to the minimal free energy, which means that we select individuals for the next generation from the parent, and the offspring population should satisfy the principle of the minimal free energy.…”

“…In general, the higher selective pressure may directly lead to the better individuals selected for the next generation. Therefore, we can measure the selective pressure by the fitness values of the selected individuals [41]. The energy E of each individual is mapped into the normalized fitness value of the corresponding individual.…”