Low Dimensional Simplex Evolution Algorithms And Their Applications

In this paper, a low dimensional simplex evolution (LDSE) algorithm and some variants of it, such as LDSE with normal struggle (NS-LDSE) and simplex evolution with variable dimension strategy (SEVD), for the global optimization of continuous variables are proposed and, a general low dimensional reproduction (LDR) strategy to improve the performance of real-coded evolutionary algorithms (REAs) is also suggested.LDSE is a kind of real-coded evolutionary algorithm, in which m-dimensional simplex operations are introduced as evolutionary operators, where m is much less than the dimension n of the problem. Different with other hybrid evolutionary algorithms, LDSE employs reformed traditional iterations as evolutionary operators conditionally and selectively, rather than using them directly. Numerical results show that LD-SEs outperform other evolutionary algorithms in the sense of convergence rate and convergence probability. By the theory of stochastic process, we have discussed the convergence properties of LDSE. It is proved by means of homogeneous finite Markov chain analysis that NS-LDSE is convergent to the global optimum for any chosen initial population. Meanwhile, the convergence speed of NS-LDSE is estimated by means of Markov stopping time.The LDR strategy keeps some (randomly chosen) components of the current vector (individual) invariant in the reproduction process. It needs no additional computation, requires only one control parameter and is very easy to use. Numerical tests show that the LDR strategy can significantly improve the performance of REAs, e.g., real-coded genetic algorithm (RGA), evolutionary programming (EP), differential evolution (DE) and particle swarm optimization (PSO). The LDR strategy can also improve the performance of LDSE significantly.Based on LDSE, we provided an algorithm for the minimum unsatisfied relationships (MIN UR) problem of inconsistency inequalities. Numerical results show that our algorithm is more effective than previous algorithms. Besides, motivated by a practical problem from molecular biology we encountered, we have studied the complicated system of inequalities with possible inconsistency (CSIPI) involving multilevel "and", "or" relationships. As in MIN UR, its objective function is non-smooth, even discontinuous, and hard for traditional optimization algorithm to cope with. We gave an effective method to solve this kind of problem with LDSE. And the practical CSIPI problem is solved successfully.