Research And Application Of Penalty Funciton Method In Constrained Optimization

Penalty function method is a classical and valuable method to solve constrained opti-mization problems. The key of this method is how to construct an effective penalty function. GA (Genetic Algorithm), which searches the optimal solution by simulating biological evo-lution, is a strong random searching algorithm. Combination of penalty function method and GA is a hot spot in current research of optimization. Based on the introduction of penalty function method and GA, we focus on how to construct penalty function. First, we introduce four kinds of frequently-used penalty functions:static penalty function, dynam-ic penalty function, annealing penalty function and adaptive penalty function and analyze their advantages and disadvantages. Then we present a new adaptive penalty function with simpler construction and prove its convergence. And then we combine it with GA to be a hybrid algorithm, and prove the new algorithm’s convergence.Several numerical experiments are also given. We compare the performance of hybrid algorithms based on static penalty function (HSSP), dynamic penalty function (KDPD), annealing penalty function (JHAP), adaptive penalty function (BHAP) and the new adap-tive penalty function (NAP). For most cases, hybrid algorithms based on adaptive penal-ty function (BHAP and NAP) have higher accuracy and stronger stability than those on HSSP, KDPD and JHAP. For high-dimension and nonlinear optimization case, compared with BHAP, hybrid algorithm based on NAP has more improvements, such as the rate of convergence is faster and the stability is stronger.To further improve the performance of hybrid algorithm based on NAP, we introduce a hybrid algorithm with constrained variable metric method. Theoretical analysis and exper-imental results indicate that new hybrid algorithm can get better accuracy without deterio-ration of stability and convergence rate.