A Derivative-free Algorithm Of Lipschitz Functions Constrained Optimization

In order to study the minimization problem of Lipschitz function,this paper uses a non-derivative method of linear search,that is,CS-DFNP algorithm.As the objective function and the constraint function may be non-smooth,the traditional optimization theory and method based on gradient concept is no longer applicable.As a result,in this paper,the Claxke-Jahn generalized directional derivative is used to study the stability of the problem.Firstly,the paper carries out a theoretical analysis and research on the problem of only some boundary constraints.And we proves that the CS-DFNP algorithm will generate Clarke-Jahn stability for such problems.For the optimization problem with complex inequality constraint function,we use the exact penalty function method to deal with it under the condition of reasonable assumptions.And according to the existing exact penalty function,the results of the research prove the validity of the algorithm in theory.When we use the penalty function method,we only punish the nonlinear inequality constraint and do not deal with the boundary constraint,which also reduces the unnecessary error and improve the accuracy of the algorithm.Finally,the paper chooses two examples,one is a smooth unconstrained problem and the other is a non-smooth optimization problem.Experimental results show that the algorithm is feasible in the range of error tolerance.For the smooth unconstrained problem,the choice of the initial value has little effect on the experimental results,but the effect is obvious for the latter.The algorithm chooses the Halton sequence when selecting the dense direction sequence,which is based on a certain method.The deviation between them is relatively small.To some extent,it can be seen as random.The sequence is treated in unit,which can also be better to deal with the step in the experiment.When doing experiments,we find that the choice of parameters has great influence on the timeliness and accuracy of the algorithm,which is the problem that we will inevitably encounter.The relatively optimal parameters can be selected by a large number of experiments.On the whole,CS-DFN algorithm is feasible,but there is still much room to be improved.