The approximation algorithm for solving a sort of non-smooth programming

An approximation algorithm for solving a sort of non-smooth programming is proposed. In the programming, the objective function is the Hölder function and the feasible region is contained in a rectangle (viz. hyper-rectangle). To establish the algorithm, the properties of the Hölder function and an approximation of the function by using Bernstein α-polynomial are studied. According to the properties of the approximation polynomial and the algorithm for solving geometric programming, the strategy for branching and bounding and the branch-and-bound algorithm are constructed to solve the programming. The convergence of the algorithm can be guaranteed by the exhaustive of the bisection of the rectangle. The feasibility of the algorithm is validated by solving an example.