The Study Of Optimality Conditions And Algorithms For Non-smooth Optimization In Hilbert Space

Non-smooth optimization which is also known as nondifferentiable optimization isan important branch of optimization theories and methods. The traditional optimalityconditions and optimization theories and methods based on the concept of differentialare not fit for the non-smooth optimization problems due to the quality of discontinuousand nondifferentiable. Therefore the classical differential concept should be spread andset up all kinds of generalized differential concept, then base on the generalizeddifferential theory to find relevant optimization theory including the optimalityconditions. Finally, the optimization algorithms should be researched.Hilbert space is a completed inner product space and in which any Cauchy columnis a convergent column. Then the most theories of calculus can be popularized fromRnspace to Hilbert space. It is a mean of formulation to Fourier series and Fouriertransform based on polynomial in arbitrary orthogonal set.This paper mainly researches the optimality conditions and algorithms ofnon-smooth optimization in Hilbert space. Firstly, it spread the subdifferential of theconvex function, the generalized gradient of part Lipschitz function and thequasidifferential of the quasidifferentiable function fromRnspace to Hilbert space.Then obtain the optimality conditions of non-smooth optimization problems in Hilbertspace. At last, it improves some steps in the algorithms of non-smooth optimizationproblems including the ε subgradient method, incremental subgradient method and theexternal cuboid algorithm in order to make them more simple and convenient.