Some New Results On The Projected Gradient Method

Projected gradient methods are special generalized elimination meth-ods and also are interior point methods, which are very useful for solv-ing the optimization problem with linear equality restrictions or linearinequality restrictions. We discuss the projected gradient method ofthe following form: min f(x) s. t. g_i(x)≥0, i∈τh_j(x)=0, j∈εIn a space H, we get three iteration formulas by classify the locationof iteration points x~k and the subset C of H, and we conclude a newalgorithm. When the problem that we discussed has solution, we provethe bound, convergence of the sequence of number x~k, and explain thenew algorithm is feasible. Finally, we compute the convergence rateand compare the convergence rate with the new and old algorithms.This paper consists of four chapters:In chapterⅠ, we introduce the background and significance of theprojected gradient method.In chapterⅡ, we give some related definitions of this paper.In chapterⅢ, based on the projected gradient method of R~n, wegive the new iteration formulas and prove the convergence of the al-gorithm.In chapterⅣ, we improve the algorithm of the projected gradientmethod in a Hilbert space, and also prove the convergence, compute the convergence rate of the algorithm. In the end, we introduce someoptimality conditions of the projected gradient method.