| A very common problem in diverse areas of mathematics and physical sciences consists of trying to find a point in the intersection of convex set.This problem is referred to as the convex feasibility problem ;It is usually found in Best approximation,Image reconstruction of discrete models,Image reconstruction of continuous models and Subgradient algorithms.One frequently employed approach in solving the convex feasibility problem is the projection algorithms.In this paper,we present an proof of convergence of algorithm which solve the convex inequalities belonging to the convex feasibility problem.The proof utilize the ideas of projection algorithm ,the frame of the descent iterative algorithm and the distinguishing feature of the convex inequalities problem.At same time ,an algorithm is presented to obtain the strict solution to the convex inequalities.In chapter 4 and chapter 5,the main results this text have introduced ,can be summarized as follow:In chapter 4: The convex inequalities is transformed into the convex indeterminate equation by maximum function.We can prove that the numerical sequence generated by given algorithm converge to the root of the equation ,which is the solution of the convex inequalities too,when the distance function is used as the descent function according to descent iterative algorithm and the geometric properties of subgradient is exploited.Several numerical examples show that the algorithm is effective.In chapter 5:In some problems ,the strict solution of the convex inequalities is expected to be found .But,we cann't find the strict solution by the proved algorithm because of itself restriction.we discover that the way which was used to calculate descent direction of non-smoothly exact penalty function by Bertsekas ( 1982 ) can be used to get the descent direction of maximum function at its root. That means merely to solve a quadratic programming for strict solution.This chape will demonstrate the main proof of that algorithms.
|
PDF Full Text Request