الگوریتم های القایی سیمپلکس برای حل رده ای از مسائل برنامه ریزی محدب

abstract: in this thesis, we focus to class of convex optimization problem whose objective function is given as a linear function and a convex function of a linear transformation of the decision variables and whose feasible region is a polytope. we show that there exists an optimal solution to this class of problems on a face of the constraint polytope of feasible region. based on this, we develop an algorithm that is inspired by the simplex method of linear programming. although the simplex method for linear programming may generate an exponential of solution but this method is an efficiency algorithm for linear programming problems so we encounter to less number of iteration. we show that this proposed method in the special case that the convex function has only a single argument is terminated in a finite number of iterations. we then use this insight to develop a second algorithm that solves the problem in a finite number of iterations for and arbitrary number of arguments in the convex function. a computational study illustrates the efficiency of the algorithm and suggests that the average-case performance of these algorithms is a polynomial of low order in the number of decision variables. in this thesis, we study the following class of nonlinear programming problems: where is a vector of decision variables, , , , and are the problem data, and is a convex function with continuous partial derivatives. in addition, we assume that the feasible region of (cp) is nonempty and bounded, so that an optimal solution to (cp) exists. for convenience, we denote the feasible region of (cp) by p. we will develop two solution methods that solve (cp) through solving a sequence of either one-dimensional or k dimensional linear and convex programming problems. the methods that we develop are partially inspired by the simplex for linear and convex programming. where we employ the fact that we can show that (cp) has an optimal solution that lies on a face of p of dimension k. note that this generalizes the well-known result that a programming problem has an extreme point optimal solution (provided an optimal solution exists.) our proposed algorithm generate a sequence of solutions on face of p of dimension no more than k. finally we generate the algorithms that solve (cp) problems and we use of them to solve the following problems: (cp) this kind of problems are solved analogous to (cp) problems except the nonbasic variables solution would be bounds of nonbasic. key words: convex optimization, simplex-inspired method, nonlinear programming, zangwill algorithm, kkt optimization conditions.