A New Generalized Gradient Projection Algorithm

Since the gradient projection method was proposed, it has been paid much attention to because it has the following three advantages: 1) its structure is very simple, 2) it is easily operated, 3) its each computional step is iterated explictly and doesn't need to solve complex linear programming or quadratic programming problems. Now the gradient projection method has been considered as one of the basic methods for solving nonlinear optimization problems. The conjugate gradient method is one of the most efficient methods for solving unconstrained optimization problems. It has played a special role for solving large-scale nonlinear optimization problems due to its simplicity and its very low memory requirement. It has been authors' idea to use the conjugate gradient method to solve constrained optimization problems for years. For researching on the above idea, in this thesis we combine the conjugate gradient formula with the gradient projection method, and get a new algorithm.In this thesis, a new hybrid conjugate gradient formula is first given. The conjugate gradient method generated by this formula is globally convergent and avoids the propensity of short steps. Its numerical results illustrate that this method outperforms PRP method. Further, by combining the new hybrid conjugate gradient formula with the generalized gradient projection method, and using the method and technology for dealing with arbitrary initial point, we get a new generalized gradient projection algorithm with arbitrary initial point for nonlinear inequality constrained optimization problems. Under weak conditions, the algorithm converges globally. And the numerical rusults show that the algorithm is effective in practical computation.