A Dimension–reducing Conic Method for Unconstrained Optimization

In this paper we present a new algorithm for finding the unconstrained minimum of a twice–continuously differentiable function f(x) in n variables. This algorithm is based on a conic model function, which does not involve the conjugacy matrix or the Hessian of the model function. The basic idea in this paper is to accelerate the convergence of the conic method choosing more appropriate points x1, x2, . . . , xn+1 to apply the conic model. To do this, we apply in the gradient of f a dimension–reducing method (DR), which uses reduction to proper simpler one–dimensional nonlinear equations, converges quadratically and incorporates the advantages of Newton and Nonlinear SOR algorithms. The new method has been implemented and tested in well known test functions. It converges in n + 1 iterations on conic functions and, as numerical results indicate, rapidly minimizes general functions.