Method of Dual Matrices for Function Minimization,

摘要

In the paper,the method of dual matrices for the minimization of functions is introduced. The method,which bypasses the one-dimensional search for the stepsize,is characterized by employing two matrices at each iteration. One matrix is such that a linearly independent set of directions can be generated,regardless of the stepsize employed. The other matrix is such that,at the point where the first matrix fails to yield a linearly independent gradient,it generates a displacement leading to the minimal point. Thus,the one-dimensional search is completely bypassed. For a quadratic function,it is proved that the minimal point is obtained at most n + 1iterations,where n is the number of variables in the function. Since the one-dimensional search is not needed,the total number of gradient evaluations for convergence is at most n + 2. This represents a saving on the total computational effort versus 2n + 1gradient evaluations required by the conventional quadratically convergent algorithms. (Author)