Descent, method of

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A method for solving the minimization problem

where is some function of the variables . The iterative sequence of the method of descent is computed by the formula

where is a vector indicating some direction of decrease of at , and is an iterative parameter, the value of which indicates the step-length in the direction . If is a differentiable function and is not an extremal point of it, then the vector must satisfy the inequality

 (*)

where is the gradient of at .

If is a sufficiently smooth function (e.g. twice continuously differentiable) and if the sequence satisfies inequality (*), then there exists a sequence such that

Under certain restrictions (see [3]) on the function and on the method of choosing the parameters and the vectors , the sequence converges to a solution of the initial problem.

The gradient method, in which the vectors are in some way expressed in terms of the vectors , is a method of descent. One of the most common cases is when

where is a symmetric matrix satisfying

for any two vectors and , with certain constants . Under additional restrictions (see [3]) on and by a special selection of , the gradient method ensures the convergence of to a solution of the initial problem with the rate of an arithmetical progression with ratio . A special case of the gradient method is the method of steepest descent (cf. Steepest descent, method of), in which the matrix is the unit matrix.

References

 [1] L.V. Kantorovich, G.P. Akilov, "Functionalanalysis in normierten Räumen" , Akademie Verlag (1964) (Translated from Russian) [2] G. Zoutendijk, "Methods of feasible directions" , Elsevier (1970) [3] B.N. Pshenichnyi, Yu.M. Danilin, "Numerical methods in extremal problems" , MIR (1978) (Translated from Russian) [4] B.T. Polyak, "Gradient methods for the minimization of functionals" USSR Comp. Math. Math. Physics , 3 : 4 (1963) pp. 864–878 Zh. Vychisl. Mat. i Mat. Fiz. , 3 : 4 (1963) pp. 643–654