Descent, method of

A method for solving the minimization problem

$$f(x^*)=\min_xf(x),$$

where $f$ is some function of the variables $(x_1,\dotsc,x_n)$. The iterative sequence $\{x_k\}$ of the method of descent is computed by the formula

$$x^{k+1}=x^k+\alpha_kg^k,$$

where $g^k$ is a vector indicating some direction of decrease of $f$ at $x^k$, and $\alpha_k$ is an iterative parameter, the value of which indicates the step-length in the direction $g^k$. If $f$ is a differentiable function and $x^k$ is not an extremal point of it, then the vector $g_k$ must satisfy the inequality

$$(f'(x^k),g^k)<0,\label{*}\tag{*}$$

where $f'(x^k)$ is the gradient of $f$ at $x^k$.

If $f$ is a sufficiently smooth function (e.g. twice continuously differentiable) and if the sequence $\{g^k\}$ satisfies inequality \eqref{*}, then there exists a sequence $\{\alpha_k\}$ such that

$$f(x^0)>\dotsb>f(x^k)>\dotsb.$$

Under certain restrictions (see [3]) on the function $f$ and on the method of choosing the parameters $\{\alpha_k\}$ and the vectors $g^k$, the sequence $\{x_k\}$ converges to a solution $x^*$ of the initial problem.

The gradient method, in which the vectors $\{g^k\}$ are in some way expressed in terms of the vectors $\{f'(x^k)\}$, is a method of descent. One of the most common cases is when

$$g^k=-B(x^k)f'(x^k),$$

where $B(x)$ is a symmetric matrix satisfying

$$m(x,x)\leq(B(y)x,x)\leq M(x,x)$$

for any two vectors $x$ and $y$, with certain constants $M\geq m>0$. Under additional restrictions (see [3]) on $f$ and by a special selection of $\{\alpha_k\}$, the gradient method ensures the convergence of $\{x^k\}$ to a solution $x^*$ of the initial problem with the rate of an arithmetical progression with ratio $g<1$. A special case of the gradient method is the method of steepest descent (cf. Steepest descent, method of), in which the matrix $B(x)$ 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

Comments

See also Coordinate-wise descent method.

References

[a1]	J.M. Ortega, W.C. Rheinboldt, "Iterative solution of non-linear equations in several variables" , Acad. Press (1970)