# Gradient method

From Encyclopedia of Mathematics

A method for the minimization of a function of several variables. It is based on the fact that each successive approximation of the function $ F $
is obtained from the preceding one by a shift in the direction of the gradient of the function:

$$ \mathbf x ^ {n + 1 } = \ \mathbf x ^ {n} - \delta _ {n} \ \mathop{\rm grad} F ( \mathbf x ^ {n} ). $$

The parameter $ \delta _ {n} $ can be obtained, e.g., from the condition of the magnitude

$$ F ( \mathbf x ^ {n} - \delta _ {n} \ \mathop{\rm grad} F ( \mathbf x ^ {n} )) \ \ \textrm{ being minimal } . $$

See also Descent, method of; Steepest descent, method of.

#### Comments

#### References

[a1] | J.E. Dennis, R.B. Schnabel, "Numerical methods for unconstrained optimization and nonlinear equations" , Prentice-Hall (1983) MR0702023 Zbl 0579.65058 |

[a2] | R. Fletcher, "Practical methods of optimization" , Wiley (1980) MR0585160 MR0633058 Zbl 0439.93001 |

[a3] | D.G. Luenberger, "Linear and nonlinear programming" , Addison-Wesley (1984) MR2423726 MR2012832 Zbl 0571.90051 |

**How to Cite This Entry:**

Gradient method.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Gradient_method&oldid=47111