Difference between revisions of "Gradient method"
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| + | $#C+1 = 4 : ~/encyclopedia/old_files/data/G044/G.0404710 Gradient method | ||
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| − | + | 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|Descent, method of]]; [[Steepest descent, method of|Steepest descent, method of]]. | See also [[Descent, method of|Descent, method of]]; [[Steepest descent, method of|Steepest descent, method of]]. | ||
====Comments==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.E. Dennis, R.B. Schnabel, "Numerical methods for unconstrained optimization and nonlinear equations" , Prentice-Hall (1983) {{MR|0702023}} {{ZBL|0579.65058}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Fletcher, "Practical methods of optimization" , Wiley (1980) {{MR|0585160}} {{MR|0633058}} {{ZBL|0439.93001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D.G. Luenberger, "Linear and nonlinear programming" , Addison-Wesley (1984) {{MR|2423726}} {{MR|2012832}} {{ZBL|0571.90051}} </TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.E. Dennis, R.B. Schnabel, "Numerical methods for unconstrained optimization and nonlinear equations" , Prentice-Hall (1983) {{MR|0702023}} {{ZBL|0579.65058}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Fletcher, "Practical methods of optimization" , Wiley (1980) {{MR|0585160}} {{MR|0633058}} {{ZBL|0439.93001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D.G. Luenberger, "Linear and nonlinear programming" , Addison-Wesley (1984) {{MR|2423726}} {{MR|2012832}} {{ZBL|0571.90051}} </TD></TR></table> | ||
Latest revision as of 19:42, 5 June 2020
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=28206
Gradient method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gradient_method&oldid=28206