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A method for solving the minimization problem
 
A method for solving the minimization problem
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d0313601.png" /></td> </tr></table>
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$$f(x^*)=\min_xf(x),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d0313602.png" /> is some function of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d0313603.png" />. The iterative sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d0313604.png" /> of the method of descent is computed by the formula
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where $f$ is some function of the variables $(x_1,\ldots,x_n)$. The iterative sequence $\{x_k\}$ of the method of descent is computed by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d0313605.png" /></td> </tr></table>
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$$x^{k+1}=x^k+\alpha_kg^k,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d0313606.png" /> is a vector indicating some direction of decrease of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d0313607.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d0313608.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d0313609.png" /> is an iterative parameter, the value of which indicates the step-length in the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136010.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136011.png" /> is a differentiable function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136012.png" /> is not an extremal point of it, then the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136013.png" /> must satisfy the inequality
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$(f'(x^k),g^k)<0,\tag{*}$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136015.png" /> is the gradient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136016.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136017.png" />.
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where $f'(x^k)$ is the gradient of $f$ at $x^k$.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136018.png" /> is a sufficiently smooth function (e.g. twice continuously differentiable) and if the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136019.png" /> satisfies inequality (*), then there exists a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136020.png" /> such that
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If $f$ is a sufficiently smooth function (e.g. twice continuously differentiable) and if the sequence $\{g^k\}$ satisfies inequality \ref{*}, then there exists a sequence $\{\alpha_k\}$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136021.png" /></td> </tr></table>
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$$f(x^0)>\ldots>f(x^k)>\ldots.$$
  
Under certain restrictions (see [[#References|[3]]]) on the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136022.png" /> and on the method of choosing the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136023.png" /> and the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136024.png" />, the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136025.png" /> converges to a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136026.png" /> of the initial problem.
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Under certain restrictions (see [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136027.png" /> are in some way expressed in terms of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136028.png" />, is a method of descent. One of the most common cases is when
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136029.png" /></td> </tr></table>
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$$g^k=-B(x^k)f'(x^k),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136030.png" /> is a symmetric matrix satisfying
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where $B(x)$ is a symmetric matrix satisfying
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136031.png" /></td> </tr></table>
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$$m(x,x)\leq(B(y)x,x)\leq M(x,x)$$
  
for any two vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136033.png" />, with certain constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136034.png" />. Under additional restrictions (see [[#References|[3]]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136035.png" /> and by a special selection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136036.png" />, the gradient method ensures the convergence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136037.png" /> to a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136038.png" /> of the initial problem with the rate of an arithmetical progression with ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136039.png" />. A special case of the gradient method is the method of steepest descent (cf. [[Steepest descent, method of|Steepest descent, method of]]), in which the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031360/d03136040.png" /> is the unit matrix.
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for any two vectors $x$ and $y$, with certain constants $M\geq m>0$. Under additional restrictions (see [[#References|[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|Steepest descent, method of]]), in which the matrix $B(x)$ is the unit matrix.
  
 
====References====
 
====References====

Revision as of 14:34, 10 August 2014

A method for solving the minimization problem

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

where $f$ is some function of the variables $(x_1,\ldots,x_n)$. 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

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

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 \ref{*}, then there exists a sequence $\{\alpha_k\}$ such that

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

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)
How to Cite This Entry:
Descent, method of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Descent,_method_of&oldid=32816
This article was adapted from an original article by Yu.A. Kuznetsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article