Elias at 14:57, 8 December 2011

2011-12-08T14:57:22Z

← Older revision		Revision as of 09:57, 8 December 2011
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	where <math>x, p \in \~~MATHSET~~{R}^{n}</math>.		where <math>x, p \in \mathbb{R}^{n}</math>.

	==Calling Syntax==		==Calling Syntax==

Elias at 14:55, 22 July 2011

2011-07-22T14:55:10Z

← Older revision		Revision as of 09:55, 22 July 2011
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			[[Category:Utility Functions]]
	==Purpose==		==Purpose==

Elias: Created page with "==Purpose== ''LineSearch ''solves line search problems of the form $\min\limits_{0<\alpha _{\min }\leq \alpha \leq \alpha _{\max }}f(x^{(k)}+\alpha p)$ where <..."

2011-07-22T12:52:27Z

Created page with "==Purpose== ''LineSearch ''solves line search problems of the form <math> \min\limits_{0<\alpha _{\min }\leq \alpha \leq \alpha _{\max }}f(x^{(k)}+\alpha p) </math> where <..."

New page

==Purpose==

''LineSearch ''solves line search problems of the form

<math>
\min\limits_{0<\alpha _{\min }\leq \alpha \leq \alpha _{\max
}}f(x^{(k)}+\alpha p)
</math>

where <math>x, p \in \MATHSET{R}^{n}</math>.

==Calling Syntax==

<pre>
Result = LineSearch(f, g, x, p, f_0, g_0, LineParam, alphaMax, pType, PriLev, varargin)
</pre>

==Inputs==

{| class="wikitable"
!Input||Description
|-
|''f''||colspan="2"|Name of m-file computing the objective function ''f ''(''x'').
|-
|''g''||colspan="2"|Name of m-file computing the gradient vector ''g''(''x'').
|-
|''x''||colspan="2"|Current iterate ''x''.
|-
|''p''||colspan="2"|Search direction ''p''.
|-
|''f_0''||colspan="2"|Function value at ''a ''= 0.
|-
|''g_0''||colspan="2"|Gradient at ''a ''= 0, the directed derivative at the present point.
|-valign="top"
|''LineParam''||colspan="2"|Structure with line search parameters (see [[TOMLAB Appendix A]]), the following fields are used:
{|
|''LineAlg''||Type of line search algorithm, 0 = quadratic interpolation, 1 = cubic interpolation.
|-valign="top"
|''fLowBnd''||Lower bound on the function value at optimum.

''sigma InitStepLength rho tau1 tau2 tau3 eps1 eps2 ''see [[TOMLAB Appendix A]].
|}
|-
|''alphaMax||colspan="2"|''Maximal value of step length ''α''.
|-valign="top"
|''pType''||colspan="2"|Type of problem:
{|
|0||Normal problem.
|-
|1||Nonlinear least squares.
|-
|2||Constrained nonlinear least squares.
|-
|3||Merit function minimization.
|-
|4||Penalty function minimization.
|}
|-valign="top"
|''PriLev''||colspan="2"|Printing level:
{|
|''PriLev > ''0||Writes a lot of output in ''LineSearch''.
|-
|''PriLev > ''3||Writes a lot of output in ''intpol2 ''and ''intpol3''.
|}
|-
|''varargin''||colspan="2"|Other parameters directly sent to low level routines.

|}

==Outputs==

{| class="wikitable"
!Output||Description
|-valign="top"
|''Result''||colspan="2"|Result structure with fields:
{|
|''alpha''||Optimal line search step ''α''.
|-
|''f_alpha''||Optimal function value at line search step ''α''.
|-
|''g_alpha''||Optimal gradient value at line search step ''α''.
|-
|''alphaVec''||Vector of trial step length values.
|-
|''r_k''||Residual vector if Least Squares problem, otherwise empty.
|-
|''J_k''||Jacobian matrix if Least Squares problem, otherwise empty.
|-
|''f_k''||Function value at ''x ''+ ''αp''.
|-
|''g_k''||Gradient value at ''x ''+ ''αp''.
|-
|''c_k''||Constraint value at ''x ''+ ''αp''.
|-
|''dc_k''||Constraint gradient value at ''x ''+ ''αp''.
|}
|}

==Description==

The function ''LineSearch ''together with the routines ''intpol2 ''and ''intpol3 ''implements a modified version of a line search algorithm by Fletcher. The algorithm is based on the Wolfe-Powell conditions and therefore the availability of first order derivatives is an obvious demand. It is also assumed that the user is able to supply a lower bound ''fLow ''on ''f''(''α''). More precisely it is assumed that the user is prepared to accept any value of ''f''(''α'') for which ''f''(''α'') ''= fLow ''. For example in a nonlinear least squares problem ''fLow ''= 0 would be appropriate.

''LineSearch ''consists of two parts, the ''bracketing phase ''and the ''sectioning phase''. In the bracketing phase the iterates ''a''(''k'') moves out in an increasingly large jumps until either ''f <nowiki><</nowiki>= fLow ''is detected or a bracket on an interval of acceptable points is located. The sectioning phase generates a sequence of brackets [''a''(''k'') '', b''(''k'')] whose lengths tend to zero. Each iteration pick a new point ''a''(''k'') in [''a''(''k'') '', b''(''k'')] by minimizing a quadratic or a cubic polynomial which interpolates ''f(a''(''k'')), ''f<nowiki>'</nowiki>(a''(''k'')), ''f''(b(''k'') ) and ''f<nowiki>'</nowiki>(b''(''k'')) if it is known. The sectioning phase terminates when ''a''(''k'') is an acceptable point.

LineSearch - Revision history

Elias at 14:57, 8 December 2011

Elias at 14:55, 22 July 2011

Elias: Created page with "==Purpose== ''LineSearch ''solves line search problems of the form \min\limits_{0<\alpha _{\min }\leq \alpha \leq \alpha _{\max }}f(x^{(k)}+\alpha p) where <..."

Elias: Created page with "==Purpose== ''LineSearch ''solves line search problems of the form $\min\limits_{0<\alpha _{\min }\leq \alpha \leq \alpha _{\max }}f(x^{(k)}+\alpha p)$ where <..."