LinRatSolve - Revision history

Elias at 12:13, 16 January 2012

2012-01-16T12:13:10Z

← Older revision		Revision as of 07:13, 16 January 2012
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	==Calling Syntax==		==Calling Syntax==

	<~~syntaxhighlight~~ lang="matlab">		<source lang="matlab">
	Result=linRatSolve(Prob,PriLev)		Result=linRatSolve(Prob,PriLev)
	</~~syntaxhighlight~~>		</source>

	==Inputs==		==Inputs==

Elias at 12:27, 7 December 2011

2011-12-07T12:27:31Z

← Older revision		Revision as of 07:27, 7 December 2011
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	<math>		<math>
	\begin{array}{cccccc}		\begin{array}{cccccc}
	\~~min_{x}~~ & \frac{c1x}{c2x} \\		\min\limits_x & \frac{c1x}{c2x} \\
	\text{subject to} & x_L & \leq & x & \leq & x_U \\		\text{subject to} & x_L & \leq & x & \leq & x_U \\
	& b_L & \leq & Ax & \leq & b_U \\		& b_L & \leq & Ax & \leq & b_U \\
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	<math>		<math>
	\begin{array}{cccccc}		\begin{array}{cccccc}
	\~~min_{x}~~ c1 z2\\\\\text{subject to} & z1_L & \leq & z1 & \leq & \infty \\& 1 & \leq & c2 z2 & \leq & 1 \\& 0 & \leq & A z2 - z1 beq & \leq & 0 \\& -\infty & \leq & A z2 - z1 b_U & \leq & 0 \\& -\infty & \leq & - A z2 + z1 b_L & \leq & 0 \\\\& 0 & \leq & A1 z2 - z1 xeq & \leq & 0 \\& -\infty & \leq & A1 z2 - z1 x_U & \leq & 0 \\& -\infty & \leq & - A1 z2 + z1 x_L & \leq & 0 \\		\min\limits_x c1 z2\\\\\text{subject to} & z1_L & \leq & z1 & \leq & \infty \\& 1 & \leq & c2 z2 & \leq & 1 \\& 0 & \leq & A z2 - z1 beq & \leq & 0 \\& -\infty & \leq & A z2 - z1 b_U & \leq & 0 \\& -\infty & \leq & - A z2 + z1 b_L & \leq & 0 \\\\& 0 & \leq & A1 z2 - z1 xeq & \leq & 0 \\& -\infty & \leq & A1 z2 - z1 x_U & \leq & 0 \\& -\infty & \leq & - A1 z2 + z1 x_L & \leq & 0 \\
	\end{array}		\end{array}
	</math>		</math>

Elias at 12:18, 7 December 2011

2011-12-07T12:18:46Z

← Older revision		Revision as of 07:18, 7 December 2011
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	<math>		<math>
	\begin{array}{cccccc}		\begin{array}{cccccc}
	\~~min\limits_x~~ & \~~multicolumn~~{5}{~~l}{ \Large (c1 x) \over (c2 x)~~ } \\\~~mbox~~{subject to} & x_L & \leq & x & \leq & x_U \\& b_L & \leq & Ax & \leq & b_U \\		\min_{x} & \frac{c1x}{c2x} \\
			\text{subject to} & x_L & \leq & x & \leq & x_U \\
			& b_L & \leq & Ax & \leq & b_U \\
	\end{array}		\end{array}
	</math>		</math>


	where <math>c1,c2,x,x_L,x_U \in \~~MATHSET~~{R}^{n}</math> , <math>b_L,b_U \in \~~MATHSET~~{R}^{m_1}</math> and <math>A \in \~~MATHSET~~{R}^{m_1 \times n}</math>.		where <math>c1,c2,x,x_L,x_U \in \mathbb{R}^{n}</math> , <math>b_L,b_U \in \mathbb{R}^{m_1}</math> and <math>A \in \mathbb{R}^{m_1 \times n}</math>.


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	</syntaxhighlight>		</syntaxhighlight>

	==Inputs===		==Inputs==

	{\| class="wikitable"		{\| class="wikitable"
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	<math>		<math>
	\begin{array}{cccccc}		\begin{array}{cccccc}
	\~~multicolumn~~{6}~~{l}{\min\limits_x~~ c1 z2}\\\\\~~mbox~~{subject to} & z1_L & \leq & z1 & \leq & \infty \\& 1 & \leq & c2 z2 & \leq & 1 \\& 0 & \leq & A z2 - z1 beq & \leq & 0 \\& -\infty & \leq & A z2 - z1 b_U & \leq & 0 \\& -\infty & \leq & - A z2 + z1 b_L & \leq & 0 \\\\& 0 & \leq & A1 z2 - z1 xeq & \leq & 0 \\& -\infty & \leq & A1 z2 - z1 x_U & \leq & 0 \\& -\infty & \leq & - A1 z2 + z1 x_L & \leq & 0 \\\end{array}		\min_{x} c1 z2\\\\\text{subject to} & z1_L & \leq & z1 & \leq & \infty \\& 1 & \leq & c2 z2 & \leq & 1 \\& 0 & \leq & A z2 - z1 beq & \leq & 0 \\& -\infty & \leq & A z2 - z1 b_U & \leq & 0 \\& -\infty & \leq & - A z2 + z1 b_L & \leq & 0 \\\\& 0 & \leq & A1 z2 - z1 xeq & \leq & 0 \\& -\infty & \leq & A1 z2 - z1 x_U & \leq & 0 \\& -\infty & \leq & - A1 z2 + z1 x_L & \leq & 0 \\
			\end{array}
	</math>		</math>

	~~where <math>$A1 \in \MATHSET{R}^{N},\; A1=speye(N)$</math>.~~

	OBSERVE the denominator ''c''2''x ''must always be positive. It is normally a good a idea to run the problem with both signs (multiply each side by -1).		where <math>A1 \in \mathbb{R}^{N},\; A1=speye(N)</math>.

			OBSERVE the denominator ''c''2''x ''must always be positive. It is normally a good a idea to run the problem with both signs (multiply each side by -1).

	==See Also==		==See Also==

	[[lpassign\|''lpAssign'']].		[[lpassign\|''lpAssign'']].

Elias at 13:26, 20 July 2011

2011-07-20T13:26:14Z

← Older revision		Revision as of 08:26, 20 July 2011
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	[[Category:Solvers]]		[[Category:Solvers]]
	~~{{cleanup\|Clean this article.}}~~

	==Purpose==		==Purpose==

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	<math>		<math>
	\begin{array}{cccccc}		\begin{array}{cccccc}

	\min\limits_x & \multicolumn{5}{l}{ \Large (c1 x) \over (c2 x) } \\\mbox{subject to} & x_L & \leq & x & \leq & x_U \\& b_L & \leq & Ax & \leq & b_U \\		\min\limits_x & \multicolumn{5}{l}{ \Large (c1 x) \over (c2 x) } \\\mbox{subject to} & x_L & \leq & x & \leq & x_U \\& b_L & \leq & Ax & \leq & b_U \\
	\end{array}		\end{array}
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	where <math>$c1,c2,x,x_L,x_U \in \~~Rdim~~{n}$</math> , <math>$b_L,b_U \in \~~Rdim~~{m_1}$</math> and <math>$A \in \~~Rdim~~{m_1 \times n}$</math>.		where <math>c1,c2,x,x_L,x_U \in \MATHSET{R}^{n}</math> , <math>b_L,b_U \in \MATHSET{R}^{m_1}</math> and <math>A \in \MATHSET{R}^{m_1 \times n}</math>.


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

			<syntaxhighlight lang="matlab">
	Result=linRatSolve(Prob,PriLev)		Result=linRatSolve(Prob,PriLev)
			</syntaxhighlight>

	==~~=Description of~~ Inputs===		==Inputs===

	{\|		{\| class="wikitable"
	\|-valign="top"		\|-valign="top"
	\|''Prob''\|\|colspan="2"\|Structure Prob. Prob must be defined. Best is to use Prob = lpAssign(.....), if using the TQ format. Prob.QP.c1/c2 matrices should then be set (Prob.QP.c ignored).		\|''Prob''\|\|colspan="2"\|Structure Prob. Prob must be defined. Best is to use Prob = lpAssign(.....), if using the TQ format. Prob.QP.c1/c2 matrices should then be set (Prob.QP.c ignored).
	\|-		\|-valign="top"
	\|''PriLev''\|\|Print level in ''linRatSolve''.		\|''PriLev''\|\|Print level in ''linRatSolve''.
	\|-
	~~\|\|\|~~= ~~0Silent~~ except for error messages.		= 0 Silent except for error messages.
	\|-
	~~\|\|\|~~''> ''0 Print summary information about problem transformation.		''> ''0 Print summary information about problem transformation.
	\|-
	~~\|\|\|~~Calls ''PrintResult ''with specified ''PriLev''.		Calls ''PrintResult ''with specified ''PriLev''.
	\|-
	~~\|\|\|~~= 2 Standard output from ''PrintResult ''(default).		= 2 Standard output from ''PrintResult ''(default).
	\|-		\|-
	\|colspan="2"\|Extra fields used in Prob:		\|colspan="2"\|Extra fields used in Prob:
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	\|}		\|}

	==~~=Description of~~ Outputs===		==Outputs==
	{\|
			{\| class="wikitable"
			\|-valign="top"
	\|''Result''\|\|Structure with results from optimization. Output depends on the solver used.		\|''Result''\|\|Structure with results from optimization. Output depends on the solver used.
	~~\|-valign="top"~~
	~~\|\|\|~~The fields ''x_k'', ''f_k'', ''x_0'', ''xState'', ''bState'', ''v_k ''are transformed back to match the original problem.		The fields ''x_k'', ''f_k'', ''x_0'', ''xState'', ''bState'', ''v_k ''are transformed back to match the original problem.
	\|-
	~~\|\|\|~~The output in Result.Prob is the result after linRatSolve transformed the problem, i.e. the altered Prob structure		The output in Result.Prob is the result after linRatSolve transformed the problem, i.e. the altered Prob structure
	\|}		\|}

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	\end{array}		\end{array}
	</math>		</math>


	The problem then becomes:		The problem then becomes:


	<math>		<math>
	\begin{array}{cccccc}		\begin{array}{cccccc}
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	</math>		</math>

	where <math>$A1 \in \~~Rdim~~{N},\; A1=speye(N)$</math>.		where <math>$A1 \in \MATHSET{R}^{N},\; A1=speye(N)$</math>.

	OBSERVE the denominator ''c''2''x ''must always be positive. It is normally a good a idea to run the problem with both signs (multiply each side by -1).		OBSERVE the denominator ''c''2''x ''must always be positive. It is normally a good a idea to run the problem with both signs (multiply each side by -1).

Elias at 12:21, 11 July 2011

2011-07-11T12:21:31Z

← Older revision		Revision as of 07:21, 11 July 2011
Line 1:		Line 1:
			[[Category:Solvers]]
			{{cleanup\|Clean this article.}}

	==Purpose==		==Purpose==

Elias: /* Calling Syntax */

2011-07-10T13:50:08Z

Calling Syntax

← Older revision		Revision as of 08:50, 10 July 2011
Line 56:		Line 56:
	\|\|\|The output in Result.Prob is the result after linRatSolve transformed the problem, i.e. the altered Prob structure		\|\|\|The output in Result.Prob is the result after linRatSolve transformed the problem, i.e. the altered Prob structure
	\|}		\|}


	==Description==		==Description==

Elias: /* Calling Syntax */

2011-07-10T13:46:50Z

Calling Syntax

← Older revision		Revision as of 08:46, 10 July 2011
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	\|''z1_L ''\|\|Lower bound for z1 (default 1e-5). See description below		\|''z1_L ''\|\|Lower bound for z1 (default 1e-5). See description below
	\|}		\|}
	~~'''~~Description of Outputs~~'''~~
			===Description of Outputs===
	{\|		{\|
	\|''Result''\|\|Structure with results from optimization. Output depends on the solver used.		\|''Result''\|\|Structure with results from optimization. Output depends on the solver used.
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	===Description===		==Description==

	The linear ratio problem is solved by linRatSolve by rewriting the problem as a linear constrained optimization problem. n+1 variables z1 and z2(2:n+1) are needed, stored as x(1:n+1). The n original variables are removed so one more variable exists in the final problem.		The linear ratio problem is solved by linRatSolve by rewriting the problem as a linear constrained optimization problem. n+1 variables z1 and z2(2:n+1) are needed, stored as x(1:n+1). The n original variables are removed so one more variable exists in the final problem.

Elias: Created page with "==Purpose== Finds a linearly constrained solution of a function of the ratio of two linear functions with the use of any suitable TOMLAB solver. Binary and integer variables ar..."

2011-07-10T13:46:07Z

Created page with "==Purpose== Finds a linearly constrained solution of a function of the ratio of two linear functions with the use of any suitable TOMLAB solver. Binary and integer variables ar..."

New page

==Purpose==

Finds a linearly constrained solution of a function of the ratio of two linear functions with the use of any suitable TOMLAB solver. Binary and integer variables are not supported.

linRatSolve solves problems of the type:

<math>
\begin{array}{cccccc}

\min\limits_x & \multicolumn{5}{l}{ \Large (c1 x) \over (c2 x) } \\\mbox{subject to} & x_L & \leq & x & \leq & x_U \\& b_L & \leq & Ax & \leq & b_U \\
\end{array}
</math>

where <math>$c1,c2,x,x_L,x_U \in \Rdim{n}$</math> , <math>$b_L,b_U \in \Rdim{m_1}$</math> and <math>$A \in \Rdim{m_1 \times n}$</math>.

==Calling Syntax==

Result=linRatSolve(Prob,PriLev)

===Description of Inputs===

{|
|-valign="top"
|''Prob''||colspan="2"|Structure Prob. Prob must be defined. Best is to use Prob = lpAssign(.....), if using the TQ format. Prob.QP.c1/c2 matrices should then be set (Prob.QP.c ignored).
|-
|''PriLev''||Print level in ''linRatSolve''.
|-
|||= 0Silent except for error messages.
|-
|||''> ''0 Print summary information about problem transformation.
|-
|||Calls ''PrintResult ''with specified ''PriLev''.
|-
|||= 2 Standard output from ''PrintResult ''(default).
|-
|colspan="2"|Extra fields used in Prob:
|-valign="top"
|''SolverRat''||Name of the TOMLAB solver. Valid names are: cplex, minos, snopt, xa and more. See SolverList('lp');
|-
|''QP.c1''||The numerator in the objective.
|-
|''QP.c2''||The denominator in the objective.
|-
|''z1_L ''||Lower bound for z1 (default 1e-5). See description below
|}
'''Description of Outputs'''
{|
|''Result''||Structure with results from optimization. Output depends on the solver used.
|-valign="top"
|||The fields ''x_k'', ''f_k'', ''x_0'', ''xState'', ''bState'', ''v_k ''are transformed back to match the original problem.
|-
|||The output in Result.Prob is the result after linRatSolve transformed the problem, i.e. the altered Prob structure
|}

===Description===

The linear ratio problem is solved by linRatSolve by rewriting the problem as a linear constrained optimization problem. n+1 variables z1 and z2(2:n+1) are needed, stored as x(1:n+1). The n original variables are removed so one more variable exists in the final problem.

<math>
\begin{array}{ccc}
\\z1 & = & 1 / (c2 x) \\
z2 & = & x z1 \\
z1 (c1 x) & = & (c1 z1 x) = c1 z2\\
\end{array}
</math>

The problem then becomes:

<math>
\begin{array}{cccccc}
\multicolumn{6}{l}{\min\limits_x c1 z2}\\\\\mbox{subject to} & z1_L & \leq & z1 & \leq & \infty \\& 1 & \leq & c2 z2 & \leq & 1 \\& 0 & \leq & A z2 - z1 beq & \leq & 0 \\& -\infty & \leq & A z2 - z1 b_U & \leq & 0 \\& -\infty & \leq & - A z2 + z1 b_L & \leq & 0 \\\\& 0 & \leq & A1 z2 - z1 xeq & \leq & 0 \\& -\infty & \leq & A1 z2 - z1 x_U & \leq & 0 \\& -\infty & \leq & - A1 z2 + z1 x_L & \leq & 0 \\\end{array}
</math>

where <math>$A1 \in \Rdim{N},\; A1=speye(N)$</math>.

OBSERVE the denominator ''c''2''x ''must always be positive. It is normally a good a idea to run the problem with both signs (multiply each side by -1).

==See Also==

[[lpassign|''lpAssign'']].