PdcoTL - Revision history

Bjorn: /* Description */

2014-08-06T14:55:05Z

Description

← Older revision		Revision as of 10:55, 6 August 2014
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	\min\limits_{x,r} &		\min\limits_{x,r} &
	{\phi(x) + \frac{1}{2}\\|D_1 x\\|^2 + \frac{1}{2}\\|r\\|^2 } \\		{\phi(x) + \frac{1}{2}\\|D_1 x\\|^2 + \frac{1}{2}\\|r\\|^2 } \\
	\\		\end{array}
			</math>

			<math>
			\begin{array}{lllcll}
	\mathrm{s/t} & x_L & \leq & x & \leq & x_U \\		\mathrm{s/t} & x_L & \leq & x & \leq & x_U \\
	{} & Ax & + & D_{2}r & = & b \\		{} & Ax & + & D_{2}r & = & b \\

Elias at 12:16, 16 January 2012

2012-01-16T12:16:53Z

← Older revision		Revision as of 08:16, 16 January 2012
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	==Calling Syntax==		==Calling Syntax==

	<~~syntaxhighlight~~ lang="matlab">		<source lang="matlab">
	Result=tomRun('pdco',Prob,...);		Result=tomRun('pdco',Prob,...);
	</~~syntaxhighlight~~>		</source>

	==Inputs==		==Inputs==

Elias at 12:04, 10 January 2012

2012-01-10T12:04:44Z

← Older revision		Revision as of 08:04, 10 January 2012
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	''pdcoTL ''solves linearly constrained convex nonlinear optimization problems of the kind		''pdcoTL ''solves linearly constrained convex nonlinear optimization problems of the kind

			====Equation: pdcoTL====
	~~<equation id~~=~~"eqn~~:~~pdco">~~
	<math>		<math>
	\begin{array}{cccccc}		\begin{array}{cccccc}
Line 13:		Line 12:
	\end{array}		\end{array}
	</math>		</math>
	~~</equation>~~


	where <math>f(x)</math> is a convex nonlinear function, <math>x,x_L,x_U \in \mathbb{R}^{n}</math> , <math>A\in \mathbb{R}^{m \times n}</math> and <math>b_L, b_U \in \mathbb{R}^{m}</math>.		where <math>f(x)</math> is a convex nonlinear function, <math>x,x_L,x_U \in \mathbb{R}^{n}</math> , <math>A\in \mathbb{R}^{m \times n}</math> and <math>b_L, b_U \in \mathbb{R}^{m}</math>.
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	''pdco ''implements an primal-dual barrier method developed at Stanford Systems Optimization Laboratory (SOL).		''pdco ''implements an primal-dual barrier method developed at Stanford Systems Optimization Laboratory (SOL).

	The problem (~~<xr id="eqn~~:~~pdco" />~~) is first reformulated into SOL PDCO form:		The problem ([[#Equation: pdcoTL]]) is first reformulated into SOL PDCO form:

Elias at 14:41, 8 December 2011

2011-12-08T14:41:32Z

← Older revision		Revision as of 10:41, 8 December 2011
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	where <math>f(x)</math> is a convex nonlinear function, <math>x,x_L,x_U \in \~~MATHSET~~{R}^{n}</math> , <math>A\in \~~MATHSET~~{R}^{m \times n}</math> and <math>b_L, b_U \in \~~MATHSET~~{R}^{m}</math>.		where <math>f(x)</math> is a convex nonlinear function, <math>x,x_L,x_U \in \mathbb{R}^{n}</math> , <math>A\in \mathbb{R}^{m \times n}</math> and <math>b_L, b_U \in \mathbb{R}^{m}</math>.

	==Calling Syntax==		==Calling Syntax==
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	\mathrm{s/t} & x_L & \leq & x & \leq & x_U \\		\mathrm{s/t} & x_L & \leq & x & \leq & x_U \\
	{} & & & Ax & = & b \\		{} & & & Ax & = & b \\
	{} & ~~\multicolumn{5}{l}~~{r \mathrm{\ unconstrained}} \\		{} & {r \mathrm{\ unconstrained}} \\
	\end{array}		\end{array}
	</math>		</math>
Line 138:		Line 138:
	\begin{array}{lllcll}		\begin{array}{lllcll}
	\min\limits_{x,r} &		\min\limits_{x,r} &
	~~\multicolumn{5}{l}~~{\phi(x) + \frac{1}{2}\\|D_1 x\\|^2 + \frac{1}{2}\\|r\\|^2 } \\		{\phi(x) + \frac{1}{2}\\|D_1 x\\|^2 + \frac{1}{2}\\|r\\|^2 } \\
	\\		\\
	\mathrm{s/t} & x_L & \leq & x & \leq & x_U \\		\mathrm{s/t} & x_L & \leq & x & \leq & x_U \\

Elias: /* Purpose */

2011-07-21T10:21:15Z

Purpose

← Older revision		Revision as of 06:21, 21 July 2011
Line 3:		Line 3:

	''pdcoTL ''solves linearly constrained convex nonlinear optimization problems of the kind		''pdcoTL ''solves linearly constrained convex nonlinear optimization problems of the kind


	<equation id="eqn:pdco">		<equation id="eqn:pdco">

Elias: /* Purpose */

2011-07-21T10:20:56Z

Purpose

← Older revision		Revision as of 06:20, 21 July 2011
Line 13:		Line 13:
	</math>		</math>
	</equation>		</equation>


	where <math>f(x)</math> is a convex nonlinear function, <math>x,x_L,x_U \in \MATHSET{R}^{n}</math> , <math>A\in \MATHSET{R}^{m \times n}</math> and <math>b_L, b_U \in \MATHSET{R}^{m}</math>.		where <math>f(x)</math> is a convex nonlinear function, <math>x,x_L,x_U \in \MATHSET{R}^{n}</math> , <math>A\in \MATHSET{R}^{m \times n}</math> and <math>b_L, b_U \in \MATHSET{R}^{m}</math>.

Elias: /* Inputs */

2011-07-21T10:12:13Z

Inputs

← Older revision		Revision as of 06:12, 21 July 2011
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	\|\|\|''eps_x''\|\|Accuracy for satisfying ''x''1 ''. * z''1 = 0'', x''2 ''.z''1 = 0, where ''z ''= ''z''1 ''- z''2 and ''z''1 '', z''2 ''> ''0.(''Prob.SOL.pdco.OptTol'')		\|\|\|''eps_x''\|\|Accuracy for satisfying ''x''1 ''. * z''1 = 0'', x''2 ''.z''1 = 0, where ''z ''= ''z''1 ''- z''2 and ''z''1 '', z''2 ''> ''0.(''Prob.SOL.pdco.OptTol'')
	\|-valign="top"		\|-valign="top"
	\|\|\|''bTol''\|\|Accuracy for satisfying ''Ax ''+ ''D''<sub>2</sub>''r ''= ''b'', ''A<sup>T</sup> y ''+ ''z ''= ''&~~Delta~~;f ''(''x'') and ''x - x''1 = ''bL , x ''+''x''2 = ''bU '', where ''x''<sub>1</sub> '', x''<sub>2</sub> ''> ''0 (''Prob.SOL.pdco.FeaTol'')		\|\|\|''bTol''\|\|Accuracy for satisfying ''Ax ''+ ''D''<sub>2</sub>''r ''= ''b'', ''A<sup>T</sup> y ''+ ''z ''= ''∇f ''(''x'') and ''x - x''1 = ''bL , x ''+''x''2 = ''bU '', where ''x''<sub>1</sub> '', x''<sub>2</sub> ''> ''0 (''Prob.SOL.pdco.FeaTol'')
	\|-		\|-
	\|\|\|''wait''\|\|0 - solve the problem with default internal parameters; 1 - pause: allows interactive resetting of parameters. (''Prob.SOL.pdco.wait'')		\|\|\|''wait''\|\|0 - solve the problem with default internal parameters; 1 - pause: allows interactive resetting of parameters. (''Prob.SOL.pdco.wait'')

Elias at 10:01, 21 July 2011

2011-07-21T10:01:21Z

← Older revision		Revision as of 06:01, 21 July 2011
Line 1:		Line 1:
	[[Category:Solvers]]		[[Category:Solvers]]
	~~{{cleanup\|Clean this article.}}~~

	==Purpose==		==Purpose==

	''pdcoTL ''solves linearly constrained convex nonlinear optimization problems of the kind		''pdcoTL ''solves linearly constrained convex nonlinear optimization problems of the kind

			<equation id="eqn:pdco">
	<math>		<math>
	\begin{array}{cccccc}		\begin{array}{cccccc}
Line 14:		Line 12:
	\end{array}		\end{array}
	</math>		</math>
			</equation>

			where <math>f(x)</math> is a convex nonlinear function, <math>x,x_L,x_U \in \MATHSET{R}^{n}</math> , <math>A\in \MATHSET{R}^{m \times n}</math> and <math>b_L, b_U \in \MATHSET{R}^{m}</math>.
	where <math>f(x)</math> is a convex nonlinear function, <math>x,x_L,x_U \in \RR^n</math> , <math>A\in \RR^{m \times n}</math> and <math>b_L, b_U \in \RR^m</math>.

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

			<syntaxhighlight lang="matlab">
	Result=tomRun('pdco',Prob,...);		Result=tomRun('pdco',Prob,...);
			</syntaxhighlight>

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

	{\|		{\| class="wikitable"
	\|''Prob''\|\|colspan="2"\|Problem description structure. The following fields are used:		!''Prob''\|\|colspan="2"\|Problem description structure. The following fields are used:
	\|-		\|-
	\|\|\|''x_0''\|\|Initial ''x ''vector, used if non-empty.		\|\|\|''x_0''\|\|Initial ''x ''vector, used if non-empty.
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	\|\|\|''eps_x''\|\|Accuracy for satisfying ''x''1 ''. * z''1 = 0'', x''2 ''.z''1 = 0, where ''z ''= ''z''1 ''- z''2 and ''z''1 '', z''2 ''> ''0.(''Prob.SOL.pdco.OptTol'')		\|\|\|''eps_x''\|\|Accuracy for satisfying ''x''1 ''. * z''1 = 0'', x''2 ''.z''1 = 0, where ''z ''= ''z''1 ''- z''2 and ''z''1 '', z''2 ''> ''0.(''Prob.SOL.pdco.OptTol'')
	\|-valign="top"		\|-valign="top"
	\|\|\|''bTol''\|\|Accuracy for satisfying ''Ax ''+ ''D''2 ''r ''= ''b'', ''AT y ''+ ''z ''= ''?f ''(''x'') and ''x - x''1 = ''bL , x ''+''x''2 = ''bU '', where ''x''1 '', x''2 ''> ''0 (''Prob.SOL.pdco.FeaTol'')		\|\|\|''bTol''\|\|Accuracy for satisfying ''Ax ''+ ''D''<sub>2</sub>''r ''= ''b'', ''A<sup>T</sup> y ''+ ''z ''= ''Δf ''(''x'') and ''x - x''1 = ''bL , x ''+''x''2 = ''bU '', where ''x''<sub>1</sub> '', x''<sub>2</sub> ''> ''0 (''Prob.SOL.pdco.FeaTol'')
	\|-		\|-
	\|\|\|''wait''\|\|0 - solve the problem with default internal parameters; 1 - pause: allows interactive resetting of parameters. (''Prob.SOL.pdco.wait'')		\|\|\|''wait''\|\|0 - solve the problem with default internal parameters; 1 - pause: allows interactive resetting of parameters. (''Prob.SOL.pdco.wait'')
	\|}		\|}

	==~~=Description of~~ Outputs===		==Outputs==

	{\|		{\| class="wikitable"
	\|''Result''\|\|colspan="2"\|Structure with result from optimization. The following fields are set by ''pdcoTL''		!''Result''\|\|colspan="2"\|Structure with result from optimization. The following fields are set by ''pdcoTL''
	\|-		\|-
	\|\|\|''x_k''\|\|Solution vector		\|\|\|''x_k''\|\|Solution vector
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	\|\|\|''v_k''\|\|Lagrangian multipliers (orignal bounds + constraints )		\|\|\|''v_k''\|\|Lagrangian multipliers (orignal bounds + constraints )
	\|-valign="top"		\|-valign="top"
	\|\|\|''y_k''\|\|Lagrangian multipliers (for bounds + dual solution vector) The full \[''z''; ''y''\] ~~vec- tor~~ as returned from ''pdco'', including slacks and extra linear constraints after rewriting constraints: ''-inf < ~~b L~~ < A * x < ~~b U~~ < inf ''; non-inf lower AND upper bounds		\|\|\|''y_k''\|\|Lagrangian multipliers (for bounds + dual solution vector) The full [''z''; ''y''] vector as returned from ''pdco'', including slacks and extra linear constraints after rewriting constraints: ''-inf < b_L < A * x < b_U < inf ''; non-inf lower AND upper bounds
	\|-		\|-
	\|\|\|''ExitFlag''\|\|Tomlab Exit status from ''pdco ''MEX		\|\|\|''ExitFlag''\|\|Tomlab Exit status from ''pdco ''MEX
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	''pdco ''implements an primal-dual barrier method developed at Stanford Systems Optimization Laboratory (SOL).		''pdco ''implements an primal-dual barrier method developed at Stanford Systems Optimization Laboratory (SOL).

	The problem (19) is first reformulated into SOL PDCO form:		The problem (<xr id="eqn:pdco" />) is first reformulated into SOL PDCO form:


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	where <math>D_1</math> and <math>D_2</math> are positive-definite diagonal matrices defined from <math>d_1</math>, <math>d_2</math> given in Prob.SOL.d1 and Prob.SOL.d2.		where <math>D_1</math> and <math>D_2</math> are positive-definite diagonal matrices defined from <math>d_1</math>, <math>d_2</math> given in Prob.SOL.d1 and Prob.SOL.d2.

	In particular, <math>d_2</math> indicates the accuracy required for satisfying each row of <math>Ax=b</math>. See ''pdco'' for a detailed discussion of <math>D_1</math> and <math>D_2</math>. Note that in ''pdco'', the objective <math>f(x)</math> is denoted <math>\phi(x)</math>, <math>bl == x\_L</math> and <math>bu == x\_U</math>.		In particular, <math>d_2</math> indicates the accuracy required for satisfying each row of <math>Ax=b</math>. See ''pdco.m'' for a detailed discussion of <math>D_1</math> and <math>D_2</math>. Note that in ''pdco'', the objective <math>f(x)</math> is denoted <math>\phi(x)</math>, <math>bl == x\_L</math> and <math>bu == x\_U</math>.

	==Examples==		==Examples==

	Problem 14 and 15 in ''tomlab/testprob/con prob.m.m ''are good examples of the use of ''pdcoTL''.		Problem 14 and 15 in ''tomlab/testprob/con prob.m ''are good examples of the use of ''pdcoTL''.

	==M-files Used==		==M-files Used==

Elias at 12:23, 11 July 2011

2011-07-11T12:23:09Z

← Older revision		Revision as of 08:23, 11 July 2011
Line 1:		Line 1:
			[[Category:Solvers]]
			{{cleanup\|Clean this article.}}

	==Purpose==		==Purpose==

Elias: Created page with "==Purpose== ''pdcoTL ''solves linearly constrained convex nonlinear optimization problems of the kind $\begin{array}{cccccc} \min\limits_x & f(x) \\ \mathrm{s/t} & x_L &..."$

2011-07-11T10:42:38Z

Created page with "==Purpose== ''pdcoTL ''solves linearly constrained convex nonlinear optimization problems of the kind <math> \begin{array}{cccccc} \min\limits_x & f(x) \\ \mathrm{s/t} & x_L &..."

New page

==Purpose==

''pdcoTL ''solves linearly constrained convex nonlinear optimization problems of the kind

<math>
\begin{array}{cccccc}
\min\limits_x & f(x) \\
\mathrm{s/t} & x_L & \leq & x & \leq & x_U \\
{} & b_L & \leq & Ax & \leq & b_U \\
\end{array}
</math>

where <math>f(x)</math> is a convex nonlinear function, <math>x,x_L,x_U \in \RR^n</math> , <math>A\in \RR^{m \times n}</math> and <math>b_L, b_U \in \RR^m</math>.

==Calling Syntax==

Result=tomRun('pdco',Prob,...);

===Description of Inputs===

{|
|''Prob''||colspan="2"|Problem description structure. The following fields are used:
|-
|||''x_0''||Initial ''x ''vector, used if non-empty.
|-
|||''A''||The linear constraint matrix.
|-
|||''b_L,b_U''||Lower and upper bounds for the linear constraints.
|-
|||''PriLevOpt''||Print level in ''pdsco ''solver. If ''> ''0: prints summary information.
|-
|||''SOL''||Structure with SOL special parameters:
|-
|||''pdco''||Options structure with fields as defined by ''pdcoSet''.
|-valign="top"
|||''d1''||Primal regularization vector. Must be a positive vector (length ''n'') or scalar, in which case ''D''1 = ''diag''(''d''1) is used. Default: 10''-''4 .
|-valign="top"
|||''d2''||Dual regularization vector. Must be a positive vector (length ''m'') or a scalar value, in which case ''D''2 = ''diag''(''d''2) is used. Default: 10''-''4 .
|-
|||''y0''||Initial dual parameters for linear constraints (default 0)
|-valign="top"
|||''z0''||Initial dual parameters for simple bounds (default 1''/N '')
''xsize'',''zsize ''are used to scale (''x, y, z''). Good estimates should improve the performance of the barrier method.
|-
|||''xsize''||Estimate of the biggest x at the solution. (default 1''/N '')
|-
|||''zsize''||Estimate of the biggest z at the solution. (default 1''/N '')
|-
|||''optParam''||Structure with optimization parameters. The following fields are used:
|-
|||''MaxIter''||Maximum number of iterations. (''Prob.SOL.pdco.MaxIter'').
|-valign="top"
|||''MinorIter''||Maximum number of iterations in ''LSQR'' (''Prob.SOL.pdco.LSQRMaxIter'').
|-
|||''eps_x''||Accuracy for satisfying ''x''1 ''. * z''1 = 0'', x''2 ''.z''1 = 0, where ''z ''= ''z''1 ''- z''2 and ''z''1 '', z''2 ''> ''0.(''Prob.SOL.pdco.OptTol'')
|-valign="top"
|||''bTol''||Accuracy for satisfying ''Ax ''+ ''D''2 ''r ''= ''b'', ''AT y ''+ ''z ''= ''?f ''(''x'') and ''x - x''1 = ''bL , x ''+''x''2 = ''bU '', where ''x''1 '', x''2 ''> ''0 (''Prob.SOL.pdco.FeaTol'')
|-
|||''wait''||0 - solve the problem with default internal parameters; 1 - pause: allows interactive resetting of parameters. (''Prob.SOL.pdco.wait'')
|}

===Description of Outputs===

{|
|''Result''||colspan="2"|Structure with result from optimization. The following fields are set by ''pdcoTL''
|-
|||''x_k''||Solution vector
|-
|||''f_k''||Function value at optimum
|-
|||''g_k''||Gradient of the function at the solution
|-
|||''H_k''||Hessian of the function at the solution, diagonal only
|-
|||''x_0''||Initial solution vector
|-
|||''f_0''||Function value at start, x = x_0
|-
|||''xState''||State of variables. Free == 0; On lower == 1; On upper == 2; Fixed == 3;
|-valign="top"
|||''bState''||State of linear constraints. Free == 0; Lower == 1; Upper == 2; Equality == 3;
|-
|||''v_k''||Lagrangian multipliers (orignal bounds + constraints )
|-valign="top"
|||''y_k''||Lagrangian multipliers (for bounds + dual solution vector) The full \[''z''; ''y''\] vec- tor as returned from ''pdco'', including slacks and extra linear constraints after rewriting constraints: ''-inf < b L < A * x < b U < inf ''; non-inf lower AND upper bounds
|-
|||''ExitFlag''||Tomlab Exit status from ''pdco ''MEX
|-
|||''Inform''||''pdco''information parameter: 0 = Solution found;
|-
|||0||Solution found
|-
|||1||Too many iterations
|-
|||2||Linesearch failed too often
|-
|||''Iter''||Number of iterations
|-
|||''FuncEv''||Number of function evaluations
|-
|||''GradEv''||Number of gradient evaluations
|-
|||''HessEv''||Number of Hessian evaluations
|-
|||''Solver''||Name of the solver ('pdco')
|-
|||''SolverAlgorithm''||Description of the solver
|}

==Description==

''pdco ''implements an primal-dual barrier method developed at Stanford Systems Optimization Laboratory (SOL).

The problem (19) is first reformulated into SOL PDCO form:

<math>
\begin{array}{cccccc}
\min\limits_x & f(x) \\
\mathrm{s/t} & x_L & \leq & x & \leq & x_U \\
{} & & & Ax & = & b \\
{} & \multicolumn{5}{l}{r \mathrm{\ unconstrained}} \\
\end{array}
</math>

The problem actually solved by ''pdco ''is

<math>
\begin{array}{lllcll}
\min\limits_{x,r} &
\multicolumn{5}{l}{\phi(x) + \frac{1}{2}\|D_1 x\|^2 + \frac{1}{2}\|r\|^2 } \\
\\
\mathrm{s/t} & x_L & \leq & x & \leq & x_U \\
{} & Ax & + & D_{2}r & = & b \\
\end{array}
</math>

where <math>D_1</math> and <math>D_2</math> are positive-definite diagonal matrices defined from <math>d_1</math>, <math>d_2</math> given in Prob.SOL.d1 and Prob.SOL.d2.

In particular, <math>d_2</math> indicates the accuracy required for satisfying each row of <math>Ax=b</math>. See ''pdco'' for a detailed discussion of <math>D_1</math> and <math>D_2</math>. Note that in ''pdco'', the objective <math>f(x)</math> is denoted <math>\phi(x)</math>, <math>bl == x\_L</math> and <math>bu == x\_U</math>.

==Examples==

Problem 14 and 15 in ''tomlab/testprob/con prob.m.m ''are good examples of the use of ''pdcoTL''.

==M-files Used==

''pdcoSet.m'', ''pdco.m'', ''Tlsqrmat.m''

==See Also==

''pdscoTL.m''

PdcoTL - Revision history

Bjorn: /* Description */

Elias at 12:16, 16 January 2012

Elias at 12:04, 10 January 2012

Elias at 14:41, 8 December 2011

Elias: /* Purpose */

Elias: /* Purpose */

Elias: /* Inputs */

Elias at 10:01, 21 July 2011

Elias at 12:23, 11 July 2011

Elias: Created page with "==Purpose== ''pdcoTL ''solves linearly constrained convex nonlinear optimization problems of the kind \begin{array}{cccccc} \min\limits_x & f(x) \\ \mathrm{s/t} & x_L &..."

Elias: Created page with "==Purpose== ''pdcoTL ''solves linearly constrained convex nonlinear optimization problems of the kind $\begin{array}{cccccc} \min\limits_x & f(x) \\ \mathrm{s/t} & x_L &..."$