Category:Problem characterization - Revision history

JonasSchulze: Text replacement - "" to ""

2016-01-23T10:11:00Z

Text replacement - "<bibreferences/>" to "<biblist />"

← Older revision		Revision as of 10:11, 23 January 2016
Line 31:		Line 31:

	== References ==		== References ==
	<~~bibreferences~~/>		<biblist />

	<!--List of all categories this category is part of. -->		<!--List of all categories this category is part of. -->

JonasSchulze: Text replacement - "\(.*)\<\/bibref\>" to ""

2016-01-20T17:54:23Z

Text replacement - "\<bibref\>(.*)\<\/bibref\>" to "<bib id="$1" />"

← Older revision		Revision as of 17:54, 20 January 2016
Line 27:		Line 27:
	The model equations are described by the functional <math>F[\cdot]</math> ([[:Category:Model characterization\|model characterization]]). The objective functional <math>\Phi[\cdot]</math>, the constraints <math>C[\cdot]</math> that may include control- and path-constraints, and the interior point constraints <math>\Gamma[x]</math> that specify also the boundary conditions are classified as [[:Category:Objective characterization \| objective characterization]] objective characterization. Characteristics of an optimal solution from a control theory point of view are listed as [[:Category:Solution characterization \| solution characterization]].		The model equations are described by the functional <math>F[\cdot]</math> ([[:Category:Model characterization\|model characterization]]). The objective functional <math>\Phi[\cdot]</math>, the constraints <math>C[\cdot]</math> that may include control- and path-constraints, and the interior point constraints <math>\Gamma[x]</math> that specify also the boundary conditions are classified as [[:Category:Objective characterization \| objective characterization]] objective characterization. Characteristics of an optimal solution from a control theory point of view are listed as [[:Category:Solution characterization \| solution characterization]].

	The formulation of optimization problems is typically not unique. Sometimes, as in the case of MPEC reformulations of state-dependent switches <~~bibref>~~Baumrucker2009</~~bibref~~>, disjunctive programming <~~bibref>~~Grossmann2002</~~bibref~~>, or outer convexification <~~bibref>~~Sager2009</~~bibref~~>, reformulations may be seen as part of the solution approach in the sense of the ''modeling for optimization paradigm'' <~~bibref>~~Oldenburg2008</~~bibref~~>. Even in obvious cases, such as a Mayer term versus a Lagrange term formulation, they may be mathematically, but not necessarily algorithmically equivalent. We propose to use either the original or the most adequate formulation of the optimization problem and list possible reformulations as variants.		The formulation of optimization problems is typically not unique. Sometimes, as in the case of MPEC reformulations of state-dependent switches <bib id="Baumrucker2009" />, disjunctive programming <bib id="Grossmann2002" />, or outer convexification <bib id="Sager2009" />, reformulations may be seen as part of the solution approach in the sense of the ''modeling for optimization paradigm'' <bib id="Oldenburg2008" />. Even in obvious cases, such as a Mayer term versus a Lagrange term formulation, they may be mathematically, but not necessarily algorithmically equivalent. We propose to use either the original or the most adequate formulation of the optimization problem and list possible reformulations as variants.

SebastianSager: Added references

2010-11-18T16:27:08Z

Added references

← Older revision		Revision as of 16:27, 18 November 2010
Line 30:		Line 30:


			== References ==
			<bibreferences/>

			<!--List of all categories this category is part of. -->
	[[Category:Main]]		[[Category:Main]]

SebastianSager at 16:23, 18 November 2010

2010-11-18T16:23:37Z

← Older revision		Revision as of 16:23, 18 November 2010
Line 1:		Line 1:
	This is a super-category of all problem description categories, i.e., it lists all subcategories that have been defined so far. Note that the category [[:Category:MIOCP]] contains a list of ''all'' mixed-integer optimal control problems ~~defined on these pages so far~~. The following graph shows the organization of categories on mintoc.de.		This is a super-category of all problem description categories, i.e., it lists all subcategories that have been defined so far. Note that the category [[:Category:MIOCP]] contains a list of ''all'' mixed-integer optimal control problems in the library. The following graph shows the (yet incomplete) organization of categories on mintoc.de.
	{{#categorytree: Problem characterization\|depth=5\|mode=pages\|showcount}}		{{#categorytree: Problem characterization\|depth=5\|mode=pages\|showcount}}
	~~This categorization is yet incomplete and extended whenever new problems are added to the database that do not fit in this scheme.~~


	<!-- ------------- -->		<!-- ------------- -->

SebastianSager at 16:21, 18 November 2010

2010-11-18T16:21:40Z

← Older revision		Revision as of 16:21, 18 November 2010
Line 12:		Line 12:
	<p>		<p>
	<math>		<math>
	\begin{array}{~~llcl~~}		\begin{array}{lrcl}
	\displaystyle \min_{x, u, v} & \Phi[x,u,v]) \\[1.5ex]		\displaystyle \min_{x, u, v} & \Phi[x,u,v]) \\[1.5ex]
	\mbox{s.t.} & 0 & = & F[x,u,v], \\		\mbox{s.t.} & 0 & = & F[x,u,v], \\

SebastianSager: Added optimization problem

2010-11-18T16:13:06Z

Added optimization problem

← Older revision		Revision as of 16:13, 18 November 2010
Line 10:		Line 10:

	On the most abstract level, we want to solve an optimization problem that can be written as		On the most abstract level, we want to solve an optimization problem that can be written as
			<p>
			<math>
			\begin{array}{llcl}
			\displaystyle \min_{x, u, v} & \Phi[x,u,v]) \\[1.5ex]
			\mbox{s.t.} & 0 & = & F[x,u,v], \\
			& 0 & \le & C[x,u,v], \\
			& 0 & = & \Gamma[x].
			\end{array}
			</math>
			</p>

	Here <math>x(\cdot): \R^\mathrm{d} \mapsto \R^{n_x}</math> denotes the differential-algebraic states in a <math>d</math>-dimensional space. Note that we use the notation common in control theory with <math>x</math> as differential states and <math>u</math> as controls, not the PDE formulation with <math>x</math> as independent variable and <math>u</math> as differential states. Until now, for most applications we have <math>d=1</math> and the independent variable time <math>t \in [t_0, t_f]</math>, the case of ordinary or algebraic differential equations. <math>u(\cdot): \R^\mathrm{d} \mapsto \R^{n_u}</math> and <math>v(\cdot): \R^\mathrm{d} \mapsto \Omega</math> are controls, where <math>u(\cdot)</math> are continuous values that map to <math>\R^{n_u}</math>, and <math>v(\cdot)</math> are controls that map to a finite set <math>\Omega</math>. We allow also constant-in-time or constant-in-space control values rather than distributed controls.		Here <math>x(\cdot): \R^\mathrm{d} \mapsto \R^{n_x}</math> denotes the differential-algebraic states in a <math>d</math>-dimensional space. Note that we use the notation common in control theory with <math>x</math> as differential states and <math>u</math> as controls, not the PDE formulation with <math>x</math> as independent variable and <math>u</math> as differential states. Until now, for most applications we have <math>d=1</math> and the independent variable time <math>t \in [t_0, t_f]</math>, the case of ordinary or algebraic differential equations. <math>u(\cdot): \R^\mathrm{d} \mapsto \R^{n_u}</math> and <math>v(\cdot): \R^\mathrm{d} \mapsto \Omega</math> are controls, where <math>u(\cdot)</math> are continuous values that map to <math>\R^{n_u}</math>, and <math>v(\cdot)</math> are controls that map to a finite set <math>\Omega</math>. We allow also constant-in-time or constant-in-space control values rather than distributed controls.
Line 17:		Line 27:
	Basically two different kinds of switching events are at the origin of hybrid systems, controllable and state-dependent ones. The first kind is due to degrees of freedom for the optimization, in particular with controls that may only take values from a finite set. The second kind is due to state-dependent switches in the model equations, e.g., ground contact of a robot leg or overflow of weirs in a distillation column. The focus in the benchmark library is on the first kind of switches, whereas the second one is of course important for a classification of the model equations, as for certain MIOCPs both kinds occur.		Basically two different kinds of switching events are at the origin of hybrid systems, controllable and state-dependent ones. The first kind is due to degrees of freedom for the optimization, in particular with controls that may only take values from a finite set. The second kind is due to state-dependent switches in the model equations, e.g., ground contact of a robot leg or overflow of weirs in a distillation column. The focus in the benchmark library is on the first kind of switches, whereas the second one is of course important for a classification of the model equations, as for certain MIOCPs both kinds occur.

	The model equations are described by the functional <math>F[\cdot]</math> ([[:Category:Model characterization \| model characterization]]). The objective functional <math>\Phi[\cdot]</math>, the constraints <math>C[\cdot]</math> that may include control- and path-constraints, and the interior point constraints <math>\Gamma[x]</math> that specify also the boundary conditions are classified as [[:Category:Objective characterization \| objective characterization]] objective characterization. Characteristics of an optimal solution from a control theory point of view are listed as [[:Category:Solution characterization \| solution characterization]].		The model equations are described by the functional <math>F[\cdot]</math> ([[:Category:Model characterization\|model characterization]]). The objective functional <math>\Phi[\cdot]</math>, the constraints <math>C[\cdot]</math> that may include control- and path-constraints, and the interior point constraints <math>\Gamma[x]</math> that specify also the boundary conditions are classified as [[:Category:Objective characterization \| objective characterization]] objective characterization. Characteristics of an optimal solution from a control theory point of view are listed as [[:Category:Solution characterization \| solution characterization]].

	The formulation of optimization problems is typically not unique. Sometimes, as in the case of MPEC reformulations of state-dependent switches <bibref>Baumrucker2009</bibref>, disjunctive programming <bibref>Grossmann2002</bibref>, or outer convexification <bibref>Sager2009</bibref>, reformulations may be seen as part of the solution approach in the sense of the ''modeling for optimization paradigm'' <bibref>Oldenburg2008</bibref>. Even in obvious cases, such as a Mayer term versus a Lagrange term formulation, they may be mathematically, but not necessarily algorithmically equivalent. We propose to use either the original or the most adequate formulation of the optimization problem and list possible reformulations as variants.		The formulation of optimization problems is typically not unique. Sometimes, as in the case of MPEC reformulations of state-dependent switches <bibref>Baumrucker2009</bibref>, disjunctive programming <bibref>Grossmann2002</bibref>, or outer convexification <bibref>Sager2009</bibref>, reformulations may be seen as part of the solution approach in the sense of the ''modeling for optimization paradigm'' <bibref>Oldenburg2008</bibref>. Even in obvious cases, such as a Mayer term versus a Lagrange term formulation, they may be mathematically, but not necessarily algorithmically equivalent. We propose to use either the original or the most adequate formulation of the optimization problem and list possible reformulations as variants.

SebastianSager: Included overview text from paper

2010-11-18T15:13:28Z

Included overview text from paper

@@ Line 2: / Line 2: @@
 {{#categorytree: Problem characterization|depth=5|mode=pages|showcount}}
 This categorization is yet incomplete and extended whenever new problems are added to the database that do not fit in this scheme.
 [[Category:Main]]
 <!--
 <graphviz border='frame' format='png'>
 digraph G {

SebastianSager at 17:04, 30 November 2008

2008-11-30T17:04:47Z

@@ Line 1: / Line 1: @@
 This is a super-category of all problem description categories, i.e., it lists all subcategories that have been defined so far. Note that the category [[:Category:MIOCP]] contains a list of ''all'' mixed-integer optimal control problems defined on these pages so far. The following graph shows the organization of categories on mintoc.de.
 <!--
@@ Line 59: / Line 63: @@
 </graphviz>
 -->

SebastianSager: Commented out graphviz, added categorytree

2008-11-30T16:50:34Z

Commented out graphviz, added categorytree

← Older revision		Revision as of 16:50, 30 November 2008
Line 1:		Line 1:
	This is a super-category of all problem description categories, i.e., it lists all subcategories that have been defined so far. Note that the category [[:Category:MIOCP]] contains a list of ''all'' mixed-integer optimal control problems defined on these pages so far. The following graph shows the organization of categories on mintoc.de.		This is a super-category of all problem description categories, i.e., it lists all subcategories that have been defined so far. Note that the category [[:Category:MIOCP]] contains a list of ''all'' mixed-integer optimal control problems defined on these pages so far. The following graph shows the organization of categories on mintoc.de.

			<!--
	<graphviz border='frame' format='png'>		<graphviz border='frame' format='png'>
	digraph G {		digraph G {
Line 57:		Line 58:
	}		}
	</graphviz>		</graphviz>
			-->

			<categorytree mode=pages>Problem characterization</categorytree>

	This categorization is yet incomplete and extended whenever new problems are added to the database that do not fit in this scheme.		This categorization is yet incomplete and extended whenever new problems are added to the database that do not fit in this scheme.

	[[Category:Main]]		[[Category:Main]]

SebastianSager at 17:34, 29 November 2008

2008-11-29T17:34:53Z

@@ Line 5: / Line 5: @@
 rankdir=LR;
 node [shape=plaintext];
 Chattering [URL="index.php/Category:Chattering"];
 "Problem characterization"->"AMPL"