Convex generalized disjunctive programming (GDP) - Revision history

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	~~Author~~: Nicholas Schafhauser, Blerand Qeriqi, Ryan Cuppernull (SysEn 5800 Fall 2020)		Authors: Nicholas Schafhauser, Blerand Qeriqi, Ryan Cuppernull (SysEn 5800 Fall 2020)

	== Introduction ==		== Introduction ==

Wc593 at 11:39, 21 December 2020

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	~~Edited By~~: Nicholas Schafhauser, Blerand Qeriqi, Ryan Cuppernull		Author: Nicholas Schafhauser, Blerand Qeriqi, Ryan Cuppernull (SysEn 5800 Fall 2020)

	== Introduction ==		== Introduction ==

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	The two most common ways of reformulating a GDP problem into an MINLP are through Big-M (BM) and Hull Reformulation (HR). BM is the simpler of the two, while HR results in tighter relaxation (smaller feasible region) and faster solution times.<ref>Trespalacios, Francisco; Grossmann, Ignacio E. (2018): Improved Big-M Reformulation for Generalized Disjunctive Programs. Carnegie Mellon University. Journal contribution. <nowiki>https://doi.org/10.1184/R1/6467063.v1</nowiki> </ref>		The two most common ways of reformulating a GDP problem into an MINLP are through Big-M (BM) and Hull Reformulation (HR). BM is the simpler of the two, while HR results in tighter relaxation (smaller feasible region) and faster solution times.<ref>Trespalacios, Francisco; Grossmann, Ignacio E. (2018): Improved Big-M Reformulation for Generalized Disjunctive Programs. Carnegie Mellon University. Journal contribution. <nowiki>https://doi.org/10.1184/R1/6467063.v1</nowiki> </ref>

	Below is an example of the the GDP problem from above reformulated into an MINLP by using the ~~Big-M~~ method.		Below is an example of the the GDP problem from above reformulated into an MINLP by using the BM method.

	<math>\begin{align} \min z=f(x)\\		<math>\begin{align} \min z=f(x)\\
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	This MINLP reformulation can now be used in well-known solvers to calculate a solution.		This MINLP reformulation can now be used in well-known solvers to calculate a solution.

	The same GDP form will now be reformulated into an MINLP by using the ~~Hull Reformulation~~ method.		The same GDP form will now be reformulated into an MINLP by using the HR method.

	<math>\begin{align} \min z=f(x)\\		<math>\begin{align} \min z=f(x)\\
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	This same idea can be scaled to larger problems with more complex branching. Figure 3 illustrates a larger process network and all of the different decision points. This problem is able to be formulated into a GDP so that the most optimal route can be calculated to take through the network.		This same idea can be scaled to larger problems with more complex branching. Figure 3 illustrates a larger process network and all of the different decision points. This problem is able to be formulated into a GDP so that the most optimal route can be calculated to take through the network.
	== Conclusion ==		== Conclusion ==
	GDP is a programming method that applies disjunctive programming to MINLP problems. This method facilitates modeling discrete or continuous optimization problems by implementing algebraic constraints and logic expressions. The formulation of a GDP consists of Boolean and continuous variables and disjunctions and logic propositions. In the case of convex functions, GDPs can be reformulated using the ~~big-M~~ and the ~~hull relaxation~~. Formulation methods also include logic based methods disjunctive branch and bound and decomposition. Once reformulated into a standard MINLP, standard MILNP solvers, such as DICOPT<ref name=":3" />, SBB<ref name=":4" />, α-ECP<ref name=":5" /> and BARON<ref name=":6" />, can be used to determine optimal solutions<ref name=":0" />. The GDP method has important applications that include the optimization of complex chemical reactions and process planning.		GDP is a programming method that applies disjunctive programming to MINLP problems. This method facilitates modeling discrete or continuous optimization problems by implementing algebraic constraints and logic expressions. The formulation of a GDP consists of Boolean and continuous variables and disjunctions and logic propositions. In the case of convex functions, GDPs can be reformulated using the BM and the HR methods. Formulation methods also include logic based methods disjunctive branch and bound and decomposition. Once reformulated into a standard MINLP, standard MILNP solvers, such as DICOPT<ref name=":3" />, SBB<ref name=":4" />, α-ECP<ref name=":5" /> and BARON<ref name=":6" />, can be used to determine optimal solutions<ref name=":0" />. The GDP method has important applications that include the optimization of complex chemical reactions and process planning.

	== References ==		== References ==
	<references />		<references />

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	== Introduction ==		== Introduction ==
	Generalized disjunctive programming (GDP) involves logic propositions (Boolean variables) and sets of constraints that are chained together using the logical OR operator ( II ). GDP is an extension of linear disjunctive programming<ref>Balas, Egon. "Disjunctive Programming." Annals of Discrete Mathematics, 1979.</ref> that can be applied to Mixed Integer Non-Linear Programming (MINLP). GDP<ref>Raman and Grossman. "Modelling and Computational Techniques for Logic Based Integer Programming." Computers & Chemical Engineering, 1994.</ref>, is a generalization of disjunctive convex programming in the sense that it also allows the use of logic propositions that are expressed in terms of Boolean variables. In order to take advantage of current mixed-integer nonlinear programming solvers (e.g. DICOPT<ref name=":3">GAMS. DICOPT, https://www.gams.com/latest/docs/S_DICOPT.html</ref>, SBB<ref name=":4" />, α-ECP<ref name=":5">GAMS. AlphaECP, 1995, https://www.gams.com/latest/docs/S_ALPHAECP.html</ref>, BARON<ref name=":6">BARON, 1996, https://minlp.com/baron</ref>, ~~Coenne~~<ref name=":7">~~Coenne~~, 2006, https://projects.coin-or.org/Couenne</ref> etc.), GDPs are often reformulated as MINLPs.<ref name=":0">P. Ruiz, Juan; Grossmann, Ignacio E. (2012): A hierarchy of relaxations for nonlinear convex generalized disjunctive programming. Carnegie Mellon University. Journal contribution. <nowiki>https://doi.org/10.1184/R1/6466535.v1</nowiki> </ref>		Generalized disjunctive programming (GDP) involves logic propositions (Boolean variables) and sets of constraints that are chained together using the logical OR operator ( II ). GDP is an extension of linear disjunctive programming<ref>Balas, Egon. "Disjunctive Programming." Annals of Discrete Mathematics, 1979.</ref> that can be applied to Mixed Integer Non-Linear Programming (MINLP). GDP<ref>Raman and Grossman. "Modelling and Computational Techniques for Logic Based Integer Programming." Computers & Chemical Engineering, 1994.</ref>, is a generalization of disjunctive convex programming in the sense that it also allows the use of logic propositions that are expressed in terms of Boolean variables. In order to take advantage of current mixed-integer nonlinear programming solvers (e.g. DICOPT<ref name=":3">GAMS. DICOPT, https://www.gams.com/latest/docs/S_DICOPT.html</ref>, SBB<ref name=":4" />, α-ECP<ref name=":5">GAMS. AlphaECP, 1995, https://www.gams.com/latest/docs/S_ALPHAECP.html</ref>, BARON<ref name=":6">BARON, 1996, https://minlp.com/baron</ref>, Couenne<ref name=":7">Couenne, 2006, https://projects.coin-or.org/Couenne</ref> etc.), GDPs are often reformulated as MINLPs.<ref name=":0">P. Ruiz, Juan; Grossmann, Ignacio E. (2012): A hierarchy of relaxations for nonlinear convex generalized disjunctive programming. Carnegie Mellon University. Journal contribution. <nowiki>https://doi.org/10.1184/R1/6466535.v1</nowiki> </ref>
	[[File:GDP Intro.jpg\|none\|thumb\|523x523px\|Figure 1: Generalized Disjunctive Programming Methods<ref>Grossman, Ignacio E: Overview of Generalized Disjunctive Programming. Carnegie Mellon University.https://www.minlp.org/pdf/GBDEWOGrossmann.pdf</ref>]]		[[File:GDP Intro.jpg\|none\|thumb\|523x523px\|Figure 1: Generalized Disjunctive Programming Methods<ref>Grossman, Ignacio E: Overview of Generalized Disjunctive Programming. Carnegie Mellon University.https://www.minlp.org/pdf/GBDEWOGrossmann.pdf</ref>]]

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	== Numerical Example ==		== Numerical Example ==
	The following example was taken from the paper titled ''Generalized Disjunctive Programming: A Framework For Formulation and Alternative Algorithms For MINLP Optimization''.''<ref name=":1">P. Ruize, Juan; Grossmann, Ignacio E.: Generalized Disjunctive Programming: A Framework For Formulation And Alternative Algorithms For MINLP Optimization. Carnegie Mellon University. http://egon.cheme.cmu.edu/Papers/IMAGrossmannRuiz.pdf</ref>''		The following example was taken from the paper titled ''Generalized Disjunctive Programming: A Framework For Formulation and Alternative Algorithms For MINLP Optimization''.''<ref name=":1">P. Ruize, Juan; Grossmann, Ignacio E.: Generalized Disjunctive Programming: A Framework For Formulation And Alternative Algorithms For MINLP Optimization. Carnegie Mellon University. http://egon.cheme.cmu.edu/Papers/IMAGrossmannRuiz.pdf</ref>''

	~~[[File:GDP numeric example 1.png\|frameless\|600x600px]]~~

	[[File:GDP numeric example 3.png\|frameless\|600x600px]]		[[File:GDP numeric example 3.png\|frameless\|600x600px]]

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Introduction

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	== Introduction ==		== Introduction ==
	Generalized disjunctive programming (GDP) involves logic propositions (Boolean variables) and sets of constraints that are chained together using the logical OR operator ( II ). ~~Generalized disjunctive programming (~~GDP) is an extension of linear disjunctive programming<ref>Balas, Egon. "Disjunctive Programming." Annals of Discrete Mathematics, 1979.</ref> that can be applied to Mixed Integer Non-Linear Programming (MINLP). GDP<ref>Raman and Grossman. "Modelling and Computational Techniques for Logic Based Integer Programming." Computers & Chemical Engineering, 1994.</ref>, is a generalization of disjunctive convex programming in the sense that it also allows the use of logic propositions that are expressed in terms of Boolean variables. In order to take advantage of current mixed-integer nonlinear programming solvers (e.g. DICOPT<ref name=":3">GAMS. DICOPT, https://www.gams.com/latest/docs/S_DICOPT.html</ref>, SBB<ref name=":4" />, α-ECP<ref name=":5">GAMS. AlphaECP, 1995, https://www.gams.com/latest/docs/S_ALPHAECP.html</ref>, BARON<ref name=":6">BARON, 1996, https://minlp.com/baron</ref>, Coenne<ref name=":7">Coenne, 2006, https://projects.coin-or.org/Couenne</ref> etc.), GDPs are often reformulated as MINLPs.<ref name=":0">P. Ruiz, Juan; Grossmann, Ignacio E. (2012): A hierarchy of relaxations for nonlinear convex generalized disjunctive programming. Carnegie Mellon University. Journal contribution. <nowiki>https://doi.org/10.1184/R1/6466535.v1</nowiki> </ref>		Generalized disjunctive programming (GDP) involves logic propositions (Boolean variables) and sets of constraints that are chained together using the logical OR operator ( II ). GDP is an extension of linear disjunctive programming<ref>Balas, Egon. "Disjunctive Programming." Annals of Discrete Mathematics, 1979.</ref> that can be applied to Mixed Integer Non-Linear Programming (MINLP). GDP<ref>Raman and Grossman. "Modelling and Computational Techniques for Logic Based Integer Programming." Computers & Chemical Engineering, 1994.</ref>, is a generalization of disjunctive convex programming in the sense that it also allows the use of logic propositions that are expressed in terms of Boolean variables. In order to take advantage of current mixed-integer nonlinear programming solvers (e.g. DICOPT<ref name=":3">GAMS. DICOPT, https://www.gams.com/latest/docs/S_DICOPT.html</ref>, SBB<ref name=":4" />, α-ECP<ref name=":5">GAMS. AlphaECP, 1995, https://www.gams.com/latest/docs/S_ALPHAECP.html</ref>, BARON<ref name=":6">BARON, 1996, https://minlp.com/baron</ref>, Coenne<ref name=":7">Coenne, 2006, https://projects.coin-or.org/Couenne</ref> etc.), GDPs are often reformulated as MINLPs.<ref name=":0">P. Ruiz, Juan; Grossmann, Ignacio E. (2012): A hierarchy of relaxations for nonlinear convex generalized disjunctive programming. Carnegie Mellon University. Journal contribution. <nowiki>https://doi.org/10.1184/R1/6466535.v1</nowiki> </ref>
	[[File:GDP Intro.jpg\|none\|thumb\|523x523px\|Figure 1: Generalized Disjunctive Programming Methods<ref>Grossman, Ignacio E: Overview of Generalized Disjunctive Programming. Carnegie Mellon University.https://www.minlp.org/pdf/GBDEWOGrossmann.pdf</ref>]]		[[File:GDP Intro.jpg\|none\|thumb\|523x523px\|Figure 1: Generalized Disjunctive Programming Methods<ref>Grossman, Ignacio E: Overview of Generalized Disjunctive Programming. Carnegie Mellon University.https://www.minlp.org/pdf/GBDEWOGrossmann.pdf</ref>]]

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	==== GDP ====		==== GDP ====
	~~Generalized Disjunctive Programming~~ provides a high level framework for solving the mixed non-linear integer programs. By provide a methodology for converting the disjunctive problems into a MINLP the problem becomes simplified and easier to solve using current processing and algorithmic capabilities. These methodologies that can not only solve both the Convex and Non-Convex Problems. A Convex GDP is when both f(x) and g(x) are convex functions. Which is defined as a graph where any line segment that passes through any 2 points of the plot will always be greater than the plot itself. This allows for simple relaxations/approximations to occur which will create a faster solving methodology.<ref>Grossmann, Ignacio. Review of Mixed-Integer Nonlinear and Generalized Disjunctive Programming Applications in Process Systems Engineering.</ref>		GDP provides a high level framework for solving the mixed non-linear integer programs. By provide a methodology for converting the disjunctive problems into a MINLP the problem becomes simplified and easier to solve using current processing and algorithmic capabilities. These methodologies that can not only solve both the Convex and Non-Convex Problems. A Convex GDP is when both f(x) and g(x) are convex functions. Which is defined as a graph where any line segment that passes through any 2 points of the plot will always be greater than the plot itself. This allows for simple relaxations/approximations to occur which will create a faster solving methodology.<ref>Grossmann, Ignacio. Review of Mixed-Integer Nonlinear and Generalized Disjunctive Programming Applications in Process Systems Engineering.</ref>

	== Methodology ==		== Methodology ==

NRB at 19:29, 13 December 2020

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	==== NLP Subproblem for a fixed <math>Y_p</math> ====		==== NLP Subproblem for a fixed <math>Y_p</math> ====
	The subproblem for a fixed yP is shown in the form below		The subproblem for a fixed <math>Y_p</math> is shown in the form below

	<math>\begin{align} \min z=f(x,y^p)\\		<math>\begin{align} \min z=f(x,y^p)\\

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

	where f(x) and g(x) are twice differentiable functions, x are the continuous variables and y are the discrete variables. There are three main types of sub problems that arise from the MINLP: Continuous Relaxation, NLP subproblem for a fix <math>\begin{align}		where f(x) and g(x) are twice differentiable functions, x are the continuous variables and y are the discrete variables. There are three main types of sub problems that arise from the MINLP: Continuous Relaxation, NLP subproblem for a fix
	\Y_p\\		<math>\begin{align}
	\end{align}</math> and the feasibility problem.		Y_p
			\end{align}</math>
			and the feasibility problem.

	==== Continuous Relaxation ====		==== Continuous Relaxation ====
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	Where <math>Y_R</math> is the continuous relaxation of Y. Not that in this sub-problem all of the integer variables y are treated as continuous. This also returns a Lower Bound when it returns a feasible solution<ref name=":2">Grossmann, Ignacio. Review of Mixed-Integer Nonlinear and Generalized Disjunctive Programming Applications in Process Systems Engineering.</ref>		Where <math>Y_R</math> is the continuous relaxation of Y. Not that in this sub-problem all of the integer variables y are treated as continuous. This also returns a Lower Bound when it returns a feasible solution<ref name=":2">Grossmann, Ignacio. Review of Mixed-Integer Nonlinear and Generalized Disjunctive Programming Applications in Process Systems Engineering.</ref>

	==== NLP Subproblem for a fixed yP ====		==== NLP Subproblem for a fixed <math>Y_p</math> ====
	The subproblem for a fixed yP is shown in the form below		The subproblem for a fixed yP is shown in the form below

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

	In this sub problem you return an upper bound for the MINLP program when it has a feasible solution. So with said you can fix ~~on of these~~ integer variables and continuously relax the others in order to get a range of feasible values.<ref name=":2" />		In this sub problem you return an upper bound for the MINLP program when it has a feasible solution. So with that being said you can fix a integer variables and continuously relax the others in order to get a range of feasible values.<ref name=":2" />

	'''Feasibility Problem'''		'''Feasibility Problem'''
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	==== GDP ====		==== GDP ====
	Generalized Disjunctive Programming provides a high level framework for solving the mixed non-linear integer programs. By provide a methodology for converting the ~~dijunctive~~ problems into a MINLP the problem becomes simplified and easier to solve using current processing and algorithmic capabilities. ~~There a~~ methodologies that can not only solve ~~this~~ both Convex and Non-Convex Problems. A Convex GDP is when both f(x) and g(x) are convex functions. Which is defined as a graph where any line segment that passes through any 2 points of the plot will always be greater than the plot itself. This allows for simple relaxations/approximations to occur which will create a faster solving methodology.<ref>Grossmann, Ignacio. Review of Mixed-Integer Nonlinear and Generalized Disjunctive Programming Applications in Process Systems Engineering.</ref>		Generalized Disjunctive Programming provides a high level framework for solving the mixed non-linear integer programs. By provide a methodology for converting the disjunctive problems into a MINLP the problem becomes simplified and easier to solve using current processing and algorithmic capabilities. These methodologies that can not only solve both the Convex and Non-Convex Problems. A Convex GDP is when both f(x) and g(x) are convex functions. Which is defined as a graph where any line segment that passes through any 2 points of the plot will always be greater than the plot itself. This allows for simple relaxations/approximations to occur which will create a faster solving methodology.<ref>Grossmann, Ignacio. Review of Mixed-Integer Nonlinear and Generalized Disjunctive Programming Applications in Process Systems Engineering.</ref>

	== Methodology ==		== Methodology ==

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	== Introduction ==		== Introduction ==
	Generalized disjunctive programming (GDP) involves logic propositions (Boolean variables) and sets of constraints that are chained together using the logical OR operator ( II ). Generalized disjunctive programming (GDP) is an extension of linear disjunctive programming (Balas, 1979) that can be applied to Mixed Integer Non-Linear Programming (MINLP). GDP (Raman and ~~Grossmann~~, 1994), is a generalization of disjunctive convex programming in the sense that it also allows the use of logic propositions that are expressed in terms of Boolean variables. In order to take advantage of current mixed-integer nonlinear programming solvers (e.g. DICOPT<ref name=":3">GAMS. DICOPT, https://www.gams.com/latest/docs/S_DICOPT.html</ref>, SBB<ref name=":4" />, α-ECP<ref name=":5">GAMS. AlphaECP, 1995, https://www.gams.com/latest/docs/S_ALPHAECP.html</ref>, BARON<ref name=":6">BARON, 1996, https://minlp.com/baron</ref>, Coenne<ref name=":7">Coenne, 2006, https://projects.coin-or.org/Couenne</ref> etc.), GDPs are often reformulated as MINLPs.<ref name=":0">P. Ruiz, Juan; Grossmann, Ignacio E. (2012): A hierarchy of relaxations for nonlinear convex generalized disjunctive programming. Carnegie Mellon University. Journal contribution. <nowiki>https://doi.org/10.1184/R1/6466535.v1</nowiki> </ref>		Generalized disjunctive programming (GDP) involves logic propositions (Boolean variables) and sets of constraints that are chained together using the logical OR operator ( II ). Generalized disjunctive programming (GDP) is an extension of linear disjunctive programming<ref>Balas, Egon. "Disjunctive Programming." Annals of Discrete Mathematics, 1979.</ref> that can be applied to Mixed Integer Non-Linear Programming (MINLP). GDP<ref>Raman and Grossman. "Modelling and Computational Techniques for Logic Based Integer Programming." Computers & Chemical Engineering, 1994.</ref>, is a generalization of disjunctive convex programming in the sense that it also allows the use of logic propositions that are expressed in terms of Boolean variables. In order to take advantage of current mixed-integer nonlinear programming solvers (e.g. DICOPT<ref name=":3">GAMS. DICOPT, https://www.gams.com/latest/docs/S_DICOPT.html</ref>, SBB<ref name=":4" />, α-ECP<ref name=":5">GAMS. AlphaECP, 1995, https://www.gams.com/latest/docs/S_ALPHAECP.html</ref>, BARON<ref name=":6">BARON, 1996, https://minlp.com/baron</ref>, Coenne<ref name=":7">Coenne, 2006, https://projects.coin-or.org/Couenne</ref> etc.), GDPs are often reformulated as MINLPs.<ref name=":0">P. Ruiz, Juan; Grossmann, Ignacio E. (2012): A hierarchy of relaxations for nonlinear convex generalized disjunctive programming. Carnegie Mellon University. Journal contribution. <nowiki>https://doi.org/10.1184/R1/6466535.v1</nowiki> </ref>
	[[File:GDP Intro.jpg\|none\|thumb\|523x523px\|Figure 1: Generalized Disjunctive Programming Methods<ref>Grossman, Ignacio E: Overview of Generalized Disjunctive Programming. Carnegie Mellon University.https://www.minlp.org/pdf/GBDEWOGrossmann.pdf</ref>]]		[[File:GDP Intro.jpg\|none\|thumb\|523x523px\|Figure 1: Generalized Disjunctive Programming Methods<ref>Grossman, Ignacio E: Overview of Generalized Disjunctive Programming. Carnegie Mellon University.https://www.minlp.org/pdf/GBDEWOGrossmann.pdf</ref>]]

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Introduction

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	== Introduction ==		== Introduction ==
	Generalized disjunctive programming (GDP) involves logic propositions (Boolean variables) and sets of constraints that are chained together using the logical OR operator ( II ). Generalized disjunctive programming (GDP) is an extension of linear disjunctive programming (Balas, 1979) that can be applied to Mixed Integer Non-Linear Programming (MINLP). GDP (Raman and Grossmann, 1994), is a generalization of disjunctive convex programming in the sense that it also allows the use of logic propositions that are expressed in terms of Boolean variables. In order to take advantage of current mixed-integer nonlinear programming solvers (e.g. DICOPT ~~(Viswanathan and Grossmann~~, ~~1990)~~, SBB ~~(Brooke et al., 1998)~~, α-ECP ~~(Westerlund and Pettersson~~, 1995), ~~Bonmin (Bonami et al~~., ~~2008)~~, ~~FilMINT (Abhishek et al~~., 2006~~), BARON (Sahinidis, 1996)~~, etc.), GDPs are often reformulated as MINLPs.<ref name=":0">P. Ruiz, Juan; Grossmann, Ignacio E. (2012): A hierarchy of relaxations for nonlinear convex generalized disjunctive programming. Carnegie Mellon University. Journal contribution. <nowiki>https://doi.org/10.1184/R1/6466535.v1</nowiki> </ref>		Generalized disjunctive programming (GDP) involves logic propositions (Boolean variables) and sets of constraints that are chained together using the logical OR operator ( II ). Generalized disjunctive programming (GDP) is an extension of linear disjunctive programming (Balas, 1979) that can be applied to Mixed Integer Non-Linear Programming (MINLP). GDP (Raman and Grossmann, 1994), is a generalization of disjunctive convex programming in the sense that it also allows the use of logic propositions that are expressed in terms of Boolean variables. In order to take advantage of current mixed-integer nonlinear programming solvers (e.g. DICOPT<ref name=":3">GAMS. DICOPT, https://www.gams.com/latest/docs/S_DICOPT.html</ref>, SBB<ref name=":4" />, α-ECP<ref name=":5">GAMS. AlphaECP, 1995, https://www.gams.com/latest/docs/S_ALPHAECP.html</ref>, BARON<ref name=":6">BARON, 1996, https://minlp.com/baron</ref>, Coenne<ref name=":7">Coenne, 2006, https://projects.coin-or.org/Couenne</ref> etc.), GDPs are often reformulated as MINLPs.<ref name=":0">P. Ruiz, Juan; Grossmann, Ignacio E. (2012): A hierarchy of relaxations for nonlinear convex generalized disjunctive programming. Carnegie Mellon University. Journal contribution. <nowiki>https://doi.org/10.1184/R1/6466535.v1</nowiki> </ref>
	[[File:GDP Intro.jpg\|none\|thumb\|523x523px\|Figure 1: Generalized Disjunctive Programming Methods<ref>Grossman, Ignacio E: Overview of Generalized Disjunctive Programming. Carnegie Mellon University.https://www.minlp.org/pdf/GBDEWOGrossmann.pdf</ref>]]		[[File:GDP Intro.jpg\|none\|thumb\|523x523px\|Figure 1: Generalized Disjunctive Programming Methods<ref>Grossman, Ignacio E: Overview of Generalized Disjunctive Programming. Carnegie Mellon University.https://www.minlp.org/pdf/GBDEWOGrossmann.pdf</ref>]]

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	==== Solvers: ====		==== Solvers: ====

	* DICOPT		* DICOPT<ref name=":3" />
	* SBB<ref>GAMS. ''SBB'', 2020, www.gams.com/latest/docs/S_SBB.html.</ref>		* SBB<ref name=":4">GAMS. ''SBB'', 2020, www.gams.com/latest/docs/S_SBB.html.</ref>
	* BARON		* BARON<ref name=":6" />
	* Couenne		* Couenne<ref name=":7" />

	== Numerical Example ==		== Numerical Example ==
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	This same idea can be scaled to larger problems with more complex branching. Figure 3 illustrates a larger process network and all of the different decision points. This problem is able to be formulated into a GDP so that the most optimal route can be calculated to take through the network.		This same idea can be scaled to larger problems with more complex branching. Figure 3 illustrates a larger process network and all of the different decision points. This problem is able to be formulated into a GDP so that the most optimal route can be calculated to take through the network.
	== Conclusion ==		== Conclusion ==
	GDP is a programming method that applies disjunctive programming to MINLP problems. This method facilitates modeling discrete or continuous optimization problems by implementing algebraic constraints and logic expressions. The formulation of a GDP consists of Boolean and continuous variables and disjunctions and logic propositions. In the case of convex functions, GDPs can be reformulated using the big-M and the hull relaxation. Formulation methods also include logic based methods disjunctive branch and bound and decomposition. Once reformulated into a standard MINLP, standard MILNP solvers, such as DICOPT, SBB, α-ECP and BARON, can be used to determine optimal solutions<ref name=":0" />. The GDP method has important applications that include the optimization of complex chemical reactions and process planning.		GDP is a programming method that applies disjunctive programming to MINLP problems. This method facilitates modeling discrete or continuous optimization problems by implementing algebraic constraints and logic expressions. The formulation of a GDP consists of Boolean and continuous variables and disjunctions and logic propositions. In the case of convex functions, GDPs can be reformulated using the big-M and the hull relaxation. Formulation methods also include logic based methods disjunctive branch and bound and decomposition. Once reformulated into a standard MINLP, standard MILNP solvers, such as DICOPT<ref name=":3" />, SBB<ref name=":4" />, α-ECP<ref name=":5" /> and BARON<ref name=":6" />, can be used to determine optimal solutions<ref name=":0" />. The GDP method has important applications that include the optimization of complex chemical reactions and process planning.

	== References ==		== References ==
	<references />		<references />