Difference between revisions of "Convex generalized disjunctive programming (GDP)"

From Cornell University Computational Optimization Open Textbook - Optimization Wiki
Jump to navigation Jump to search
 
(9 intermediate revisions by 2 users not shown)
Line 1: Line 1:
Edited By: Nicholas Schafhauser, Blerand Qeriqi, Ryan Cuppernull
+
Authors: Nicholas Schafhauser, Blerand Qeriqi, Ryan Cuppernull (SysEn 5800 Fall 2020)
  
 
== 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 ). 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>]]
  
Line 16: Line 16:
 
\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 ====
Line 33: Line 35:
 
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 <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)\\
Line 43: Line 45:
 
\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'''
Line 60: Line 62:
  
 
==== 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>
+
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 ==
Line 81: Line 83:
 
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)\\
Line 101: Line 103:
 
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)\\
Line 118: Line 120:
 
==== 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 ==
 
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]]
Line 141: Line 141:
 
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 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 />

Latest revision as of 07:40, 21 December 2020

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

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[1] that can be applied to Mixed Integer Non-Linear Programming (MINLP). GDP[2], 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[3], SBB[4], α-ECP[5], BARON[6], Couenne[7] etc.), GDPs are often reformulated as MINLPs.[8]

Figure 1: Generalized Disjunctive Programming Methods[9]

Theory

The general form of an MINLP model is as follows

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 and the feasibility problem.

Continuous Relaxation

The sub problem of continuous relaxation takes the form of

Where 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[10]

NLP Subproblem for a fixed

The subproblem for a fixed is shown in the form below

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.[10]

Feasibility Problem

When the fixed MINLP subproblem is not feasible the following feasibility problem is considered.

Where J is the index set for inequalities and the feasibility problem attempts to minimize the infeasibility of the solution with the most violated constraints.[10]

GDP

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.[11]

Methodology

Below is a GDP problem that will be used for demonstration purposes in this section.


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.[12]

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


Notice that the boolean term from the original GDP has been converted into a numerical {0,1}. The logic relations have also been converted into linear integer constraints (Hy)[8].

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 HR method.

HR significantly increases the number of variables that are required in the same BM variant. The decrease in time needed to solve computations could very well be argued to be worth the reduced simplicity that one can get from BM.[13]

Solvers:

Numerical Example

The following example was taken from the paper titled Generalized Disjunctive Programming: A Framework For Formulation and Alternative Algorithms For MINLP Optimization.[14]

GDP numeric example 3.png

http://egon.cheme.cmu.edu/Papers/IMAGrossmannRuiz.pdf

http://egon.cheme.cmu.edu/Papers/IMAGrossmannRuiz.pdf

Applications

GDP formulations are useful for real-world applications where multiple branches are available when making decisions. Solving the GDP in these instances will allow the user to calculate which decisions should be made at each branching point in order to get the optimal solution. This disjunctive formulation is common in complex chemical reactions and production planning.

Figure 2: Process Network Example. Each decision point represents another disjunctive set. [14]

The process network depicted in the Figure 2 depicts multiple decisions that could be made to all end up at the goal (B) in a chemical reaction. This problem is able to be formulated into a GDP in order to figure out which route should be taken in order to maximize the profit.

Figure 3: A more complex process network.[14]

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

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[3], SBB[4], α-ECP[5] and BARON[6], can be used to determine optimal solutions[8]. The GDP method has important applications that include the optimization of complex chemical reactions and process planning.

References

  1. Balas, Egon. "Disjunctive Programming." Annals of Discrete Mathematics, 1979.
  2. Raman and Grossman. "Modelling and Computational Techniques for Logic Based Integer Programming." Computers & Chemical Engineering, 1994.
  3. 3.0 3.1 3.2 GAMS. DICOPT, https://www.gams.com/latest/docs/S_DICOPT.html
  4. 4.0 4.1 4.2 GAMS. SBB, 2020, www.gams.com/latest/docs/S_SBB.html.
  5. 5.0 5.1 GAMS. AlphaECP, 1995, https://www.gams.com/latest/docs/S_ALPHAECP.html
  6. 6.0 6.1 6.2 BARON, 1996, https://minlp.com/baron
  7. 7.0 7.1 Couenne, 2006, https://projects.coin-or.org/Couenne
  8. 8.0 8.1 8.2 P. Ruiz, Juan; Grossmann, Ignacio E. (2012): A hierarchy of relaxations for nonlinear convex generalized disjunctive programming. Carnegie Mellon University. Journal contribution. https://doi.org/10.1184/R1/6466535.v1
  9. Grossman, Ignacio E: Overview of Generalized Disjunctive Programming. Carnegie Mellon University.https://www.minlp.org/pdf/GBDEWOGrossmann.pdf
  10. 10.0 10.1 10.2 Grossmann, Ignacio. Review of Mixed-Integer Nonlinear and Generalized Disjunctive Programming Applications in Process Systems Engineering.
  11. Grossmann, Ignacio. Review of Mixed-Integer Nonlinear and Generalized Disjunctive Programming Applications in Process Systems Engineering.
  12. Trespalacios, Francisco; Grossmann, Ignacio E. (2018): Improved Big-M Reformulation for Generalized Disjunctive Programs. Carnegie Mellon University. Journal contribution. https://doi.org/10.1184/R1/6467063.v1
  13. Trespalacios, Francisco; Grossmann, Ignacio E. (2015): Algorithmic Approach for Improved Mixed-Integer Reformulations of Convex Generalized Disjunctive Programs. Carnegie Mellon University. Journal contribution. https://doi.org/10.1184/R1/6466700.v1
  14. 14.0 14.1 14.2 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