- Email: [email protected]
Alternative mixed-integer reformulation of Generalized Disjunctive Programs
Anton Friedl, Jiří J. Klemeš, Stefan Radl, Petar S. Varbanov, Thomas Wallek (Eds.) Proceedings of the 28th European Symposium on Computer Aided Process Engineering June 10th to 13th, 2018, Graz, Austria. © 2018 Elsevier B.V. All rights reserved. https://doi.org/10.1016/B978-0-444-64235-6.50097-8
Alternative mixed-integer reformulation of Generalized Disjunctive Programs Miloš Bogataj a,*, Zdravko Kravanja a a
Faculty of Chemistry and Chemical Engineering, University of Maribor, Smetanova 17, SI-2000 Maribor, Slovenia [email protected]
Abstract In this work, we propose an alternative mixed-integer (MI) reformulation of Generalized Disjunctive Programs (GDP) and compare its efficiency against the Big-M and hull reformulation (HR). We show that the reformulation yields a continuous relaxation equal to the HR for linear GDPs and comparable to the one obtained by HR for nonlinear GDPs. A particular advantage of the proposed reformulation is that the usage of the perspective function can be avoided when reformulating nonlinear GDPs. The reformulation is validated on a set of numerical examples using Branch and Bound algorithm. Results obtained thus far indicate that the proposed reformulation represents a promising alternative to the established ones. Keywords: Disjunctive Programming, Mixed-Integer Programming, Big-M Relaxation, Hull Relaxation.
1. Introduction A common approach to process synthesis is usually based on a superstructure representation of competitive/plausible alternatives. The most intuitive way for representing it in a mathematical language is the one in the form of GDP (Raman and Grossmann, 1994; Türkay and Grossmann, 1996). While, among other advantages, the GDP framework facilitates the modelling stage, solving these problems is not a straightforward task. In most cases, a GDP problem is reformulated to an MI problem and then solved using the established solvers. The GDP problems can be formulated as mixed-integer (non)linear problems (MI(N)LP) in different ways. This, however, leads to models of different sizes (in a number of variables and constraints) and of different tightness (i.e. relation of the feasible region of a problem to the feasible region of its continuous relaxation). Therefore, the efficiency of the solvers to solve these problems depends on the type of the reformulation. Several procedures can be utilized to achieve the MI formulation - the most instinctual being the Big-M relaxation (Nemhauser and Wolsey, 1988). While it produces smallsized models, its relaxation strongly depends on the selection of Big-M parameters. Often, even when the smallest Big-M parameters are selected, the Big-M reformulation results in a weak continuous relaxation. Recently, an improved Big-M reformulation for GDP has been presented by Trespalacios and Grossmann (2015). In contrast to the traditional one, the latter uses multiple Big-M parameters for each constraint. Consequently, it produces tighter continuous relaxations and reduces the computational effort needed to solve discrete-continuous optimization problems.
M. Bogataj and Z. Kravanja
550
To obtain tighter relaxations, Lee and Grossmann (2003) proposed an HR for (non)linear convex GDPs. It produces at least as tight relaxation as the Big-M reformulation, however, it requires additional variables and constraints. It has been shown that HR, although it produces larger-sized models, dominates the Big-M reformulation in the majority of cases, primarily because of the tight relaxation it produces. The concept of HR in process synthesis problems was utilized and further developed by Ropotar and Kravanja (2009). The authors presented an HR based on MI translation of variables, The reformulation tends to exploit the benefits of the fact that the variables in engineering problems are often bounded by nonzero positive lower and upper bounds. For additional insights into a hierarchy and advances in relaxations and reformulations for GDPs, the reader is referred to Sawaya and Grossmann (2012), and Ruiz and Grossmann (2012). The remainder of the manuscript is organized as follows. In section 2 we present an overview of a GDP model and present the alternative MI reformulation. In section 3 we present an illustrative example. Section 4 is dedicated to testing of the proposed reformulation on a set of process synthesis examples and presentation of the results.
2. Alternative MI reformulation of GDP Consider the general GDP presented as follows: = Z min
∑c ik
ik
+ f ( x)
hg ( x) ≤ 0
s.t.
Yik ∨ g ik ( x ) ≤ 0 k ∈ K i∈Dk c = γ ik ik ∨ Yik k∈K
(GDP)
i∈Dk
Ω (Y ) = True
x ∈ , cik ∈ , Yik ∈ {True, False} n
In (GDP) the objective is a function of continuous variables, h g ( x) ≤ 0 are global (non)linear constraints, which must hold true regardless of the discrete decision. Each of the disjunctions k ∈ K contains disjunctive terms Dk , linked by an OR operator (∨ ) . A Boolean variable Yik is assigned to each of the disjunctive terms, which contain a set of local (non)linear constraints ( gik ( x) ≤ 0 ) describing the alternatives. The local constraints within a given term are only enforced when a corresponding Boolean variable takes a value of True. On the other hand, if the Boolean variable equals False, the local constraints are ignored. Ω (Y ) = True represents a set of logical propositions. The proposed MI reformulation of (GDP) based on MI translation of variables is given by (AMI). Similar to the HR presented by Lee and Grossman (2003), the Boolean variables in (AMI) are transformed into 0-1 variables yik , the continuous variables x disaggregated into variables xikD and the logical propositions reformulated into a set of linear constraints Ey ≤ e . The main difference between the reformulations is that the disaggregated variables xikD are not required to be forced to 0 when yik = 0 .
Alternative mixed-integer reformulation of Generalized Disjunctive Programs
∑y γ
= min Z
ik
+ f ( x)
h g ( x) ≤ 0
s.t. = x
ik ik
551
∑x
i∈Dk
D ik
− (1 − yik )xikF
k∈K
g ik ( xikD ) − (1 − yik ) g ik ( xikF ) ≤ 0
i ∈ Dk , k ∈ K
D ik
D,LO ik ik
F ik
i ∈ Dk , k ∈ K
D ik
D,UP ik ik
F ik
i ∈ Dk , k ∈ K
x ≥x x ≤x
y + (1 − yik ) x
y + (1 − yik ) x
(AMI)
Ey ≤ e
∑y
i∈Dk
ik
=1
k∈K
x LO ≤ x ≤ x UP , yik ∈ {0,1}
Instead, they are forced to arbitrary fixed values xikF (vanishing points -VPs), which are usually chosen between nonzero lower and upper bounds of variables xikD . For additional explanation of the reformulation based on MI translation of variables, the reader is referred to Ropotar and Kravanja (2009). In their work, the authors have shown that if local constraints ( gik ( x) ≤ 0 ) are reformulated using the perspective function i.e. yik ( g ik (( xikD − (1 − yik ) xikF ) / yik ) ) ≤ 0 , the above reformulation produces relaxation identical
to HR.
If local constraints in (GDP) are linear, it can be easily shown that
yik ( g ik (( xikD − (1 − yik ) xikF ) / yik ) ) = g ik ( xikD ) − (1 − yik ) g ik ( xikF ) . Therefore, the reformulation as
given by (AMI) is a hull relaxation for GDPs containing linearly constrained disjunctive feasible regions. If the local constraints are nonlinear or combination of linear and nonlinear constraints, the relaxation of (GDP) as given by (AMI) is generally weaker as the one produced by the HR. However, using an initialization procedure, described in section 2, the relaxation obtained by (AMI) is expected to be considerably tighter than the one obtained by the Big-M reformulation.
3. Illustrative example We illustrate the proposed reformulation on an example taken from Trespalacios and Grossmann (2015). The example deals with optimization of linear objective function over a set of disjoint circles. Its GDP representation is given by (1). The optimal solution of the example is Z = –9.472, Y = (False, False, True), x = (5.789, 2.106). If we apply the (AMI) model to (1) we obtain the MINLP reformulation (2). The solution of the relaxed MINLP, assuming the VPs are the centers of the circles (i.e. (0,0), (1,5) and (4,3)), is ZR = –9.816, y = (0.027, 0.038, 0.935) and x = (5.859, 1.902). The continuous relaxation is stronger as the one obtained by the Big-M reformulation (ZR = –10.493) and slightly weaker than the one obtained by the improved Big-M reformulation (ZR = –9.735). However, if we select the VPs as (0.680, 1.934), (1.372, 4.431) and (5.788, 2.015), the relaxation is identical to the one obtained by the HR and relaxed MINLP problem produces integer solution: ZR = Z = –9.472, y = (0, 0, 1), x = (5.789, 2.106). It is evident that the selection of VPs in (AMI) plays an important role in providing tight continuous relaxations when the constraints in the disjunctions are nonlinear.
M. Bogataj and Z. Kravanja
552 min Z = ∑ cv xv v∈V
Yi ∨ 2 i∈Dk ( x − a ) + ( x − b ) 2 − r ≤ 0 i i 2 1 i ∨ Yi =True
(1)
i∈Dk
−1 ≤ x1 ≤ 6, −1 ≤ x2 ≤ 7; c= (−2,1), a = (0,1, 4), b = (0,5,3), r = (1, 2, 4)
min Z = ∑ cv xv v∈V
= xv
∑ xvD,i − (1 − yi ) xvF,i
v ∈V
i∈Dk
( x1,Di − ai ) 2 + ( x2,Di − bi ) 2 − ri − (1 − yi )(( x1,Fi − ai ) 2 + ( x2,F i − bi ) 2 − ri ) ≤ 0 xvD,i ≥ xvD,LO yi + (1 − yi ) x1,Fi ,i
v ∈ V , i ∈ Dk
xvD,i ≤ xvD,UP yi + (1 − yi ) x1,Fi ,i
v ∈ V , i ∈ Dk
∑y
i∈Dk
i
i ∈ Dk
(2)
=1
xvD,LO = (−1, −1), xvD,UP = (6,7) ,i ,i
v ∈ V , i ∈ Dk
c= (−2,1), a = (0,1, 4), b = (0,5,3), r = (1, 2, 4)
3.1. Algorithm for selection of VPs To study the effect of how the selection of VPs affects the continuous relaxation of (2), we have generated 500 random VPs within each of the disjoint feasible regions. The results show the average value of ZR = –10.542 ±0.764 to be approximately 10 % weaker than the one obtained by the HR. Nevertheless, we have observed that the relaxed problem does produce relaxation identical to the HR in several instances. Currently, the theoretical background for whether there always exists a set of VPs that guarantees as tight continuous relaxation as the one obtained by the HR and, if it does, how to determine such VPs is still under investigation. However, what has shown to be a promising path is to first solve a reduced version of (AMI) and set the optimal values of disaggregated variables as VPs for (AMI). The reduced model (redAMI) is given as follows: min Z = ∑ f ( xikD ) i,k
s.t. g ik ( xikD ) ≤ 0 D,LO ik
x
D ik
i ∈ Dk , k ∈ K
(redAMI)
D,UP ik
≤x ≤x
By solving (redAMI), we optimize the continuous part of the objective function over the feasible region constrained solely by local constraints. Note that the only type of variables present in the model is the disaggregated continuous variables. Testing the procedure on the 500 randomly generated VPs has revealed that in all the instances (redAMI) produces VPs that lead to the tightest possible continuous relaxation. We must note that solving (redAMI) introduces another step into the optimization
Alternative mixed-integer reformulation of Generalized Disjunctive Programs
553
procedure; however, since the model is significantly smaller, the impact of solving (redAMI) on the solution time is expected to be low. As a final note, the effect of VPs on the tightness of the continuous relaxation is only present if the disjunctions contain nonlinear or combination of linear and nonlinear constraints.
4. Process Synthesis Examples In this section, we compare the efficiency of the proposed reformulation against Big-M and HR reformulation. The process network synthesis is a classical optimization problem in which the goal is to select a process network that maximizes profit while considering the costs of raw materials and the cost of the process units. The numerical examples Proc8, Proc10 and Proc12 are variations of the example introduced by Türkay and Grossmann (1996). A general GDP model is described by (3). The examples were solved using SBB solver. The results for the three instances are given in Tables 1-2. min Z = ∑ ci + ∑ p j x j + γ i
s.t.
j
∑ rj , n x j ≤ 0
∀n ∈ N
j
Yi ¬Yi d (e x j / ti , j − 1) − s x ≤ 0 ∨ x = 0, j ∈ J i ∑j i , j j j i, j ∑ j ci = γ i ci = γ i True Ω(Y ) =
(3)
x j , ci ≥ 0, Yi ∈ {True, False}
Table 1: Comparison of relaxations and total computational times. Big-Ma
HRb
tCPU /s
LB
redAMI + AMI
Ex./Nbins
Opt.
LB
tCPU /s
LB
tCPU /s
Proc8/8
68.01
-829.88
0.23
67.7334
0.13
67.7334
0.11
Proc10/10
-27.93
-1146.88
0.84
-40.1313
0.16
-40.1314
0.14
Proc12/12
-69.21
-894.52
2.01
-73.8895
0.14
-73.8895
0.09
a – Value of 50 was used for all the Big-M parameters. | b – In the HR reformulation, value of 1e-6 was used for ε.
Table 2: SBB solver statistics. Big-M Ex.
Iter
Nds
HR tNLP/s
Iter
Nds
redAMI + AMI tNLP/s
Iter
Nds
tNLP/s
Proc8
801
34
0.23
117
1
0.12
53
1
0.09
Proc10
2820
205
0.84
236
4
0.15
78
5
0.05
Proc12
7347
463
1.97
174
3
0.14
99
3
0.08
554
M. Bogataj and Z. Kravanja
As can be seen in Table 1, the HR and the proposed approach (redAMI +AMI) produce significantly tighter relaxations than the Big-M reformulation in all the instances. Both the HR and the proposed reformulation exhibit faster computational times. The proposed approach produces equally tight continuous relaxations as HR in two instances (Proc8 and Proc12), while the relaxation of Proc10 is slightly weaker, consequently causing an increase in the number of nodes searched to find the optimal solution. Interestingly, the proposed approach requires the least computational time to find optimal solutions and the least solver iterations (Table 2).
5. Conclusions Over the last decades, a number of reformulation strategies for obtaining tighter continuous relaxations of discrete-continuous problems have been proposed. In this work, we present a reformulation that utilizes MI translation of variables and avoids the usage of the perspective function. When dealing with nonlinear GSPs and their MINLP representations, the first property, as suggested by Ropotar and Kravanja (2009), tends to reduce the search space of nonlinear subproblems. The second property might be important for producing more robust nonlinear subproblems as the proposed reformulation does not introduce additional nonlinearities to the model. Based on the results presented in this work, we can conclude that the proposed reformulation provides results comparable to those obtained by the HR, both in terms of tightness of continuous relaxation as well as of the computational effort needed to obtain optimal solutions. Future work will focus on research dedicated to providing the answers regarding the procedures for obtaining VPs, and testing of the proposed reformulation on large-scale synthesis problems. We believe that the findings of the future work will enable us to fully evaluate the benefits as well as drawbacks of the proposed reformulation.
References S. Lee, I. E. Grossmann, 2003, Generalized Convex Disjunctive Programming: Nonlinear Convex Hull Relaxation. Computational Optimization and Applications, 26 (1), 83–100. G. L. Nemhauser, L. A: Wolsey, 1988, Integer and combinatorial optimization. WileyInterscience, Willey. R. Raman, I. E. Grossmann, 1994, Modeling and Computational Techniques for Logic Based Integer Programming. Computers and Chemical Engineering, 18 (7), 563–578. M. Ropotar, Z. Kravanja, 2009, Translation of Variables and Implementation of Efficient LogicBased Techniques in the MINLP Process Synthesizer MIPSYN, AIChE Journal, 55 (11), 2896– 2913. J. P. Ruiz, I. E. Grossmann, 2012, A hierarchy of relaxations for nonlinear convex generalized disjunctive programming, European Journal of Operation Research, 218, 38–47. N. Sawaya, I. E. Grossmann, 2012. Reformulations, relaxations and cutting planes for linear generalized disjunctive programming. European Journal of Operational Research 216, 70–82. F. Trespalacios, I. E. Grossmann, 2015, Improved Big-M reformulation for generalized disjunctive programs. Computers and Chemical Engineering. 76, 98–103. M. Türkay, I. E. Grossmann, 1996, Logic-Based Algorithms for the Optimal Synthesis of Process Networks. Computers and Chemical Engineering, 20 (8), 959–978, 1996.
Alternative mixed-integer reformulation of Generalized Disjunctive Programs Miloš Bogataj a,*, Zdravko Kravanja a a
Faculty of Chemistry and Chemical Engineering, University of Maribor, Smetanova 17, SI-2000 Maribor, Slovenia [email protected]
Abstract In this work, we propose an alternative mixed-integer (MI) reformulation of Generalized Disjunctive Programs (GDP) and compare its efficiency against the Big-M and hull reformulation (HR). We show that the reformulation yields a continuous relaxation equal to the HR for linear GDPs and comparable to the one obtained by HR for nonlinear GDPs. A particular advantage of the proposed reformulation is that the usage of the perspective function can be avoided when reformulating nonlinear GDPs. The reformulation is validated on a set of numerical examples using Branch and Bound algorithm. Results obtained thus far indicate that the proposed reformulation represents a promising alternative to the established ones. Keywords: Disjunctive Programming, Mixed-Integer Programming, Big-M Relaxation, Hull Relaxation.
1. Introduction A common approach to process synthesis is usually based on a superstructure representation of competitive/plausible alternatives. The most intuitive way for representing it in a mathematical language is the one in the form of GDP (Raman and Grossmann, 1994; Türkay and Grossmann, 1996). While, among other advantages, the GDP framework facilitates the modelling stage, solving these problems is not a straightforward task. In most cases, a GDP problem is reformulated to an MI problem and then solved using the established solvers. The GDP problems can be formulated as mixed-integer (non)linear problems (MI(N)LP) in different ways. This, however, leads to models of different sizes (in a number of variables and constraints) and of different tightness (i.e. relation of the feasible region of a problem to the feasible region of its continuous relaxation). Therefore, the efficiency of the solvers to solve these problems depends on the type of the reformulation. Several procedures can be utilized to achieve the MI formulation - the most instinctual being the Big-M relaxation (Nemhauser and Wolsey, 1988). While it produces smallsized models, its relaxation strongly depends on the selection of Big-M parameters. Often, even when the smallest Big-M parameters are selected, the Big-M reformulation results in a weak continuous relaxation. Recently, an improved Big-M reformulation for GDP has been presented by Trespalacios and Grossmann (2015). In contrast to the traditional one, the latter uses multiple Big-M parameters for each constraint. Consequently, it produces tighter continuous relaxations and reduces the computational effort needed to solve discrete-continuous optimization problems.
M. Bogataj and Z. Kravanja
550
To obtain tighter relaxations, Lee and Grossmann (2003) proposed an HR for (non)linear convex GDPs. It produces at least as tight relaxation as the Big-M reformulation, however, it requires additional variables and constraints. It has been shown that HR, although it produces larger-sized models, dominates the Big-M reformulation in the majority of cases, primarily because of the tight relaxation it produces. The concept of HR in process synthesis problems was utilized and further developed by Ropotar and Kravanja (2009). The authors presented an HR based on MI translation of variables, The reformulation tends to exploit the benefits of the fact that the variables in engineering problems are often bounded by nonzero positive lower and upper bounds. For additional insights into a hierarchy and advances in relaxations and reformulations for GDPs, the reader is referred to Sawaya and Grossmann (2012), and Ruiz and Grossmann (2012). The remainder of the manuscript is organized as follows. In section 2 we present an overview of a GDP model and present the alternative MI reformulation. In section 3 we present an illustrative example. Section 4 is dedicated to testing of the proposed reformulation on a set of process synthesis examples and presentation of the results.
2. Alternative MI reformulation of GDP Consider the general GDP presented as follows: = Z min
∑c ik
ik
+ f ( x)
hg ( x) ≤ 0
s.t.
Yik ∨ g ik ( x ) ≤ 0 k ∈ K i∈Dk c = γ ik ik ∨ Yik k∈K
(GDP)
i∈Dk
Ω (Y ) = True
x ∈ , cik ∈ , Yik ∈ {True, False} n
In (GDP) the objective is a function of continuous variables, h g ( x) ≤ 0 are global (non)linear constraints, which must hold true regardless of the discrete decision. Each of the disjunctions k ∈ K contains disjunctive terms Dk , linked by an OR operator (∨ ) . A Boolean variable Yik is assigned to each of the disjunctive terms, which contain a set of local (non)linear constraints ( gik ( x) ≤ 0 ) describing the alternatives. The local constraints within a given term are only enforced when a corresponding Boolean variable takes a value of True. On the other hand, if the Boolean variable equals False, the local constraints are ignored. Ω (Y ) = True represents a set of logical propositions. The proposed MI reformulation of (GDP) based on MI translation of variables is given by (AMI). Similar to the HR presented by Lee and Grossman (2003), the Boolean variables in (AMI) are transformed into 0-1 variables yik , the continuous variables x disaggregated into variables xikD and the logical propositions reformulated into a set of linear constraints Ey ≤ e . The main difference between the reformulations is that the disaggregated variables xikD are not required to be forced to 0 when yik = 0 .
Alternative mixed-integer reformulation of Generalized Disjunctive Programs
∑y γ
= min Z
ik
+ f ( x)
h g ( x) ≤ 0
s.t. = x
ik ik
551
∑x
i∈Dk
D ik
− (1 − yik )xikF
k∈K
g ik ( xikD ) − (1 − yik ) g ik ( xikF ) ≤ 0
i ∈ Dk , k ∈ K
D ik
D,LO ik ik
F ik
i ∈ Dk , k ∈ K
D ik
D,UP ik ik
F ik
i ∈ Dk , k ∈ K
x ≥x x ≤x
y + (1 − yik ) x
y + (1 − yik ) x
(AMI)
Ey ≤ e
∑y
i∈Dk
ik
=1
k∈K
x LO ≤ x ≤ x UP , yik ∈ {0,1}
Instead, they are forced to arbitrary fixed values xikF (vanishing points -VPs), which are usually chosen between nonzero lower and upper bounds of variables xikD . For additional explanation of the reformulation based on MI translation of variables, the reader is referred to Ropotar and Kravanja (2009). In their work, the authors have shown that if local constraints ( gik ( x) ≤ 0 ) are reformulated using the perspective function i.e. yik ( g ik (( xikD − (1 − yik ) xikF ) / yik ) ) ≤ 0 , the above reformulation produces relaxation identical
to HR.
If local constraints in (GDP) are linear, it can be easily shown that
yik ( g ik (( xikD − (1 − yik ) xikF ) / yik ) ) = g ik ( xikD ) − (1 − yik ) g ik ( xikF ) . Therefore, the reformulation as
given by (AMI) is a hull relaxation for GDPs containing linearly constrained disjunctive feasible regions. If the local constraints are nonlinear or combination of linear and nonlinear constraints, the relaxation of (GDP) as given by (AMI) is generally weaker as the one produced by the HR. However, using an initialization procedure, described in section 2, the relaxation obtained by (AMI) is expected to be considerably tighter than the one obtained by the Big-M reformulation.
3. Illustrative example We illustrate the proposed reformulation on an example taken from Trespalacios and Grossmann (2015). The example deals with optimization of linear objective function over a set of disjoint circles. Its GDP representation is given by (1). The optimal solution of the example is Z = –9.472, Y = (False, False, True), x = (5.789, 2.106). If we apply the (AMI) model to (1) we obtain the MINLP reformulation (2). The solution of the relaxed MINLP, assuming the VPs are the centers of the circles (i.e. (0,0), (1,5) and (4,3)), is ZR = –9.816, y = (0.027, 0.038, 0.935) and x = (5.859, 1.902). The continuous relaxation is stronger as the one obtained by the Big-M reformulation (ZR = –10.493) and slightly weaker than the one obtained by the improved Big-M reformulation (ZR = –9.735). However, if we select the VPs as (0.680, 1.934), (1.372, 4.431) and (5.788, 2.015), the relaxation is identical to the one obtained by the HR and relaxed MINLP problem produces integer solution: ZR = Z = –9.472, y = (0, 0, 1), x = (5.789, 2.106). It is evident that the selection of VPs in (AMI) plays an important role in providing tight continuous relaxations when the constraints in the disjunctions are nonlinear.
M. Bogataj and Z. Kravanja
552 min Z = ∑ cv xv v∈V
Yi ∨ 2 i∈Dk ( x − a ) + ( x − b ) 2 − r ≤ 0 i i 2 1 i ∨ Yi =True
(1)
i∈Dk
−1 ≤ x1 ≤ 6, −1 ≤ x2 ≤ 7; c= (−2,1), a = (0,1, 4), b = (0,5,3), r = (1, 2, 4)
min Z = ∑ cv xv v∈V
= xv
∑ xvD,i − (1 − yi ) xvF,i
v ∈V
i∈Dk
( x1,Di − ai ) 2 + ( x2,Di − bi ) 2 − ri − (1 − yi )(( x1,Fi − ai ) 2 + ( x2,F i − bi ) 2 − ri ) ≤ 0 xvD,i ≥ xvD,LO yi + (1 − yi ) x1,Fi ,i
v ∈ V , i ∈ Dk
xvD,i ≤ xvD,UP yi + (1 − yi ) x1,Fi ,i
v ∈ V , i ∈ Dk
∑y
i∈Dk
i
i ∈ Dk
(2)
=1
xvD,LO = (−1, −1), xvD,UP = (6,7) ,i ,i
v ∈ V , i ∈ Dk
c= (−2,1), a = (0,1, 4), b = (0,5,3), r = (1, 2, 4)
3.1. Algorithm for selection of VPs To study the effect of how the selection of VPs affects the continuous relaxation of (2), we have generated 500 random VPs within each of the disjoint feasible regions. The results show the average value of ZR = –10.542 ±0.764 to be approximately 10 % weaker than the one obtained by the HR. Nevertheless, we have observed that the relaxed problem does produce relaxation identical to the HR in several instances. Currently, the theoretical background for whether there always exists a set of VPs that guarantees as tight continuous relaxation as the one obtained by the HR and, if it does, how to determine such VPs is still under investigation. However, what has shown to be a promising path is to first solve a reduced version of (AMI) and set the optimal values of disaggregated variables as VPs for (AMI). The reduced model (redAMI) is given as follows: min Z = ∑ f ( xikD ) i,k
s.t. g ik ( xikD ) ≤ 0 D,LO ik
x
D ik
i ∈ Dk , k ∈ K
(redAMI)
D,UP ik
≤x ≤x
By solving (redAMI), we optimize the continuous part of the objective function over the feasible region constrained solely by local constraints. Note that the only type of variables present in the model is the disaggregated continuous variables. Testing the procedure on the 500 randomly generated VPs has revealed that in all the instances (redAMI) produces VPs that lead to the tightest possible continuous relaxation. We must note that solving (redAMI) introduces another step into the optimization
Alternative mixed-integer reformulation of Generalized Disjunctive Programs
553
procedure; however, since the model is significantly smaller, the impact of solving (redAMI) on the solution time is expected to be low. As a final note, the effect of VPs on the tightness of the continuous relaxation is only present if the disjunctions contain nonlinear or combination of linear and nonlinear constraints.
4. Process Synthesis Examples In this section, we compare the efficiency of the proposed reformulation against Big-M and HR reformulation. The process network synthesis is a classical optimization problem in which the goal is to select a process network that maximizes profit while considering the costs of raw materials and the cost of the process units. The numerical examples Proc8, Proc10 and Proc12 are variations of the example introduced by Türkay and Grossmann (1996). A general GDP model is described by (3). The examples were solved using SBB solver. The results for the three instances are given in Tables 1-2. min Z = ∑ ci + ∑ p j x j + γ i
s.t.
j
∑ rj , n x j ≤ 0
∀n ∈ N
j
Yi ¬Yi d (e x j / ti , j − 1) − s x ≤ 0 ∨ x = 0, j ∈ J i ∑j i , j j j i, j ∑ j ci = γ i ci = γ i True Ω(Y ) =
(3)
x j , ci ≥ 0, Yi ∈ {True, False}
Table 1: Comparison of relaxations and total computational times. Big-Ma
HRb
tCPU /s
LB
redAMI + AMI
Ex./Nbins
Opt.
LB
tCPU /s
LB
tCPU /s
Proc8/8
68.01
-829.88
0.23
67.7334
0.13
67.7334
0.11
Proc10/10
-27.93
-1146.88
0.84
-40.1313
0.16
-40.1314
0.14
Proc12/12
-69.21
-894.52
2.01
-73.8895
0.14
-73.8895
0.09
a – Value of 50 was used for all the Big-M parameters. | b – In the HR reformulation, value of 1e-6 was used for ε.
Table 2: SBB solver statistics. Big-M Ex.
Iter
Nds
HR tNLP/s
Iter
Nds
redAMI + AMI tNLP/s
Iter
Nds
tNLP/s
Proc8
801
34
0.23
117
1
0.12
53
1
0.09
Proc10
2820
205
0.84
236
4
0.15
78
5
0.05
Proc12
7347
463
1.97
174
3
0.14
99
3
0.08
554
M. Bogataj and Z. Kravanja
As can be seen in Table 1, the HR and the proposed approach (redAMI +AMI) produce significantly tighter relaxations than the Big-M reformulation in all the instances. Both the HR and the proposed reformulation exhibit faster computational times. The proposed approach produces equally tight continuous relaxations as HR in two instances (Proc8 and Proc12), while the relaxation of Proc10 is slightly weaker, consequently causing an increase in the number of nodes searched to find the optimal solution. Interestingly, the proposed approach requires the least computational time to find optimal solutions and the least solver iterations (Table 2).
5. Conclusions Over the last decades, a number of reformulation strategies for obtaining tighter continuous relaxations of discrete-continuous problems have been proposed. In this work, we present a reformulation that utilizes MI translation of variables and avoids the usage of the perspective function. When dealing with nonlinear GSPs and their MINLP representations, the first property, as suggested by Ropotar and Kravanja (2009), tends to reduce the search space of nonlinear subproblems. The second property might be important for producing more robust nonlinear subproblems as the proposed reformulation does not introduce additional nonlinearities to the model. Based on the results presented in this work, we can conclude that the proposed reformulation provides results comparable to those obtained by the HR, both in terms of tightness of continuous relaxation as well as of the computational effort needed to obtain optimal solutions. Future work will focus on research dedicated to providing the answers regarding the procedures for obtaining VPs, and testing of the proposed reformulation on large-scale synthesis problems. We believe that the findings of the future work will enable us to fully evaluate the benefits as well as drawbacks of the proposed reformulation.
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