Modeling of discrete/continuous optimization problems: characterization and formulation of disjunctions and their relaxations

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1 Computers and Chemial Engineering (00) 4/448 Modeling of disrete/ontinuous optimization problems: haraterization and formulation of disjuntions and their relaxations Aldo Vehietti a, Sangbum Lee b, Ignaio E. Grossmann b, * a INGAR, Instituto de Desarrollo y Diseño, UTN, Faultad Regional Santa Fe, Argentina b Department of Chemial Engineering, Carnegie Mellon University, Pittsburgh, PA , USA Abstrat This paper addresses the relaxations in alternative models for disjuntions, big-m and onvex hull model, in order to develop guidelines and insights when formulating Mixed-Integer Non-Linear Programming (MINLP), Generalized Disjuntive Programming (GDP), or hybrid models. Charaterization and properties are presented for various types of disjuntions. An interesting result is presented for improper disjuntions where results in the ontinuous spae differ from the ones in the mixed-integer spae. A utting plane method is also proposed that avoids the expliit generation of equations and variables of the onvex hull. Several examples are presented throughout the paper, as well as a small proess synthesis problem, whih is solved with the proposed utting plane method. # 00 Elsevier Siene Ltd. All rights reserved. Keywords: Disrete/ontinuous optimization; Mixed-integer nonlinear programming; Generalized disjuntive programming; Big-M relaxation; Convex hull relaxation 1. Introdution Developing optimization models with disrete and ontinuous variables is not a trivial task. The modeler has often several alternative formulations for the same problem, and eah of them an have a very different performane in the effiieny on the problem solution. In the area of Proess System Engineering models ommonly involve linear and nonlinear onstraints and disrete hoies. The traditional model that has been used in the past orresponds to a mixed-integer optimization program whose representation an be expressed in the following equation form (Grossmann & Kravanja, 199): min Z f (x)d T y s:t: g(x) 50 (PA) r(x)ly 50 Ay]a * Corresponding author. Tel.: / ; fax: / addresses: aldove@eride.gov.ar (A. Vehietti), sangbum@mu.edu (S. Lee), grossmann@mu.edu (I.E. Grossmann). x R n ; y f0; 1g q where f(x), g(x) and r(x) are linear and/or nonlinear funtions. In the model (PA) the disrete hoies are represented with the binary variables y involving linear terms. More reently, generalized disjuntive programming (Raman & Grossmann, 1994; Türkay & Grossmann, 199) has been proposed as an alternative to the model (PA). A generalized disjuntive program an be formulated as follows: min Z X k K k f (x) s:t: g(x) 50 Y ik 4h ik (x)505 k k g ik k K (GDP) V(Y)True x R n ; Y ik ftrue; Falseg m ; k ]0 where the disrete hoies are expressed with the Boolean variables Y ik in terms of disjuntions, and logi propositions V(Y). The attrative feature of Generalized Disjuntive Programming (GDP) is that it allows a /0/$ - see front matter # 00 Elsevier Siene Ltd. All rights reserved. PII: S ( 0 ) X

2 44 A. Vehietti et al. / Computers and Chemial Engineering (00) 4/448 symboli/quantitative representation of disrete and ontinuous optimization problems. A modeling language for GDP problem has been disussed by Vehietti and Grossmann (000). An approah that ombines the previous two models is a hybrid model proposed by Vehietti and Grossmann (1999) where the disrete hoies an be modeled as mixed-integer onstraints and/or disjuntions. In this way we an potentially exploit the advantages of the two previous formulations by expressing part of it only in algebrai form, and the other in a symboli/quantitative form. The hybrid formulation is as follows: min Z X k f (x)d T y k K s:t: g(x) 50 r(x)ly 50 Ay]a (PH) Y ik 4h ik (x)505 k K k k y ik V(Y)True x R n ; y f0; 1g q ; Y ik ftrue; Falseg m ; k ]0 where r(x)/ly 5/0 are general mixed-integer onstraints that an be linear/nonlinear equations/inequalities. These terms an be seen as disjuntions transformed into mixed-integer form. Ay ]/a represents general integer equalities/inequalities transformed from former logi propositions. An issue that is unlear is how the modeler should express the disrete hoies, either as a symboli disjuntion, or in a mixed-integer form (Bokmayr & Kasper, 1998). One possible guideline for this deision is the gap between the optimal value of the ontinuous relaxation and the optimal integer value. Sine several algorithms involve the solution of the relaxed problem, we will investigate in this paper the tightness of different relaxations for a disjuntive set: the big-m formulation (Nemhauser & Wolsey, 1988), the Beaumont surrogate (Beaumont, 1990) and the onvex hull relaxation (Balas, 199; Lee & Grossmann, 000). The big-m formulation and the Beaumont surrogate an be regarded as obvious onstraints. However, the onvex hull relaxation of a disjuntion is tighter, and an be transformed into a set of mixed-integer onstraints. The advantage of the onvex hull relaxation is that the tight lower bound helps to redue the searh effort in the branh and bound proedure, in both nonlinear and linear problems (for examples of signifiant node redutions see Lee & Grossmann, 000; Jakson & Grossmann, 00). But the drawbak with the onvex hull formulation is that it inreases the number of ontinuous variables and onstraints of the original problem. This an potentially make a problem more expensive to solve, espeially in large problems. The big-m relaxation is more onvenient to use when the problem size does not inrease substantially when ompared with the onvex hull relaxation (see Yeomans & Grossmann, 1999, who found the big-m to be more effetive). But generally the lower bound by the big-m relaxation is weaker, whih may require longer CPU time than the onvex hull relaxation. Therefore, depending on the ase, there is a trade-off between the best possible relaxation and the problem size. In order to exploit the tightness of the onvex hull relaxation, but without the substantial inrease of the onstraints, it will be shown that utting planes an be used that orrespond to a faet of the onvex hull. In this paper we first introdue the definition and properties of a disjuntive set. We then present the different relaxations and their properties. Finally, a utting plane method is disussed, and illustrated with several small example problems. The goal of this paper is not to perform a detailed omputational study, but rather to provide insights into the modeling and solution of disjuntive problems.. Definitions and properties of a disjuntive set A disjuntive set F an be expressed as a set of onstraints separated by the or ( /) operator: F [h i (x)50] x R n (1) It is assumed that h i (x) is a ontinuous onvex funtion. F an be onsidered as a logial expression, whih enfores only one set of inequalities. The feasible region of eah disjuntive term an be expressed as the set of points that satisfy the inequality. R i fx½h i (x)50g () A disjuntive set an be expressed in other forms that are logially equivalent. F an also be expressed as the union of the feasible regions of the disjuntive terms, whih is alled Disjuntive Normal Form (DNF): i (x)50] x R n () R i (4) If the union of the feasible regions of the disjuntive terms is equal to one of its terms, R j, whih is the largest feasible region, then the disjuntive set is alled improper. Otherwise the disjuntive set is alled proper (Balas, 1985). The improper disjuntive set an be written as follows: F R i R j The improper disjuntive set has also the following

3 A. Vehietti et al. / Computers and Chemial Engineering (00) 4/ property: R i R j Öi"j () whih means that the feasible regions i (i "/j) in the disjuntive set F are inluded in the jth feasible region. Sine F is expressed as the union of the different terms, an improper disjuntive set an be redued to: F fx½h j (x)50g () On the other hand, a proper disjuntive set is the one in whih either the intersetion of the feasible regions is empty, or else it is non-empty, but Eq. (5) does not apply. Therefore, for a proper disjuntive set, either there is no intersetion among the feasible regions: S R i (8) or else, there is some intersetion, but no set R j ontains all of them: R i " ; R i "R j (9) Beaumont relaxation Beaumont (1990) proposed a valid inequality for the disjuntive set Eq. (10). A valid M i value must be alulated as in Eq. (1). Bydividing eah onstraint i / D in Eq. (11) by M i and summing over i /D, the Beaumont surrogate, whih interestingly does not involve binary variables is given as follows: X h i (x) 5N 1 (1) M i where N/jDj in Eq. (10). Beaumont showed that Eq. (1) yields an equivalent relaxation as the big-m relaxation Eq. (11) projeted onto the ontinuous x spae when the onstraints in Eq. (10) are linear... Convex hull relaxation The onvex hull relaxation for the disjuntive set Eq. (10) an be written as follows (Lee & Grossmann, 000): x X v i 0 x; v i R n. Relaxations of a disjuntive set vi y i h i 50; y i Given a disjuntive set as ondition Eq. (1) there are a number of relaxations that an be derived, the big-m, the Beaumont surrogate and the onvex hull relaxations. We onsider below the ase of onvex nonlinear onstraints, whih easily simplifies to the linear ase. X y i 1 05y i 51 05v i 5v U i y i ; (14).1. Big-M relaxation Consider the following nonlinear disjuntion: F [h i (x)50] x R n (10) where h i (x) is a nonlinear onvex funtion. For simpliity, and without loss of generality, it is assumed that eah term in the disjuntion Eq. (10) has only one inequality onstraint. The big-m relaxation of Eq. (10) is given by: h i (x)5m i (1y i ) X y i 1 05y i 51; (11) The tightest value for M i an be alulated from: M i maxfh i (x)½x L 5x5x U g (1) where v i U is a valid upper bound for the disaggregated variables v i, usually hosen as x U. The Eq. (14) define a onvex set in the (x, v, y) spae provided the inequalities h i (x)5/0, i /D are onvex and bounded. The onvex hull in Eq. (14) an be proved to be tighter or at least as tight as the big-m relaxation (see Appendix A). Also, for ase of linear disjuntions, F [a T i x5b i ] x R n ; Eq. (14) redues to the equations by Balas (199, 1988): x X v i 0 a T i v i b i y i 50; X y i 1 05y i 51; x; v i R n 05v i 5y i v up i ; (15)

4 4 A. Vehietti et al. / Computers and Chemial Engineering (00) 4/ Example 1 Consider the following nonlinear disjuntion: [(x 1 1) (x 1) 51] [(x 1 4) (x ) 51] [(x 1 ) (x 4) 51] where 05/x 1 5/5 and 05/x 5/5. The feasible region is shown in Fig. 1. Figs. and show the feasible region of the big-m and the onvex hull relaxations, respetively. The big-m relaxation is given by: (x 1 1) (x 1) 511(1y 1 ) (x 1 4) (x ) 514(1y ) (x 1 ) (x 4) 514(1y ) Fig.. Big-M relaxation of example 1. y 1 y y 1 05x 1 ; x 55; 05y i 51; i1; ; (1) where the big-m parameters are alulated by Eq. (1). The onvex hull of Fig. is given by the equations: x 1 v 11 v 1 v 1 x v 1 v v (y 1 o) v11 y 1 o 1 v1 y 1 o (y o) (y o) v1 y o 4 v1 y o y 1 y y 1 v y o v y o Fig.. Convex hull relaxation of example 1. 05y i 51; i1; ; 05v ji 55y i Öi; Öj (1) Note that to avoid division by zero o is introdued in the nonlinear inequalities as a small tolerane (Lee & Grossmann, 000). Typial values for o are / From Figs. and it is lear that the onvex hull relaxation of the disjuntive set is tighter than the big-m relaxation for this example. 4. Impat of nature of disjuntions on relaxations in x spae Fig. 1. Feasible region of example 1. Our aim in this setion is to analyze different types of disjuntions for whih it may be onvenient or not to transform them into the onvex hull formulation or a big-m formulation or a Beaumont surrogate. Sine the big-m formulation is as tight as the Beaumont surrogate, and it is more frequently used, we will ompare the

5 A. Vehietti et al. / Computers and Chemial Engineering (00) 4/448 4 onvex hull with only the big-m formulation. We will analyze the following ases: (a) improper disjuntion; (b) proper disjuntion. Within this last ase we will analyze when the intersetion of the feasible regions is empty and when it is non-empty. If we denote the feasible region of the onvex hull relaxation in the ontinuous x spae as R CH, the feasible region of the big-m relaxation as R BM, and the feasible region of the Beaumont surrogate as R B, then aording to the properties shown in the previous setion, the following an be established: R CH R BM (18) Beaumont (1990) has shown for the linear ase that R BM /R B where R B is defined by onstraint Eq. (1). In the Appendix A we show that R BM /R B for nonlinear ase. Therefore, the following property holds: R BM R B (19) It should be noted that properties Eqs. (18) and (19) apply in the spae of the ontinuous variables x Improper disjuntion When the disjuntive set is improper, the property in Eq. () holds. Sine the feasible region of one term ontains the feasible regions of the other terms, the relaxations of the onvex hull and of the big-m an be seleted to be idential. The reason is that the redundant terms an be dropped and the disjuntive set an be represented by the term with the largest feasible region R j. For example, suppose we have the following problem: min Z (x 1 :5) (x 4:5) s:t: 415x x 54 4 Y 5x 1 55 (0) 5x 54 The feasible region is shown in Fig. 4. Choosing the term with the largest feasible region, whih is the first one, and solving the problem as an NLP we obtain the optimal solution x/(,4) and Z/0.5. If we are not aware that the feasible regions are overlapped we an generate the big-m relaxation for this problem. If we use M i /0.5, i/1,, and solve the relaxed Mixed-Integer Non-Linear Programming (MINLP) problem, the solution is x/(.5,4.5), Z /0.15, y /(0.5,0.5). If we hoose M i /1 and solve the relaxed MINLP then the solution is x /(.5,4.5), Z /0, y /(0.5,0.5). Therefore, it is lear that arbitrary hoie of M i an yield a relaxation whose feasible region is larger than the disjuntive term with the largest feasible region. For the onvex hull formulation it is lear that the resulting relaxation oinides with the region of the largest term in the x spae, but at the expense of expressing it through disaggregated variables and additional onstraints. 4.. Proper disjuntion Non-empty interseting feasible regions When the feasible regions of the disjuntive terms have an intersetion, it is not lear whether or not the onvex hull and the big-m formulation ould yield the same relaxation. Suppose we have disjuntions whose feasible regions are shown in Figs. 5 and. InFig. 5 it is lear that the big-m relaxation, with a good seletion of the M i values an yield the same relaxation as the onvex hull. For the ase of Fig. the onvex hull will yield a tighter relaxation Disjoint disjuntion If the feasible region defined by eah term in the disjuntion has no intersetion with others, then the disjuntion is disjoint and proper. Fig. shows an example of disjoint disjuntion. In this ase, it is lear that the onvex hull relaxation should generally be Fig. 4. Feasible region of disjuntive set Eq. (0). Fig. 5. Interseting disjuntion.

6 48 A. Vehietti et al. / Computers and Chemial Engineering (00) 4/448 Fig.. Interseting disjuntion. Fig. 9. Disjoint disjuntion with zero point. y 1 h 1 x y x5x U y 1 05y 1 51 (1) whih inludes the zero point as a feasible point. The above also applies to linear ase. 5. Relaxation in x /y spae Fig.. Disjoint disjuntion (general ase). tighter than the big-m relaxation (an exeption is the partiular ase shown in Fig. 8). Also, in the speial ase shown in Fig. 9, where a disjuntion has two terms with linear onstraints and one of them yields zero point as a feasible region, the onvex hull yields a one with the zero point as the vertex. In this ase, the onvex hull relaxation an be simplified by not requiring disaggregated variables as given by the following: The previous setion analyzed the relation of relaxations for different types of disjuntions in the x spae. When applying the big-m onstraints Eq. (11) or the onvex hull Eq. (14) these are written in the x /y spae. Therefore, an interesting question is whether or not the properties we noted in the previous setion still apply in the x /y spae. Let us onsider the following example, whih has an improper disjuntion Example min Z (x 1 1:1) (x 1:1) 1 s:t: 4x 1 x x 1 x x 1 ; x 51; 05 1 Fig. 8. Disjoint disjuntion (partiular ase). ftrue; falseg () The optimal solution is x/(0.0,0.0), /true and Z/1.09. The feasible region is shown in Fig. 10 and the feasible region of the seond term, whih is (0,0), is inluded in the feasible region of the first term. Aording to the previous setion sine this is an improper disjuntion in the x spae, it ought to be suffiient to use the first term only. However, when

7 A. Vehietti et al. / Computers and Chemial Engineering (00) 4/ Fig. 10. Feasible region of example in the x spae. Fig. 11. Convex hull relaxation of example in the x /y spae. expressed algebraially, the big-m relaxation and the onvex hull relaxation of the disjuntion in Eq. () involve the additional variable y 1 as a ontinuous variable. In the ase of the onvex hull, we apply Eq. (1) to the first term. Rearranging the inequality y 1 [(x 1 / y 1 ) /(x /y 1 ) /1]5/0 yields: min Z (x 1 1:1) (x 1:1) y 1 s:t: x 1 x 5y 1 05x 1 5y 1 05x 5y 1 05y 1 51 () The big-m relaxation of Eq. (1) for the first term is given by: min Z (x 1 1:1) (x 1:1) y 1 s:t: x 1 x 5y 1 05x 1 ; x 51; 05y 1 51 (4) Figs. 11 and 1 show the onvex hull relaxation and the big-m relaxation of Eq. (1) in the x /y spae, respetively. It is lear that Eqs. () and (4) are not idential due to the differene in the right hand side of the nonlinear inequality. In fat, the solution of Eq. () is (x, y) /(0.0, 0.0, 1) and Z /1.09. Sine the relaxed value of y 1 is 1, this solution is the optimal solution of Eq. (), whih is also shown in Fig. 11. On the other hand, the solution of Eq. (4) is (x, y)/(0.55, 0.55, 0.05) and Z /1.1 whih is weaker than the onvex hull relaxation. This result an be seen by omparing Figs. 11 and 1. There is no differene between the feasible set of Eq. () and the feasible set of Eq. (4) projeted in the x spae as shown in Fig. 10. The differene, however, takes plae in the x /y spae. Note that the nonlinear onstraint in Eq. (4), x 1 /x 5/ Fig. 1. Big-M relaxation of example in the x /y spae. y 1, whih is shown in Fig. 1, is weaker than x 1 /x 5/y 1 in Eq. () for 05/y 1 5/1. Therefore, even though the disjuntion in Eq. () is improper in x spae, the onvex hull yields tighter relaxation than big-m relaxation in the x /y spae. Thus, this example demonstrates that for the ase of improper nonlinear disjuntions, the onvex hull may be tighter than the big-m onstraint in the x /y spae even if they are idential in the projeted x spae. For the linear ase, we hange the nonlinear onstraint in the first term of the disjuntion Eq. () by the following linear onstraint: min Z (x 1 1:1) (x 1:1) 1 s:t: 4x 1 x 515 4x 1 x x 1 ; x 51; 05 1 ftrue; falseg (5) where the disjuntion is improper in the x spae. The optimal solution is x/(0.5,0.5), /true and Z/1..

8 440 A. Vehietti et al. / Computers and Chemial Engineering (00) 4/448 The onvex hull of the disjuntion Eq. (5) yields a linear onstraint: x 1 x 5y 1 () After replaing the disjuntion Eq. (5) with onvex hull relaxation Eq. (), the solution is x /(0.5,0.5), y 1 /1 and Z/1., whih is exatly the optimal solution of Eq. (5). Sine the disjuntion Eq. (5) is improper in x spae, only the first term is suffiient for the relaxation. The big-m relaxation of Eq. (5) is given by: x 1 x 15M 1 (1y 1 ) () This relaxation learly depends on M 1 value. For example, if M 1 /1 is used, then the relaxation yields x/(0.,0.), y 1 /0. and Z/1.04, whih is weaker than the onvex hull relaxation. The best M 1 value in this ase is /1, whih yields exatly the same solution as the onvex hull relaxation. As shown with this example, even for the linear improper disjuntion the big-m relaxation may have weaker relaxation than the onvex hull depending on the big-m parameter value.. Cutting plane method The two previous setions have analyzed the issue of determining in what ases it is worth to formulate disjuntions with the onvex hull relaxation in order to obtain tighter relaxations when ompared with the big- M relaxation. In this setion, we present a numerial proedure for generating utting planes, whih potentially has the advantage of requiring muh fewer variables and onstraints than the onvex hull relaxation. Cutting planes, whih orrespond to faets of the onvex hull, an improve the tightness of the big-m relaxation. The proposed utting planes an be used within a branh and ut enumeration proedure (Stubbs & Mehrotra, 1999), or as a way to strengthen an algebrai MINLP model before solving it with one of the standard methods. Using as a basis the GDP model, the general form of the strengthened MINLP model (PC n ) at any iteration n will be as follows: min Z X X g ik y ik f (x) k K k s:t: g(x) 50 h ik (x)5m ik (1y ik ); k ; k K (PC n ) X y ik 1; k K k Ay5a b T n x5b n ; n1; ;...; N x R n ; y ik f0; 1g where b n T x 5/b n is the utting plane at the iteration n. Let us denote the solution of the ontinuous relaxation of (PC n )asx R BM,n. In order to generate the utting plane we onsider the following separation problem, whih has as an objetive to find the point within the onvex hull that is losest to the point x R BM,n. This separation problem is given by the NLP: min f(x)(xx BM;n R ) T (xx BM;n R ) s:t: g(x) 50 x X k v ik ; k K y ik h ik vik X y ik 50; k ; k K (SP n ) y ik 1; k K k Ay5a b T n x5b n ; n1; ;...; N x; v ik R n ; 05y ik 51 Let the solution of the separation problem (SP n )be x S,n. A utting plane b T n x 5/b n an then be obtained from: (x S;n x BM;n R ) T (xx S;n )]0 (8) where the oeffiient of x is a subgradient of the objetive funtion of (SP n )atx S,n (for derivation, see Stubbs & Mehrotra, 1999). Fig. 1 shows an example of a utting plane generated with the points x S,n and x BM,n R. The utting plane method an then be stated as follows: 1) Solve ontinuous relaxation of (PC n ). ) Solve separation problem (SP n ). a) If jjx S,n /x BM,n R jj5/o, stop. Fig. 1. Cutting plane generated by separation problem.

9 A. Vehietti et al. / Computers and Chemial Engineering (00) 4/ b) Else set b n1 //(x S,n /x R BM,n ) and b n1 // (x S,n /x R BM,n )x S,n. Set n/n/1, return to Step 1. This proedure an be used either in a Branh and Cut enumeration method where a speial ase is to solve the separation problem only at the root node, or else it an be used to strengthen the MINLP model before applying methods suh as Outer-Approximation (OA), Generalized Benders Deomposition (GBD), and Extended Cutting Plane (ECP). It is also interesting to note that utting planes an be derived in the x /y spae. In example, when we onsider the utting plane in the x spae, the big-m relaxation solution, x /(0.55, 0.55) annot be separated from the onvex hull sine it is feasible to the onvex hull onto the x spae. But when we onsider the utting plane in the x /y spae, then the big-m relaxation solution, (x, y) /(0.55, 0.55, 0.05) an be separated from the onvex hull sine this point is infeasible to the onvex hull relaxation Eq. (). This suggests that the appliation of utting planes in the x / y spae may be more effetive than in the x spae only for utting off the big-m relaxation point from the onvex hull. Another appliation of the separation problem is for deiding whether it is advantageous to use the onvex hull formulation. If the value of jjx S,n /x R BM,n jj is large, then it is an indiation that this is the ase. A small differene between x S,n and x R BM,n would indiate that it might be better to use the big-m relaxation. It should be also noted that the proposed utting plane method an be extended to nononvex disjuntive onstraints using the global optimization proedure by Lee and Grossmann (001). In this method the nononvex onstraints are replaed by onvex under/overestimators, with whih the onvex hull relaxation or big- M relaxation an be used. Therefore, one an use the utting plane method to tighten the relaxation of the bounding onvex onstraints. s:t: (x 1 4) (x ) 50:5 Y (x 1 ) (x 4) 51 Y (x 1 1) (x 1) 51:5 05x 1 ; x 55 (9) The feasible region is shown in Fig. 14. Note that the point (,4), whih is the minimizer of the objetive funtion, lies outside the onvex hull of the disjuntion. The optimal solution is x /(4,4), Z /4.0, Y /(false, true, false). To illustrate the utting plane proedure, first we solve the big-m relaxation of Eq. (9) with M/(19.5, 4, 0.5) from Eq. (1). The solution is x BM /(5, 4), Z BM /1.0, y BM /(0.09, 0.51, 0.0). Then we solve the separation problem (SP n ) with the relaxation point x BM /(5, 4): min Z (x 1 5) (x 4) s:t: x 1 v 11 v 1 v 1 x v 1 v v (y 1 o) (y o) (y o) v11 y 1 o 4 v1 y o v1 y o 1 y 1 y y 1 05y i 51; i1; ; v1 y 1 o v y o 4 v y o 1 0: :5 50. Disjuntive programming examples In this setion we present a number of examples to illustrate the appliation of the main onepts in this paper..1. Example min Z (x 1 ) (x 4) Fig. 14. Feasible region of example.

10 44 A. Vehietti et al. / Computers and Chemial Engineering (00) 4/448 05v ji 55y i Öi; Öj 05x 1 ; x 55 (0) The solution of problem Eq. (0) is x S /(4.1,.0) with the objetive value of Therefore, the utting plane is given as follows: T 4:15:0 x 1 4:1 ]0 (1) :04:0 x :0 whih an be simplified as /0.84(x 1 /4.1)/0.(x /.0)]/0. We add Eq. (1) to the big-m relaxation and solve it again. The solution of this augmented big-m relaxation is x CP /(4.,.4), Z CP /., y CP /(0.94, 0., 0.09). For omparison, we solve the onvex hull relaxation, obtaining x CH /(4.,.4), Z CH /., y CH /(0.44, 0.558, 0). Note that the solution x CP and the objetive value Z CP are idential to x CH and Z CH. The differene in (x BM, Z BM ) and (x CH, Z CH ) is a lear indiation that the onvex hull is signifiantly tighter than big-m relaxation. For this example, only one utting plane yields the same tightness of the relaxation as the onvex hull. The numerial results are shown in Table 1. Note that the big-m relaxation yields the lowest objetive value to the optimal solution, 4.0. Fig. 15 shows the onvex hull and utting plane. As shown in Fig. 15, the utting plane is a faet of the onvex hull. From Table 1 it an be seen that the big-m relaxation with a utting plane yields a ompetitive relaxation ompared with the onvex hull... Cutting planes in x /y spae: example Let us revisit example. If we apply the separation problem (SP n ) to the big-m relaxation solution x BM R / (0.55,0.55), the objetive value of the separation problem is zero sine x BM R is feasible to the onvex hull relaxation of Eq. () in the x spae. However, if we treat the binary variable y as ontinuous variable and then extend the dimension of the solution to the x /y spae, we have the following separation problem with (x, y) BM R /(0.55, 0.55, 0.05): min Z [(x 1 0:55) (x 0:55) (y 1 0:05) ] s:t: x 1 x 5y 1 (SP1) 05x 1 5y 1 Fig. 15. Convex hull and utting plane for example. 05x 5y 1 05y 1 51 The solution is Z/0.015 and (x, y) S /(0.489, 0.489, 0.91), whih means that (x, y) BM R is infeasible in the onvex hull relaxation Eq. () in the x /y spae. The utting plane is now given by (0.489/0.55)(x 1 /0.489)/ (0.489/0.55)(x /0.489)/(0.91/0.05)(y 1 /0.91)]/ 0.When this utting plane is added to the big-m relaxation Eq. (4), the optimal solution is (x, y) / (0.0, 0.0, 1) and Z /1.09, whih is idential to the solution of the onvex hull relaxation Eq. () and is also the optimal solution of Eq. (). This shows that the utting plane method applied to the x /y spae an yield tighter relaxations than the utting plane in the x spae only... Example 4 Consider the synthesis of a proess network (Türkay & Grossmann, 199) where the following disjuntive set is used to model the problem: Y k Y k 4h ik (x)05 4B ik x05 k ; k K () k g k k 0 It means that if the kth unit is seleted (Y k /true) then the first term of the disjuntion applies, if it is not (/Y k ) then a subset of the x variables is set to zero. Table 1 Comparisons of the relaxations for example Relaxation M x 1 x y 1 y y Z Big-M (19.5, 4, 0.5) Convex hull / Cutting plane / Optimal solution /

11 A. Vehietti et al. / Computers and Chemial Engineering (00) 4/ Fig. 1 shows the superstruture of example 4, whih has eight units. The orresponding GDP model is as follows: Fig. 1. Proess superstruture of example 4. Y 4 Y 4 41:5(x 1 x 14 )x x 1 x 1 x min Z X8 k1 s.t. Mass balanes: k a T x1 x 1 x x 4 ; x x x 8 x x 5 x x 11 x 11 x 1 x 15 ; x 1 x 19 x 1 x 9 x 1 x 5 x 1 x 0 x x ; x x 14 x 4 Speifiations: x 10 0:8x 1 50; x 10 0:4x 1 ]0 x 1 5x 14 50; x 1 x 14 ]0 Disjuntons: 4exp(x )1x 505 4x x Y x5 Y 4exp 1x : 5 4x 4 x Y Y 41:5x 9 x 10 x x 9 0; x 8 x Y 5 Y 5 4x 15 x x 15 x Y x0 Y 4exp 1x :5 5 4x 19 x Y Y 4exp(x )1x x 1 x Y 8 Y 8 4exp(x 18 )1x 10 x x 10 x 1 x () Logi propositions: [/Y /Y 4 /Y 5 Y 5 [/Y 8 Y [/Y /Y 4 /Y 5 Y [/Y 4 Y [/ /Y Y [/Y 4 Y [/Y 8 Y 8 [/Y /Y 5 /(/Y ffl//y 5 ) Y 4 [/ /Y /Y Y 4 [/Y /Y Y 4 /Y 5 Y 5 [/ /Y Y /Y Problem data: a T /(a 1 /0, a /10, a /1, a 4 /1, a 5 //15, a /0, a /0, a 8 /0, a 9 //40, a 10 /15, a 11 /0, a 1 /0, a 1 / 0, a 14 /15, a 15 /0, a 1 /0, a 1 /80, a 18 //5, a 19 /

12 444 5, a 0 //0, a 1 /5, a //80, a /0, a 4 /0, a 5 //5); x j lo /0, Ö/j. Before introduing the big-m relaxation, it should be noted that in the disjuntions we have the following properties: i) The disjuntions are improper sine the feasible region of the seond term belongs to the feasible region of the first term in x spae (exept the ost term). ii) In the seond term of the disjuntions a subset of the ontinuous variables x are zero. Beause of these properties, it is possible to rewrite the disjuntions as follows: exp(x )1x 50 x5 exp 1x : 1:5x 9 x 10 x 8 0 1:5(x 1 x 14 )x 1 0 x 15 x 1 0 x0 exp 1x :5 exp(x )1x 1 50 x18 exp 1x 10 x :5 Disjuntions: 05x 5x up 405x 5x up 5 4x x Y 05x 4 5x up Y 4 405x 5 5x up 5 4x 4 x Y Y 405x 9 5x up 9 5 4x Y 4 05x 1 5x up 1 Y 4 05x 1 5x up 1 405x 14 5x up 5 4x 1 x 1 x Y 5 05x 15 5x up Y x 1 5x up 5 4x 15 x A. Vehietti et al. / Computers and Chemial Engineering (00) 4/448 Y 05x 19 5x up Y x 0 5x up 5 4x 19 x Y 05x 1 5x up Y 1 405x 5x up 5 4x 1 x Y 8 05x 10 5x up 1 Y 8 05x 1 5x up 1 405x 18 5x up 5 4x 10 x 1 x (4) It should be noted that onstraints Eq. (4) onsist of global onstraints (nonlinear) and disjuntions (linear). The onvex hull of the above disjuntions an be redued to linear onstraints yielding the tight big-m relaxation: 05x j 5x up j y k ; j J; k K (5) k g k y k ; k K 05y k 51; k K whih means that if the first term of the disjuntion is true (y k /1) then the ontinuous variables x j an have a value between its bounds and the fixed ost is ativated, else if the seond term is true (y k /0) then the ontinuous variables beome zero that still satisfies the global onstraints (ondition i). The GDP problem Eq. () is solved with the onvex hull relaxation. The upper bounds used are x up /, x up 5 /, x up 9 /, x up 10 /1, x up 14 /1, x up 1 /, x up 19 /, x up 1 /, x up 5 /, and for the rest of the variables, x up j /.5. The objetive funtion value Z /4.8 was obtained from the onvex hull relaxation, and the orresponding NLP requires 0.0 CPU s with CONOPT/GAMS. Applying the big-m relaxation to the modified GDP formulation Eq. (4) and the same bounds, we obtained Z /49.9 as the solution value. Therefore, the onvex hull relaxation of the original GDP model yields a muh tighter lower bound. The differene between these two relaxation values omes from the fat that the feasible region by the onvex hull relaxation of nonlinear disjuntions Eq. () in the x /y spae is tighter than the feasible region by big-m relaxation of Eq. (4). However, it should be noted that their projetions onto the x spae are idential sine the disjuntions are improper. If the disjuntions are linear, then both relaxations an be idential in the x /y spae if appropriate big-m parameters are used. Sine the onvex hull relaxation yields a signifiant inrease in the number of additional onstraints and variables, we onsider the generation of utting planes to strengthen the big-m relaxation. As outlined in

13 A. Vehietti et al. / Computers and Chemial Engineering (00) 4/ Setion, a separation problem is solved. And the solution of the separation problem is used to build a utting plane as in example 4. The big-m relaxation of Eq. (4) is then solved again with this utting plane. Sine the utting plane is a faet of the onvex hull, it will tighten the lower bound. Table shows the inrease of the lower bound as utting planes are added to the big-m relaxation. The first olumn shows the number of utting planes added. The seond olumn shows the relaxation value. Note that the optimal solution of example 4 is The third olumn shows the objetive value of the separation problem. As more utting planes are added, the objetive value of the separation problem dereases, implying that the solution point of the augmented big-m relaxation gets loser to the onvex hull. The fourth olumn shows the CPU time of the separation problem. The fifth and sixth olumn show the MINLP solution results by DICOPT// with the orresponding uts. In all ases, the optimal solution is found in the seond major iteration. Sine this problem is a onvex MINLP, the Outer-Approximation (OA) algorithm stops when the rossover ours. The CPU time is less than 1 s on a Pentium III PC 00 MHz with 18 Mbytes RAM memory. After adding seven utting planes, the lower bound improved signifiantly ompared with the ase when no utting plane is used (.5 vs. 49.9). The advantage of the utting plane method is that only one linear onstraint is added to the big-m relaxation at eah step. However, there is a ost for building a utting plane and that is to solve a separation problem, whih is a onvex NLP problem (SP n )..4. Example 5 To illustrate the appliation of the utting plane method with a branh and bound algorithm, we have onstruted the following GDP problem with linear/ nonlinear proper disjuntions. min Z X9 k1 k a T x 0: log(x 1 1)0:8(x 1 8) 0: exp(x 14 1) 0:5 log(x 15 ) s.t. Mass balanes: x 1 x 5 x ; x 4 x x 8 x 10 x 19 x 0 ; x 11 x 1 x 18 x 14 x 1 x ; x 9 x x 4 x 1 x 5 x Speifiations: x 1 x x x 4 50 x 9 x 10 x x 1 x 1 x 14 x 15 x 1 50 x 9 51: log(x x 5 1) x 9 ]0:10:x 5 4x 5 ]x 5 4x x 5 x Y x 10 0:9x 0:8x Y 15x x 4x ]x 5 4x x x Y 1:5x 11 x x 8 Y x x 8 4x 11 ]1 5 4x x 8 x Y 4 x 5 5log(x 1)0:1 Y 4 4x 5 ]1 5 4x x :5 4 0 Y 5 x 51:5 log(x 4 1) Y 5 4x ]1 5 4x 4 x Table Numerial results of utting plane method for example 4 Number of utting planes Big-M relaxation Separation problem solution Separation CPU (s) DICOPT// major iterations CPU (s)

14 44 Y (x 1 4) (x 1 4) 51 Y 4x 1 ]1 5 4x 1 x 1 05 : 0 Y x 1 51:(x 0 ) Y x 58(x 0 ) 4x 0 ]1 5 4x 1 x 0 x 05 0 :4 Y 8 x 15 51: log(x 19 ) Y 8 x 15 ]10:x 19 4x 19 ]1 5 4x 15 x :5 Y 9 x 1 x 18 ]5 Y 9 x 1 5 log(x 18 1) 4x 18 ]1 5 4x 1 x () : Logi proposition: Y Y ( ffly ffly ) Y 4 Y 5 [Y 4 Y 5 Y 4 [ Y 5 [ Y [Y Y 8 Y [Y Y 9 Y [Y Y 8 Y 9 Y 9 [Y A. Vehietti et al. / Computers and Chemial Engineering (00) 4/448 [Y 4 Y 5 ][[Y Y 8 Y 9 ] [Y 4 ffly 5 ][[Y ffly 8 ] [Y 8 ffly 9 ] [Y ffly 9 ] 05x j 59 j 1;...; ; 05 k ; Y k ftrue; falseg; k1;...; 9 The optimal solution is Z //19., Y,Y,Y,Y,Y 9 /true and x/(1.15,0,1.5,.,0, 1.15,1.5,1.15,0,.,1.5,0,.8,9,0,.8,0.5,1,0,., 4.0,4.98,0,0,0,0). The big-m relaxation of Eq. () yields a lower bound of /.4. The onvex hull relaxation of problem Eq. () yields a lower bound of /09. Table shows the results of utting plane method applied to big-m relaxation of Eq. (). As more utting planes are added, the lower bound of big-m relaxation inreases and the objetive value of the separation problem dereases. After adding ten utting planes, the lower bound signifiantly improved (/19.). Table 4 shows the branh and bound searh results when utting Table Numerial results of utting plane method for example 5 Number of utting planes planes are added before starting the branh and bound searh. First, the big-m MINLP problem is solved with branh and bound searh. Nineteen nodes are searhed and the optimal solution /19. is found. Seondly, four utting planes are added to big-m MINLP problem at the root node of branh and bound tree. Note that the relaxation value, whih is the objetive value at the root node, is /9.5 and 1 nodes are searhed to find the optimal solution. The derease in the number of searh nodes is due to the tighter relaxation value. When eight utting planes are added, the relaxation value is /1.4 and only seven nodes are searhed. For omparison, the onvex hull relaxation of Eq. () is solved and the number of nodes is seven, whih is same as in the ase of eight utting planes. The CPU time for eah ase is also shown in Table 4 and less CPU time is spent with fewer number of nodes. The CPU time for generating eight utting planes is about s. This example learly shows that the utting planes an tighten the relaxation and thus redue the number of searh nodes in branh and bound method. Although the example presented is rather small, the proposed utting plane method should be promising for solving larger problems. This will be the subjet of our future work. 8. Conlusions Big-M relaxation solution 0 / / / / / / /4.4.9 / / / / Convex hull relaxation / Separation problem solution The purpose of this paper has been to analyze the different alternatives of modeling the disrete hoies as disjuntions or as mixed-integer (0 /1) inequalities, in order to provide guidelines on this deision. The resulting model an orrespond to one of the three formulations: mixed-integer onstraints (PA), disjuntive onstraints (GDP) or hybrid (PH). For the analysis, we onsidered three different possible relaxations of a disjuntive set, the onvex hull, the big-m relaxation and

15 A. Vehietti et al. / Computers and Chemial Engineering (00) 4/ Table 4 Comparisons of branh and bound searh results for example 5 Model Big-M MINLP no utting planes Big-M MINLP /4 utting planes Big-M MINLP /8 utting planes Convex hull relaxation Relaxation value /.4 /9.5 /1.4 /09.0 Optimal solution /19. /19. /19. /19. Number of nodes 19 1 CPU s.9.5 a 1.5 a 1. a CPU time for generating utting planes is not inluded. the Beaumont surrogate. The analysis was performed mainly on the first two sine the big-m formulation is widely used. Although it was proved that the onvex hull relaxation yields a tighter relaxation than the traditional 0 /1 big-m relaxation, there are several ases when the big-m relaxation an ompete with the onvex hull relaxation. As a general rule, the big-m model is ompetitive when good bounds an be provided for the variables, and for large problems where it is important to keep the number of equations and variables as small as possible. For onvex improper disjuntion both the onvex hull and the big-m model give the same relaxation in the x spae, but this may not be true in the x /y spae as was demonstrated with examples. For proper disjuntions where the feasible regions have some intersetion, the objetive funtion plays an important role, if the minimizer of the objetive funtion is inside the feasible region of the disjuntive set, both the big-m and the onvex hull relaxation may yield the same relaxation value. Otherwise the onvex hull should be generally better, but the big-m onstraints with appropriate bounds an be ompetitive. For proper disjuntions with an empty intersetion on the feasible regions (disjoint terms) the onvex hull is generally better than the big-m relaxation. Although these onlusions are not general, we believe they help to provide some insight in the modeling of disrete/ontinuous optimization problems. Finally, to address the problem of formulating tight models without generating the expliit equations of the onvex hull, a utting plane algorithm has been proposed. A number of examples have been presented to illustrate the various ideas in this paper as well as the utting plane method. Aknowledgements The authors would like to aknowledge finanial support from the NSF Grants ACI and INT Appendix A: Property of relaxations Property 1. Let R BM be the feasible set of big-m relaxation of a given disjuntive set projeted onto the x spae. Let R CH be the feasible set of onvex hull relaxation projeted onto the x spae. Let R B be the feasible set of the Beaumont surrogate that is defined in the x spae. Then R CH /R BM /R B. Proof. First onsider R BM /R B. For the linear ase, Beaumont (1990) proved that R BM /R B. Therefore, R BM /R B holds. For the nonlinear ase, we onsider one disjuntion for simpliity. Given a nonlinear disjuntive set: F [h i (x)50] x R n (A1) where h i (x) are assumed to be onvex bounded funtions. The big-m relaxation of Eq. (A1) is as follows: h i (x)5m i (1y i ); (A) X y i 1 (A) 05y i 51; (A4) where M i /max{h i (x)jx L 5/x 5/x U }. Let R F BM (x, y) be the feasible set defined by Eqs. (A), (A) and (A4). The Beaumont surrogate of Eq. (A1) is given by: X h i (x) 5N 1 (A5) M i where N/jDj and M i are assumed to be same as in Eq. (A). Let R F B (x, y) be the feasible set defined by Eqs. (A5) and (A4). Sine Eq. (A5) is given by a linear ombination of Eqs. (A) and (A), any feasible point (x*, y*) /R F BM (x, y) also satisfies Eqs. (A5) and (A4). Hene, (x*, y*) /R F B (x, y). Therefore, R F BM (x, y) / R F B (x, y). Sine R BM and R B are the projetion of R F BM (x, y) and R F B (x, y) onto the x spae, it follows that: R BM R B (A) Seondly, we onsider R CH /R BM for linear and nonlinear ase. The onvex hull relaxation of Eq. (A1) is given by:

16 448 A. Vehietti et al. / Computers and Chemial Engineering (00) 4/448 x X v i 0 x; v i R n (A) vi y i h i 50; (A8) y i X y i 1 (A9) 05y i 51; (A10) 05v i 5v U i y i ; (A11) Let R CH (x, y, n) be the feasible set defined by Eqs. (A), (A8), (A9), (A10) and (A11). Consider any feasible point (x*, y*, n*) /R F CH(x, y, n). From Eq. (A), there exist m i suh that: y i m i v i ; (A1) h i (m i )50; (A1) Sine h i (x) are onvex funtions, for any l /D: X h l (x)h l y i m i 5 X y i h i (m i ) (A14) For h l (m l )5/0 and h l (m i ) i "l 5/M l, it follows from Eqs. (A14), (A9) and (A10): h l (x)5 X y i M l M l (1y l ) (A15) ;i"l Eq. (A15) is idential to Eq. (A) in the big-m relaxation for l /D. Hene, any feasible point (x*, y*, n*) /R F CH (x, y, n) has a orresponding feasible point (x*, y*) whih satisfies Eqs. (A), (A) and (A4). Therefore, (x*, y*) /R F BM (x, y). Sine R BM and R CH are the projetion of R F BM (x, y) and R F CH (x, y, n) onto the x spae, it follows that: R CH R BM (A1) From Eqs. (A) and (A1), R CH /R BM /R B. This ompletes the proof. Referenes Balas, E. (199). Disjuntive programming. Disrete optimizations II, annals of disrete mathematis, vol. 5. Amsterdam: North Holland. Balas, E. (1985). Disjuntive Programming and a hierarhy of relaxations for disrete optimization problems. SIAM Journal of Algebri and Disrete Mathematis (), 4/485. Balas, E. (1988). On the onvex hull of the union of ertain polyhedra. Operations Researh Letters, 9/84. Beaumont, N. (1990). An algorithm for disjuntive programs. European Journal of Operational Researh 48, /1. Bokmayr, A., & Kasper, T. (1998). Branh-and-infer: a unifying framework for integer and finite domain onstraint programming. INFORMS Journal on Computing 10 (), 8/00. Grossmann, I. E., & Kravanja, Z. (199). Mixed-integer nonlinear programming: a survey of algorithms and appliations. In L. T. Biegler, T. F. Coleman, A. R. Conn & F. N. Santosa (Eds.), Largesale optimization with appliations, part II: optimal design and ontrol (pp. /100). Springer. Jakson, J. R., & Grossmann, I. E. (00). High-level optimization model for the retrofit planning of proess networks. Industrial Engineering and Chemial Researh 41 (1), /0. Lee, S., & Grossmann, I. E. (000). New algorithms for nonlinear generalized disjuntive programming. Computers and Chemial Engineering 4 (9/10), 15/141. Lee, S., & Grossmann, I. E. (001). A global optimization algorithm for nononvex generalized disjuntive programming and appliations to proess systems. Computers and Chemial Engineering 5 (11 /1), 15/19. Nemhauser, G. L., & Wolsey, L. A. (1988). Integer and ombinatorial optimization. Wiley. Raman, R., & Grossmann, I. E. (1994). Modeling and omputational tehniques for logi based integer programming. Computers and Chemial Engineering 18 (), 5/58. Stubbs, R., & Mehrotra, S. (1999). A branh-and-ut method for 0/1 mixed onvex programming. Mathematial Programming 8 (), 515/5. Türkay, M., & Grossmann, I. E. (199). Logi-based algorithms for the optimal synthesis of proess networks. Computers and Chemial Engineering 0 (8), 959/98. Vehietti, A., & Grossmann, I. E. (1999). LOGMIP: a disjuntive0/1 nonlinear optimizer for proess system models. Computers and Chemial Engineering, 555/55. Vehietti, A., & Grossmann, I. E. (000). Modeling issues and implementation of language for disjuntive programming. Computers and Chemial Engineering 4 (9 /10), 14/155. Yeomans, H., & Grossmann, I. E. (1999). Nonlinear disjuntive programming models for the synthesis of heat integrated distillation sequenes. Computers and Chemial Engineering (9), 115/ 1151.