Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979
343
Optimum Design of Multipurpose Chemical Plants Ignacio E. Grossmann and Roger W. H. Sargent" Department of Chemical Engineering, Imperial College, London, S W7 2BY, England
The optimum design of chemical plants that are used in different ways at different times is considered. The subset of plants with piecewise constant operating conditions is formulated as a nonlinear program. The design of a multicrude pipeline and the design of sequential multiproduct batch plants are given as examples. The latter implies the solution of a mixed-integer problem for which an efficient method for obtaining good suboptimal solutions is proposed. The results show that with the approach developed in the paper, it is possible to design chemical plants that have a specified flexibility and that are economically optimum.
Introduction There are often situations in which a given plant or set of equipment is used in different ways at different times. The applications range from the sequential batch processing of entirely different products, as occurs in the fine chemicals industry, to the continual realignment of conditions in continuous processing to accommodate changes in feedstocks or market requirements, as occurs in oil refineries. An essential feature of a multipurpose plant is that it must have enough flexibility to meet specifications which vary in the time domain. Assuming that the specifications are given, the optimum design of a multipurpose plant will correspond to an optimum control problem, as the state of the process must be represented by a system of differential equations. However, in a large number of cases, the dynamics of the process can be neglected and then it is possible to formulate the optimum design problem as a nonlinear program, as will be shown in this paper. The design of a multicrude pipeline and of a multiproduct batch plant are given as examples. The mathematical properties of the latter are analyzed in detail because of the complications caused by the presence of integer constraints on a subset of variables. Theory It is assumed that the plant will be subject to piecewise constant operating conditions in n successive time periods as shown in Figure 1. The dynamic effect of switching from one state to the other will be small in the time domain if the length of each period t k , k = 1,2, ..., n, is sufficiently long. The performance of the plant can then be represented in each period by the system of nonlinear equations
0 k = 1,2, ..., n (1) where u,uk are vectors of design variables associated with the sizing of the units and x k is the vector of state and control variables. The vector uk represents a set of variables defining the portion of the plant utilized in period k, while u is the remaining set of design variables which have fixed values throughout. Since the plant must have enough capacity to meet the requirements in every time period, a vector of limiting design variables, 0, must be chosen when designing the plant, such that fk(u,uk,xk,tk)
=
k = 1, 2, ..., n (2) A similar formulation is obtained for various other types of variable conditions; for example, if the operating pressure varies from period to period the plant must be designed to withstand the maximum operating pressure. 0 L vk
0019-7882/79/1118-0343$01.00/0
There will be other design or operating constraints in every period hLkIhk(u,uk,xk,tk) I huh k = 1, 2, ..., n (3) as well as other general constraints SL Is ( u , O , u 1 , u 2 ,
..., v n , d , x 2 , ..., X " , t l , t 2 , ..., t") 5 su
(4)
As the cost of the plant, C, is given by C = f " ( U , O , d , U 2 , ..., c n , d , x 2 , ...,x",tl,t2, ..., t") ( 5 ) the optimum design problem will then consist in choosing u , 0, v k , x k , t k ,k = 1, 2, ..., n, to minimize C subject to the constraints (l),(2), (3), and (4). Assuming that f", s, f k , hk,k = 1, 2, ..., n, are continuously differentiable functions, and that all the variables involved in them are continuous, the problem to be solved will correspond to a nonlinear program (NLP) for which efficient algorithms are available. There are, however, some special features in this design problem which are worth mentioning. In general, C will increase strictly monotonically with the O,, i = 1,2, ..., m, and hence the bounds in (2) will be attained for each element for some period k unless constraints (4) prevent this. This follows from the Kuhn-Tucker conditions for 0, for fixed values of u,v k , x k , t k , k = 1, 2, ..., n, assuming that constraints (4)are inactive aC " i = 1, 2, ..., m =p i k = 0; X i k 2 0; 0; - L'ik 2 0 (7) 1,2, ..., m; k = 1, 2, ...,n Since for each i = 1, 2, ..., m, aC/aOi > 0, it is clear from (6) that Xik > 0 for at least one h. This in turn implies from ( 7 ) that a t least one constraint must be active for each component of 0, and hence Oi = max { u t k ) i = 1, 2, ..., ml (8) k = 1 , 2 , ...,n Xik(Oi
- )0;
i
=
It should be noted that it is the formulation of (2) that should be used as the derivatives in (8) with respect to u; are discontinuous, implying possible failure of the Kuhn-Tucker conditions and consequent numerical difficulties. I t must also be pointed out that care has to be exercised when formulating design problems of multipurpose plants, as the vector u reduces effectively the degrees of freedom of (1) when compared with the single-period problem. Additional degrees of freedom can always be introduced, either by specifying outlet conditions as inequalities, or by modifying the flowsheet with the addition of bypasses, stand-by equipment, or other control devices. In the next sections examples of multipurpose plants will be presented. 1979 American Chemical Society
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Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979 state
1
-
I
Figure 3. Sequential multiproduct batch plant. Figure 1. State of the plant with piecewise constant operating conditions.
where P, is the minimum pressure which is specified for the outlet. Eliminating 19), (lo), and (12), the optimization problem is to choose bHP, Dj,j = 1, 2, 3, F , i = 1, 2, ...,n, in order to minimize the annual cost 3
C = Cc&jDj ,i=l-
+cBbHP + 144(Pi - PIi)
Figure 2. Multicrude pipeline. i=l
Table I. Data for the Multicrude Pipeline Problema crude 1 2 3 4
Q,'
Q,'
Q,'
Pi
pi
ei
700 380 570 320
300 150 200 120
400 230 370 200
48 50 51.1 53.3
1.4 1.8 2.5 2.9
2500 1800 2000 1700
subject to the constraints
a P , i = 50;P,,'=
lOO;P,,'= l O O ; q i = 0 . 6 ( i = 1 , 2 , = 4000; L, = 3000. C, = 0.05; CD = 0.1; C B = 250; y g = 0.86. D I L = 18; D I U = 32; D,L = D,L = 8 ; D,' = D,' = 16.
..., 4). L, = 8000; L ,
i
Design of a Multicrude Pipeline The horizontal pipeline shown in Figure 2 must be designed to transport n crudes with different flows and properties. The allocation of time of the crudes is given and minimum outlet pressures are specified. The problem is to determine the diameters of the three sections ( u ) and the required horsepower of the pump (0) in order to minimize the total annual cost. Assuming turbulent flow and taking the Fanning friction factor as f = 0.04/Reo,l6,the required horsepower for each crude, bHPi, is given by Qlip1 144(Pi - PIi) K1i/D14.84 bHP = 330007; Pi i =1, 2, ..., n ~
[
+
where
(
K,' = 0.1076675(Qj')''84Lj
(10)
$16
.
I
i = 1, 2, ..., n;j = 1, 2, 3 and Q is the flow rate (ft3/min),L the length (ft),D the diameter (in.), p the density (lb/ft3), p the viscosity (cP), 3 the efficiency of the pump, and P the pressure (psia). The installed horsepower (bHP)of the pump must be such that bHP I bHPi i = 1, 2, ..., n (11) The outlet pressures are given by
i = 1, 2, ..., n;j = 2, 3 and they must satisfy the inequalities i = 1, 2, ..., n; j = 2, 3 P! L P,'
(13)
= 1, 2,
D k IDj IDju
..., n j = 1,2,3
where
B, is the allocation of time in one year for crude i(h),c, is the cost of electricity ($/kWh), DF, DIUare lower and upper bounds of the diameters (in.); cD, cB, y are cost parameters. It is convenient, as suggested by Murtagh (1972), to use the transformed variable d, = l/D?& so that all the constraints are linear. The problem shown in Table I has been solved using the variable metric projection (VMP) method of Sargent and Murtagh (1973). The results are shown in Table I1 and they define a pipeline which is both flexible and economical. Design of a Multiproduct B a t c h P l a n t Formulation of t h e Problem. The design problem formulated by Sparrow et al. (1975) will be considered in this work. As shown in Figure 3, the plant consists of a sequence of M batch processing stages which are used to manufacture N different products. In each stage j , N, units operate independently in parallel and all the units within a given stage j have the same size VI (1). The time required to process one batch of product i in stage j is given by TI, = t,, + c,,B,YJ i = 1, 2, ..., N, j = 1, 2 , ..., M (16)
where t, 2 0, c, 2 0 and y, are constants, and B, (kg) is the batch size for product i. The required unit size, VI,,for processing product i in stage j is given by i = 1, 2, ..., N, j = 1, 2, ..., M (17) V I ,= B,S,,
Table 11. Results of Multicrude Pipeline Problem starting point solution
bfiP 450 309.1
Dl 28 24.6
D, 14 15.6
D3
14 16
P' 110 105.6
PZ
P3
P4
C
110 102.2
110 105.6
110 101.9
149 833 130 555
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979 M
-
2
Product
Stage 1
I
1
i
Stoge 2
-
(
,=1
-Unit 1
I
I
1
C= ~a,V,~~N,
++ 5i+! 4 L + ;i i 1 ; I
I
l
/
1
-Unt
2
l
I
1
345
1
1
1
-Unt 2 -Unit
1
stage 3
*
(24)
subject to the constraints (16)-(ZO), (22), and (23). It is clear from Figure 4 that the number of batches (n,) treated in a given arbitrary period H need not be integer-valued, but the number of units (N,) in each stage must of course be positive integers so the problem is a mixed integer nonlinear program (MINLP). This class of problem is very difficult to solve, and no general method yet exists for its efficient solution. Sparrow et al. (1975) assumed that the sizes of the units were not available in continuous ranges and that the difference in standard sizes was significant. The problem was then formulated as a discrete one and a branch and bound technique was used to solve it coupled with a heuristic estimate. Their method becomes less efficient when the cycling times depend on the batch size, that is, when clI is nonzero in (17). In fact, in their paper all the problems were solved with c,, = 0 in (17). With the present MINLP formulation the dependency of the cycling times on the batch sizes is taken into account without any difficulty. The Relaxed Subproblem. The solution of the MINLP may be obtained by solving a sequence of relaxed subproblems in which all variables are treated as continuous. By elimination of some of the intermediate variables, this relaxed problem may be formulated as follows. Choose the B,, TL1,V I ,N, to minimize M
C = C~.,V,~JN, ,=1
subject to 1=1
.tl,
VI 1 B,S,,; TL,N, 1 t , + c,, B,’J i = l , 2,..., N j = l , 2 ,..., M VILIVI 5 V,”; N I LI N, 5 N,” j = 1, 2 , ..., M B, 1 0 I = 1, 2, ..., N It will be seen later that the bounds NIL,N,” are systematically adjusted in the sequence of problems. Initially the lower bound N I Lis set to unity for all j , since there must clearly be a t least one unit in each stage. It is also clear that all the constraints in (25) can be satisfied, with strict inequality in all cases, by choosing the B, sufficiently small and the N,” sufficiently large. Since the objective function and constraint functions are continuous, and the values of the variables are bounded, the objective function attains its minimum over the feasible set of variable values. Thus problem (25) has a t least one optimal solution, and it is of interest to establish whether it has more than one. When 0 < 6, < 1 , O < y,< 1,which is usually the case, problem (25) has a non-convex objective function and a non-convex set of feasible solutions, and under these circumstances the possibility of multiple local minima cannot normally be excluded. However the problem can be reformulated as follows. Choose the B,, TLl, V I ,N, to minimize M
C = EcY;V,’JN; j= 1
subject to
i = 1, 2 , ..., N ; j = 1, 2, ..., M
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Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979
j = 1, 2 , ..., M Vj>O; N,>O j = 1 , 2,..., M
B, > 0; TL,> 0 i = 1, 2 , ..., N It can then be seen that it corresponds to a geometric program, as studied by Duffin et al. (19671, of the general form minimize fo(t) subject to k = 1,2 , ..., p f k ( t )5 1 t,>O j = l , 2,..., m
where
t=
{tl,k?,
.e.,
tml
(27)
The t, are the variables, the c, and a, are constants with all c, > 0, i = 1, 2, ...,n, and the J(k),k = 0, 1, 2 , ...,p , are given index sets with P
u J ( h ) = {1,2,...,n\ k=O The Appendix gives a theorem which provides sufficient conditions for such a program to have a unique KuhnTucker point, and hence a unique minimizer. It has already been shown that problem (26) has a t least one minimizer and that there is a point satisfying the constraints with strict inequality (i.e., the feasible set has a non-empty interior). It is also easy to see that its exponent matrix [a,] has linearly independent columns, so the theorem is applicable to problem (261, showing that problem (25) has a unique Kuhn-Tucker point which is the global minimizer. It has often been argued (cf. Duffin et al., p 97, 1967) that rather than solve the primal problem (27) directly, it is preferable to solve the linearly constrained geometric dual program
The optimal primal variables are then given by the system of equations n
ciIItja” = 6[u(6’)
iEJ(O)
j=1
iEJ(k) (29) where k ranges for X,(S’) > 0 and 6‘ maximizes u. For problem (26) the dual implies solving a NLP with 3MN + 5M + N variables 6isubject to 2M + 2N + 1 linear equality constraints. Also (26) is a canonical problem because no posynomial term can approach zero without causing at least one of the other terms to approach infinity. Therefore no dual gap exists and the solution of the primal can be obtained through the dual (Duffin et al., 1967). On = 6,’/Xk(6’)
the other hand, one can solve the primal problem directly as a NLP, which was the approach adopted in this work. In the case of (25) 2M + 2N variables are involved subject to 2MN + 1 constraints, MN + 1 of them nonlinear. Although it is not very clear what the best alternative is, several disadvantages can be pointed out when solving the dual problem. The number of variables can become rather large, in fact 3MN + 3M - N variables more than in the primal problem. Because not all of the constraints are necessarily active at the solution, numerical difficulties can be expected unless slack variables are used in the form of penalty (Peterson, 1973). Finally, a feasible point in the dual space will in general be infeasible in the primal space, except for the optimum solution. This complicates the numerical solution as it has to be very accurate, a rather difficult task for a large number of variables. The disadvantage with the primal formulation is that the number of constraints is rather large and a subset of them are nonlinear. It is worth noting that although nonlinear programming algorithms usually find a Kuhn-Tucker point rather than a minimizer, this causes no difficulty in the present case since the unique KuhnTucker point is in fact the unique global minimizer. This property will be exploited for solving the MINLP as shown in the next section. Solution of the MINLP. The MINLP can be solved using a branch and bound technique coupled with the NLP algorithm. In general the steps to follow are the following. 1. Solve problem ( 2 5 ) . If all NJRare integer, stop. 2. A feasible solution to the mixed problem, which is a good estimate of the optimum, can be obtained by taking as a starting point the solution in step 1 and solving ( 2 5 ) adding the following constraints M
C ( ( N ] R- )NJ)([NJR] - N J )=
J=1
(30) [N;] 5 N , I( N J R ) j = 1, 2, ..., M where [XI= largest integer 5 x and ( x ) = smallest integer I x.
3. A rooted tree is enumerated assigning to each branch one inequality constraint on a particular N, until feasible solutions to the MINLP are obtained. At any node a NLP subproblem is defined in which the inequality constraints, corresponding to the path connecting the node with the root, are added to the relaxation problem. The solution of this subproblem represents a lower bound on the optimum cost of the mixed integer problem. This bound increases monotonically along any path in the tree, as the uniqueness of the solution of the subproblems is guaranteed and the feasible space is reduced a t each node. Therefore, nodes which exceed the current estimate need not be expanded further, whereas terminal nodes that do not exceed the estimate are taken as the new current estimate. In the first step integer solutions for N,, j = 1, 2, ...,M , should not be infrequent as these variables tend to lie a t their lower bounds because of the form of (24). Thus in the branch-and-bound strategy a sequence of NLP must be solved. Even when many possibilities can be rejected, the computing requirements can become excessive. Therefore it is important to have a good initial estimate, such as the one obtained in the second step, which will probably be the global optimum in many cases. As the optimum cost of the relaxed problem, CR, is known, and it is a valid lower bound, the maximum tolerance e for the suboptimal cost determined in step 2, CE, is given by CE - CR e=(31) CR
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979
Table 111. Examples of t h e Multiproduct Batch Plant Problem l a M = 3 N = 2 4=6000 2 3 4 8 20 8 S= [ 4 6 31 t = [16 4 41 c = O Q = CY;
=
40 000
20 000 250 8; = 0.6 1 < N ; < 3 250 < Vi < 2500 (j= 1 : 2,3) Problem I b . Sarre as l a except 4 10 4
]
t=
c = [“‘5 2 2 0.45 0.67 0.18 yj = 0.3333 ( j ‘ = 1,2,3) Problem IC. Same as l a except
Problem 2
4.2 4.3 2.2 6.4
8 2
4 5
4.4 7.1 6.5 2.0
aj =
250
Bj =
c= 0 0.6 1 < Nj < 3 250 < Vj < 2500 ( j = 1,2,..., 4)
Problem 3
c=
=
L2.34 10.3 5.7 1.15 9.86 5.28 5.95 7.01 7.0 3.96 6.01 5.13 r . 2 0.24 0.4 0.5 0.15 0.35 0.7 0.34 0.5 0.85 0.3
250
Bj
5.981 1.2 1.081
0.66
= 0.6 1 < Nj Q 3 250 < ( j = 1,2,..., 4)
Table IV. Results of the Examples of Table I11
Vi
[ ]
Vj < 10 000
When the tolerance 6 is small the suboptimal solution should suffice for practical purposes and the tedious calculations of step 3 can be avoided. Numerical Results. The examples shown in Table I11 have been solved with the VMP method and the results are shown in Table IV. In all the cases the solutions were obtained successfully with the computing time ranging from 20 to 80 s CDC 6400 time. In the examples where noninteger values were obtained in the first step, only the suboptimal solution was determined solving the second step. It can be seen that t is very small, making the application of the branch and bound procedure hardly worth the effort. Conclusions The problem of optimum design of multipurpose processes has been formulated as a NLP. The design of the multicrude pipeline has shown the usefulness of the approach to design flexible systems which are commercially attractive. It should be noted that the NLP corresponding to the multicrude pipeline also corresponds to a geometric program with posynomial terms satisfying the conditions of the theorem in the Appendix, and hence the solution obtained is a global optimum. The design of multiproduct batch plants was formulated as a MINLP. The proposed method of solution deter-
347
Bi
Ni
TLi
c
Problem l a
480 720 960
240 120
357.7 494.5 644.3
161.1 14.37 82.42 10.01
1
20 16
38 499.8
1 1
Problem l b
1 1 1
30 968
Problem IC. Noninteger Solution
1250 1875 2500
625 312.5
9.959 8.79
1.8201 106 346 2.008 1 Problem IC. Integer Solution; E = 0.0039 1200 600 10 2 106 769 1800 300 8 2 1 2400 Problem 2 682.4 162.48 8 1 61 300 1010.11 208.45 6 1 1008.1 155.1 12 1 1042.2 1 Problem 3. Noninteger Solution 8536.1 1030.9 6.29 1.503 349 471 10000.0 1236.1 6.45 1.943 5533.9 970.8 6.17 2.002 7626.7 1 Problem 3. Integer Solution; E = 0.0122 9247.9 1030.1 6.29 2 353 744 10000 1234.9 6.44 2 5533.9 970.9 6.17 2 1 7619.4
mined either the exact solution or a very good suboptimal solution so that it was possible to circumvent a tedious branch and bound procedure. The general technique developed for multipurpose plants can obviously also be applied to design plants which will meet a variety of possible (rather than actual) situations, and hence provides a rational basis for designing flexible plants when the specifications or design data are subject to uncertainties. This approach is developed in a related paper (Grossmann and Sargent, 1978). Appendix Posynomial Programs. Consider the posynomial program min vO(t)ltES,J where S , = (tit > 0;f k ( t )I 1 h = 1, 2, ..., p ) (Al)
t = {t,,tz, ..., t,) ER” f k ( t )=
c
C,ImIt)a” i E J ( k ) 1=1
h = 0, 1, 2, ..., p
the a, and c, are constants, with c, > 0, i = 1, 2, ...,n, and the J ( h ) ,h = 0, 1, 2, ..., p , are given index sets with P
U J ( h ) = {l,2,...,n ]
k =O
Theorem. If the columns of the matrix [a,,], iEJ(O), j = 1, 2, ..., m, are linearly independent, the feasible set S, has a non-empty interior, and P ( t )attains its minimum on S,, then program (Al) has a unique Kuht-Tucker point in S,, which is the global minimizer of P ( t ) on S,. Proof. Define the vectors x = (xl, x 2 , ...,x n } and z = (zl, z 2 , ..., z,) by the relations z, = In t, j = 1, 2, ..., m
348
Ind. Eng. Chem. Process Des. Dev., Vol. 18, No. 2, 1979 m X,
= Ca,z, ]=1
i = 1, 2, ..., n
(A21
m
C c, exp(Za,,z,) iEJ(k) J=1 h = 0 , 1 , 2,..., p
=
C c, exp(x,)
minl@O(z) (z~S,l
(A31
where
k
I 1
= 1, 2,
..., p )
In particular it follows that S , has a non-empty interior, and since P ( t )attains its minimum on S, and the mapping t z is strictly monotone, @O(z) attains its minimum on
-
sz.
Now if fES, is a Kuhn-Tucker point of problem ( A l ) , there exist scalars Xk 1 0, h = 1, 2, ..., p , such that D
D
where
But from (A2) we have
and since t > 0 for tES, it is clear that there is a one-to-one correspondence between the Kuhn-Tucker points of (Al) and those of (A3). Since the columns of the matrix [a,], iEJ(O),are linearly independent, the linear mapping from z to its columnspace is one-to-one, and since exp(x) is a strictly convex function of x and c, > 0, i = 1,2 , .,., n, it follows that the &z), k = 0,1, 2 , ...,p , are convex functions of z with $O(z) strictly convex (see Lemmas 1 and 2 below). Thus the set S, is convex and has a non-empty interior, so problem (A3) has a unique Kuhn-Tucker point which is the global minimizer of ~ O ( Z )on S , (see Lemma 3 below). From above it follows that problem (AI) has a unique Kuhn-Tucker point which is the global minimizer of p ( t ) on S,. QED. The above proof uses the following properties of convex functions and convex programs. Lemma 1. If C,(z), i = 1, 2, ..., n, are strictly convex functions of z E R m , and c,, i = 1, 2, ..., n, are positive scalars, then Cnl,lc,C,(z) is a strictly convex function of 6.
i=l
i=l
?Ci{XCi(Z1) + (1 - X)Ci(Z2)}> ~CiCi(XZ,+
i=l
XI22
i=l
rEJ(k)
Clearly the functions f k ( t ) and @ ( z ) are continuously differentiable. For t > 0 the variables zI are real-valued, and problem ( A l ) maps into the problem
s, = (+#k(z)
n
XCCiCi(Z1) + (1 - X)CCiCi(Z2) =
with the corresponding functions f k ( t )= 4 k ( z )=
Proof. Suppose z1 # z2 and 0 < X < 1. Then n
QED Lemma 2. If x = L ( z ) is a one-to-one linear mapping from zER" to x E X C R " and C(x) is a strictly convex function of x , then $42) = C ( L ( z ) )is a strictly convex function of z . Proof. Since x = L ( z ) is' one-to-one, z1 # 2 2 implies L(zl) # L ( Z ~and ) , from the strict convexity of C(x) it follows that, for all X E (0,1) XC(L(Z1)) + (1 - X)C(L(Z2)) > C(XL(Z1) + (1 - X)L(z,)) But since x = L ( z ) is a linear mapping XL(Z1) + (1 - X)L(z2) = L(X.21 + (1 - X)ZJ and the result follows. $ED. Note that the results of Lemmas 1 and 2 also hold if strict convexity is everywhere related to convexity, and then x = L ( z ) need not be one-to-one. Lemma 3. Suppose that ~ O ( X ) is a strictly convex, continuously differentiable function defined on a convex set SCR", defined by
s = { x l @ k ( x ) I1
h = 1, 2, ..., p }
where the 4 k ( x ) are continuously differentiable functions. Then if S has a non-empty interior and @O(x)attains its minimum on S , there is a unique Kuhn-Tucker point in S which is the global minimizer of 4O(x) on S. Proof. It follows from the Kuhn-Tucker theorem (cf. Duffin et al., p 66, 1967) that i E S is a local minimizer of qbo(x) on S if, and only if, it is a Kuhn-Tucker point (cf. eq A4). There is a t least one local minimum of 4O(x) in S since @O(x)attains its minimum on S. Suppose, contrary to the assertion of the theorem, that xl and x2 are distinct local minimizers, with 40(x1) 1. 4O(xZ). Then since S is convex, x = Axl + (1 - X)x2ES for all AE(0,l) and from the strict convexity of 4O(x) we have @O(x) < @O(xl), contrary to the hypothesis that x1 is a local minimizer. Thus there can only be one local minimizer, and hence only one Kuhn-Tucker point, which is the global minimizer of ~ O ( X )on S . QED. Literature Cited Duffin, R. J., Peterson, E. L., Zener, C., "Geometric Programming Theory and Application", Wiley, New York, N.Y., 1967. Grossman, I . E., Sargent, R. W. H., AIChE J . , 24(6), 1021 (1978). Murtagh, B. A., Chem. Eng. Sci., 27, 1131 (1972). Peterson, E. L., "Optimization and Design", M . Avriel, M. J. Rijkaert, D. J. Wikle, Ed., p 228 Prentice-Hail, Englewood Cliffs, N.J., 1973. Sargent R. W. H.. Murtagh, B. A,, Math. Prog., 4, 245 (1973). Sparrow, R . , Forder, G., Rippin, D., 1nd. Eng. Chem. Process Des. Dev., 14, 197 (1975).
Received f o r review June 20, 1977 Accepted August 29, 1978
