P = absolute pressure, cm Hg at 0°C T = absolute temperature, O K Literature Cited Barrer. R. M . , Edge, A. V. J.. Proc. Roy. SOC.,Ser. A, 300, 1 (1967). Barrer, R. M., Ruzicka. D. J., Trans. Faraday Soc., 58, 2262 (1962). Barrer, R. M., Stuart, W. I., R o c . Roy. Soc., Ser. A, 249, 464 (1959). Deaton, W. M., Frost, E. M., Jr., U. S. Bur. Mines, Monogr., 8 (1946). de Forcrand, R . , Compt. Rend., 134, 835 (1902a). de Forcrand. R., Compt. Rend.. 135, 959 (1902b).
King, M. B.. “Phase Equilibrium in Mixtures,” pp 160-1 73, Pergamon. New York, N. Y., 1961. Marshall, D. R., Saito, S., Kobayashi, R.,A.l.Ch.E. J., 10, 202 (1964). Stackleberg, M. yon, Muller, H. R., Z.Elekfrochem., 58, 25 (1954). van der Waals, J. H.. Platteeuw, J. C., Advan. Chem. Phys., 2, 1 (1969). Viilard, P., Compt. Rend., 106, 1602 (1888).
Received for review October 5 , 1973 Accepted March 5,1974 This research was sponsored by the Department of the Interior, Bureau of Mines, under Grant No. G0111343.
A Stronger Version of the Discrete Minimum Principle Arthur W. Westerberg* and George Stephanopoulos Department of Chemical Engineering, University of Florida, Gainesville, Florida 3267 7
The discrete form of Pontryagin’s Minimum Principle proposed by a number of authors has been shown by .others in the past to be fallacious; only a weak result can be obtained. Due to the mathematical character of the objective function and the stage transformation equations, only a small class of chemical engineering problems have been solved by the strong discrete minimum principle. This paper presents a method to overcome the previous shortcomings of the strong principle. An algorithmic procedure is developed which uses this new version. Numerical examples are provided to clarify the approach and demonstrate its usefulness.
1. Introduction
Pontryagin’s minimum principle (Pontryagin, et al., 1962) is a well-known method to solve a wide class of extremal problems associated with given initial conditions. A discrete analog of the minimum principle, where the differential equations are substituted by difference equations, is not valid in general but only in certain almost trivial cases. Rozonoer (1959) first pointed out this fact. Katz (1962) and Fan and Wang (1964) later on developed a discrete minimum principle which was shown to be fallacious by Horn and Jackson (1965a), by means of simple counterexamples. As was pointed out by Horn and Jackson (1965a,b) and lucidly presented by Denn (1969), the failure of a strong minimum principle lies in the fact that we cannot deduce the nature of the stationary values of the Hamiltonian from a consideration of first-order variations only. Inclusion of the second-order terms does not help to draw a general conclusion about the nature of the stationary points in advance. A weak minimum principle which relates the solution of the problem to a stationary point of the Hamiltonian exists and is valid (Horn, 1961; Jackson, 1964). In the case of control systems described by differential equations, time, by its evolution on a continuum, has a “convexifying” effect (Halkin, 1966) which does not make necessary the addition of some convexity assumptions to the specification of the problem. Thus a strong minimum principle can be applied for these problems, requiring the minimization of the Hamiltonian even in the case that a continuous problem is solved by discretizing it with respect to the time and using a strong discrete minimum principle. For discrete, staged systems described by difference equations, the evolution of the system does not have any “convexifying” effect and, in order to obtain a minimum principle, we must add some convexity assumptions to the problem specification or reformulate the problem in
an equivalent form which possesses inherently the convexity assumptions. This present work belongs to the second class. In the present work we propose to show a strong version of the minimum principle which relates the solution of the problem to a minimum point, rather than a stationary point, of the Hamiltonian. This is attained through the use of the Hestenes’ method of multipliers, a technique used effectively by the authors (Stephanopoulos and Westerberg, 1973a,b) to resolve the dual gaps of the twolevel optimization method. This method turns a stationary point of the Hamiltonian into a minimum point; thus minimum seeking algorithms can be used. In section 2, the Lagrangian formulation of the problem is given and from this the discrete minimum principle and the two-level optimization procedure are developed. In section 3, the main theorems which constitute the basis of the proposed stronger discrete minimum principle are presented. In section 4, an algorithmic procedure is developed, and in section 5, numerical examples are solved. Finally, in section 6, the various algorithms are compared and some relationships between the discrete minimum principle and the two-level optimization procedure are pointed out. 2. Statement of the Problem, a n d Its Lagrangian Formulation As a basis for the description of the minimum principle, the two-level optimization approach, their success and failure, and the development of a strong minimum principle, consider the following sequential unconstrained problem. (Constraints and recycles do not change the following results (Westerberg, 1973), and we want to keep the presentation here as simple as possible.) s
min F
=C41(x,,u,) 1-1
Ind. Eng. Chern., Fundam., Vol. 13, No. 3, 1974
(PI) 231
subject to
(9)
x,+1= f,{x,,u,)
x,
xl0(given)
For every i = 2, . . . , N the vector valued function f , ( x , , u,) is given and satisfies the following conditions: (a) the function f, is defined for all x , and u,;(b) for every u, the function f , ( x , , u,)is twice continuously differentiable with respect to x , : and (c) the f,(x,, u,) and all its first and second partial derivatives are uniformly bounded. These conditions correspond to the usual "smoothness" assumptions. The Lagrangian function for this problem (PI)is given by
N
~ ( ~ , ( x 0-uA JI T X 1
+ X I + l ~ f , ( X , , u ' ) l+
XITXl0
Thus,' in order to solve problem P1 by the discrete minimum principle, we require that each stage Hamiltonian be at a stationary point. Consider the point ( a , &, . , . , R K ; iil, ii2, . . . , i i ~and ) the variations 6u1, 6u2, . . . , 6 u around ~ the previous point with respect to the controls, such that Ibutj( 5 e , where the second index j denotes the j t h element of the control vector u,. 6 x 1 will be taken as zero. The variational equations corresponding to the connection constraints of the problem P1, with up to the second-order terms included, are
- AN+lTX'v+l=
i-1
.v
C ~ , ( x , , u , h , A + 1+) X I T X l 0 - ~ , + I T X , + l
(1)
1-1
The solution to the problem (Pl) is a stationary point of the Lagrangian function L. The necessary conditions for a stationary point of the Lagrangian are
_ a L_ - ah -ax,
A,
ax,
-aL - x,
ax,
a f+, + -"A,
(i = 1, ...)N )
=0
For the considered sequential system, the solution of the above system with respect to the 6xL's, i = 1, . . . , N, is straightforward and yields the following general formula in terms of the variations in the controls only
(2)
8x1
- fI-l(xL,u,)= 0
(i
= 1, ..., N
)
(4)
From eq 2 we have the defining equations for the multipliers
with the natural boundary condition
(6) Equation 4 simply necessitates the satisfaction of the connection constraints. The Lagrangian approach constitutes a unifying and general presentation of the necessary conditions which must be satisfied at the solution of a problem. For the solution of the necessary conditions different strategies have been developed. In a tutorial presentation (Westerberg, 1973) the relationship of the different strategies to solve problem PI, with the Lagrangian approach, is established and it is shown that methods such as sensitivity analysis, discrete minimum principle, and the two-level optimization method are simply different techniques to solve the same necessary conditions, eq 1,2, and 3. 2.1. Discrete Minimum Principle. Let us define the stage Hamiltonian H,as h + l
=0
H , = 4t(x,,ut)+ A,+lTf,(x,,u,) ( i and the overall Hamiltonian by 'V
H
=C
= 1,..., N )
Then =
'Y
'V
1-1
1-1
C H , - ~ X , x-I,-A I T ~ , '
and the necessary conditions (2), (3) yield
232
Substituting into the last expression for 6 ~ ~ (i' s= 2, . . . , N ) with their equals from eq 10 and, noting that the stage Hamiltonians H,(i = 1, . . . , N ) ,are given by eq 7 , we find
(7)
H , + XlTx;
,=1
L
The variation in the objective function F caused by the variations in the controls is given by
Ind. Eng. Chern., Fundam., Vol. 13, No. 3, 1974
Unlike the continuous case (Denn, 1969; Halkin, 1966) there is no general way in which the last two terms in eq 11 can be made to vanish. Therefore, the considered variations in the controls may well produce 6F < 0, or 6F > 0, or 6F = 0. Thus it is evident from the above that a strong minimum principle, which requires that the solution to the problem (PI) minimizes the stage Hamiltonians ITL,( I = 1, . . . , N)is not generally available (Denn, 1969, Halk-
in, 1966; Horn and Jackson, 1965b). A weaker form of the discrete minimum principle can be used and requires that the solution to the problem (PI) makes the stage Hamiltonians stationary. The examples presented by Horn and Jackson, which counter the strong minimum principle, are such that the stage Hamiltonians do not possess a minimum stationary point whereas the problem itself does. However, there exist special cases where the strong minimum principle can be applied (Denn, 1969; Horn and Jackson, 1965b). At this point a further clarification is required. The strong discrete minimum principle fails for physically staged steady-state systems. Halkin (1966) has shown that it always succeeds for discrete time systems, obtained as an approximation to the continuous time systems, since the time increment At can become arbitrarily small and make the last two terms in eq 11 vanish. The success or the failure of the strong discrete minimum principle can also be explained in terms of convexity (or directional convexity) or lack of it for the sets of the reachable states of the system (Halkin, 1966; Holtzman and Halkin, 1966). Since this constitutes an important characteristic and the basis for the development of a strong version of the discrete minimum principle, we will return to it later in this work. The two-level optimization procedure is an infeasible decomposition strategy of a Lagrangian nature which solves problem P1. The similarity between the Lagrangian and the weak discrete minimum principle is a well known fact and in a recent note (Schock and Luus, 1972) the similarity between the two-level optimization procedure and the discrete minimum principle has been pointed out. Further insight in the relationship between the last two methods will be given later in this paper, and a stronger version of the discrete minimum principle will be developed, based on a method to overcome the shortcomings of the two-level optimization procedure developed by the authors (Stephanopoulos and Westerberg, 1973a,b). Therefore, we feel it is necessary to make a short presentation of the two-level optimization procedure, its shortcomings, and their resolution before proceeding to the development of a stronger form of the discrete minimum principle. 2.2. Two-Level Optimization Procedure. Dual Gaps and Their Resolution. Consider again problem P1. The Lagrangian for this problem is given by eq 1, which, with the natural boundary condition (6), yields .Y
L
=
m x t & , A J l + J + AITxlo i=1
Next we define problem P2 as
and finally the general Lagrange problem P3
*
+-
Figure I. Problem with dual gap and its resolution.
solves problem P3. The two-level optimization procedure fails if no saddle point exists for the Lagrangian. To provide a further insight in the two-level optimization procedure and its shortcomings, consider the following class of problems N
min F = c4;(x,,ul)
P4)
1-1
subject to x,+1
(i
- fl(x,,u,> = 2,+1 XI
= 1, ...j
h ?
- xp = 21.
Let w(z) =
min x,,ut,l-l”
,*v
{Fix,,, - f, = z [ + ~i, = 1, ... N ; 3
x1
- x:
=
zll
It has been shown by Everett (1963) that i f f , , riL ( i = 1, . . . , N ) minimize the Lagrangian of a problem from the class of problems (P4), for given multipliers, then there exist a supporting hyperplane of the set R = {(zo, z ) l z o 2 w ( z ) ) a t the point (w(z), z ) , where w ( z ) = F ( i 1 , . . . , i , v , 61. . . , E N ) and i,+l- fi(i,, LzL) = z,+1 ( i = 1, . . . , N)and - x l o = 21. If there are no multipliers X,I . . . , X,V such that there exists a supporting hyperplane of the set R a t the point ( u ; ( O ) , O ) , the two-level method fails to give the solution to problem P1. Figure 1 shows the failure of the two-level approach since there is no such supporting hyperplane for the set R a t the point (w(O),O). To overcome this shortcoming of the two-level method, the authors have developed a strategy (Stephanopoulos and Westerberg, 1973a,b) which makes use of the Hestenes method of multipliers (Hestenes, 1969). Let us define a new problem, which we shall call problem P5, as follows.
max h(A,,..., A,) A,.
.A,
The two-level method involves solving P3 by first guessing the multipliers X,I . . . , X,V, solving the subproblem minimizations in (P2), and then adjusting the multipliers until h(X1, . . . , A,) is maximized. The important question concerning this procedure relates to the existence of a saddle point for the Lagrange function of the problem. A number of theorems which provide the theoretical basis and give some answers to the saddle point existence question can be found in the work of Lasdon (1970). Here we can simply mention that if the point (21, . . . , fs,&, . . . , a s ; X,I . . . , X.V) is a saddle point of the Lagrangian then the point 21,. . . , i ~ v61, , . . . , Cis) solves problem P1 and (XI, . . . , X,V)
K ( x , - x:)T(xl
-
XI0)
(P5)
subject to x,+1
- fl= 0 - x; x1
(i = 1, ..’, N ) =
0
K>O The Lagrangian of this problem is
Ind. Eng. Chern., Fundarn., Vol. 13, No. 3,1974
233
Theorem 2. Let u = ii be a local isolated minimum of F with all the connection constraints satisfied. Then there exists a real valued parameter KO,such that
Now
where
(i.e., the matrix is positive definite) for K > KO. Proof. In Theorem 1it was shown that For permissible variations Aq ( i e . , satisfying the connection constraints), AL* > 0 always, and for nonpermissible variations, where one might have AL < 0, there is a K > 0 such that AL* > 0, and, a t a stationary point, L* is always at minimum. The introduction of the quadratic terms causes w ( z ) to move upward and, under certain conditions, the existence of a supporting hyperplane at the point ( u ( O ) , O ) is guaranteed for finite K . The introduction of the quadratic terms destroys the separability of the system since when expanded they produce cross-produl). However, one can recover a comuct terms putational separability in trade for an iteration by expanding this term in a Taylor series up to the first-order terms.
3. Strong Discrete Minimum Principle In this section we will develop a stronger version of the discrete minimum principle using Hestenes' method of multipliers and following a procedure similar to the one that was developed by the authors to overcome the dual gaps of the two-level optimization procedure. As a basis for the presentation we will use problem P1. Consider now the augmented problem P5 N
Y
K(x, - X ; ) ~ ( X ,
- xlo)
(P5)
subject to = fl(x,,u,) x1 =
with K
(i
= 1, .*.,
N)
(given)
> 0. The Hamiltonian for this problem is given by N
H* = F*
+ ~ X , + , T f l ( x , , u ,+) XlTxIo i==i
In the following theorems we will establish some important properties of H*. Theorem 1. If (GI,. . . , Zis) is a stationary point of the Hamiltonian H for the problem P1, then it is also a stationary point for the Hamiltonian H* of the augmented problem P5 for any real value of the parameter K . Proof. From the equations defining H and H* we take =
H
+ K C. ,_[ x , + ,- f J T X [xi+, - f , ] + K(xi -
The solution of problem P1 is a stationary point of the Hamiltonian H . The nature of the stationary point for the discrete case cannot be predetermined and thus a2H/au2 5 or > 0. Assuming that we have a nonsingular problem
and we conclude that
for a large enough K so that the second term which is always positive prevails over any negativity of the first term. Q.E.D. The second theorem implies that any stationary point of the Hamiltonian H can be made a minimum point by choosing the parameter K large enough. Theorems 1 and 2 establish the following result. For a large enough K , a local solution to problem P5 which is also a local solution to problem P1, minimizes the Hamiltonian H* of the augmented problem P5. The above result requires that we minimize the overall problem Hamiltonian, H*. The available weak minimum principle permits us to solve the overall problem by solving one problem per stage per iteration since H decomposes into a sum of stage Hamiltonians. Unfortunately H* does not decompose, as we shall now see. Let us see now how we can simplify the above result and relate the solution of problem P1 or P5 to stage Hamiltonians. Consider the penalty term in the objective function of problem P5 &XI+,
- flP[.,+l - f , ] =
f+l
b
H*
is independent of the constant K: therefore, it will be valid and for K > KO. Let us consider next the matrix a2H*/au2
i=1
-
~10) (E)
Differentiation with respect to the controls yields
Evaluating the derivatives at the point (GI,. . . , z i , ~ ) , since
andx,+l - f l = Ofori = 1 , .. . , N , we find aH* -_ - aau-H - 0 au 234
Ind. Eng. Chem., Fundam., Vol. 13,No. 3, 1974
We note that each member of the summation includes separable terms, e.g., x 1 + l T , x [ + 1 , and f L T f l , and nonseparable terms such as the cross-product x 1 + l T f L .The following develops a strategy to decompose computationally the problem of minimizing H*. This approach is motivated by the authors' work on solving nonconvex problems using the two-level approach. Expand the cross-product term in a Taylor series and consider the following linear approximation around the point X l + l , ^ f r = ^fr(xL,GI)
Then the Hamiltonian of the augmented problem takes the following form H*for H*
BN.
N
N
H*
Z4i(xi,ui) + K C [ X , + + ~f~l ~Xf+,i +2;,+:7z ~ I-1
%,+,Tf,
Step 1: Assume a value for the penalty constant K. Step 2: Assume values for the control variables B1, . . . ,
-
1-1
- b1+,T7J+ K ( x , - x,")'(x,
Step 3: Using the values Bl, . . , , Bx, solve the state equations
- x1O.l +
x, = x; x,+, = f,(x,,u,)
(i = 1, .*.,N ) Let 22, , . . , i~ be the found
fonvard and find x p , . . . , XN. values (i, = x10, given constant). Step 4: Using the values B1, . . . , 2.v and solve the adjoint equations
wherelo= x l 0 and we have defined a stage Hamiltonian XN+1
Let us now establish the following result which also constitutes the basis of the proposed algorithm (given later in this paper). Theorem 3. If the point LiT = ( r l l T , a p r , . . . , ii.yT) minimizes the overall Hamiltonian H*, then each stage Hamiltonian Hl*, i = 1, . . . , N,resulting after the Taylor series linear approximation of the cross-product terms is minimized with respect to the corresponding control variable u Lat the point i i L . Proof. At a solution point we require feasibility, i.e .
aH* aH* ~A 0=(i = 1, .'., A')
au, au, and therefore a stationary point of H* with respect to uLis a stationary point of H L * with respect to u L .The Hessian matrix of the second derivatives of H* must be positive definite a t the point Li, i . e . 0
. . . , iA\,
=0
backwards. Let il,. . . , X N + ~be the found values. Step 5: Formulate the Hamiltonian H* of the augmented problem P5 and expand the cross-product terms around the point (il,. . . , i h ; Bl, . . . , 12s).Formulate the stage Hamiltonian HI*, i = 1, . . . , N.Minimize all H1*, i = 1, . . . , N and find optimal values for the controls, say a,, i = 1, . . . , N.If the minimization procedure fails to produce a minimum for a t least one HL*increase K and go to step 2. Step 6: If laLJ- CLJ1 < t for a l l j and for i = 1, . . . , NO' denotes the j t h component of the vector u , ) stop, the solution is found, otherwise assume new values for the controls putting
- -
new u, = u , (This requirement is evident from the algorithm to be presented in the next section.) From eq 13 it is clear then that
i1,
(i = l, .*.> N)
and go back to step 3. The effective use of the above algorithm depends largely on the value of the constant K. Very small values of K will not produce the necessary convexity of the stage Hamiltonians, while a very large K will mask the objective function and will make the algorithm insensitive to the descent direction of the objective function. Miele, et al. (1972), have suggested a method for choosing a proper value for the constant K in the context of the Hestenes method of multipliers. As will be discussed in section 6, the two methods are related and this method for choosing K should be satisfactory here. 5 . Numerical Examples
Example 1. Consider the following two-stage example, Figure 2 , taken from Horn and Jackson (1965a). This example is the conterexample which demonstrated the fallacy of the strong minimum principle and is described by
A=
1
lo
min x12 8'8'
subject to
must be positive definite. Thus, all the submatrices on the diagonal must be positive definite, i . e . , d2H*/au12, a2H*/au22, . . . , a2H*/aun2 must be positive definite, but a2H*/aUL2 = a2HL*/auL2;therefore all the G H * / d u L 2 (i = 1, . . . , N ) are positive definite. Theorem 3 implies that the minimization of the H* with respect to the control variables can be replaced by the problems min H,*
i = 1, ..., N
= X1o
- 281 - 1
,(e1)2
x; = x20
xl* = x , l
+
+
(x21)2
e1
+
(e212
xZ2= arbitrary
xio =
x20
1
I
ui
4. The Algorithm a n d Computational Characteristics
The theorems of section 3 imply the following algorithm which constitutes a stronger version of the now available discrete minimum principle.
The solution to this problem is 81 = O2 = 0, which gives the smallest value of x12. It, however, does not minimize the Hamiltonian for stage 1 since this Hamiltonian has only one stationary point, which is a maximum. Let us now apply the procedure of the strong discrete miniInd. Eng. Chern., Fundam., Vol. 13, No. 3,1974
235
xl* = X''(1
+ x 2 ' ) - T(1
x,2 = 4x:
x: = 1 where u1 and u2 are to be chosen subject to
Figure 2. Two-dimensional, two-stage example
0
mum principle developed in the previous sections. Thus we have
[~ 1 -0 28' -
;(01,?]'
2K(xI1)[xIo- 28' K(;'l)[ x :
- 2;'
+ K[x,O + &I2 -
k(@)2]- 2K(g2'Xx; + 8'1 + - i(iii)2]+ K(;,')[x?O+ 8'1 t
-
1
- 2x118' - ,x1l(8')*
X,'x,'
+ X,'x,o + X,'8'
and
H,*= xi1 - ( x ~ ' 4-) ~(6*)*4- K(xIi)2+ K(x,')' 2K(x,:i[ xlo
-
K(;ll)[xl@ - 28'
=
1+ 2 K ( x l 1) 2K x: - 28 -
[,
A,' = 2 ( x , ' )
~ 2 ' ) ~
~ K ( x , ') *16Kx,'x2'
$@)*I 1
1 K H2* = - x l L - -A '(u')* X,'U* + K x 1 2 -;i-(~')~ + 2 ' K x 1 2 ( u 2 ) K(x12)'4 K ( u 2 )- 2Kx2'u2 - p(u* - u') -I-
+
+ [-4KxI' + 2Kx2' + 4 K ( x l 1 )- 2K(X,') - 2X1' + A,']
+
=0
and dH2*/8O2 = 2(02)= 0 which gives O2 = 0. 1. Put K = 1. 2. Assume el = 1 and = 0. 3. Find x l l = 1 - 2 - l/2 = - 3h and xzl = 1 1 = 2. 4. Also Xll = 1 and X 2 l = 2(2) = 4. 5 . Minimize HI* and H2* which give: Sl = 0.947 and = 0 ; - @ = /0.053/ > t = 0.001. Go back to step 2. 2a. Assume e l = 0.947 and O2 = 0. 3a. Find x l l = - 1.343 and xzl = 1.947. 4a. Find = 1 and X 2 l = 3.894. 5a. Minimize HI* and H2* and nfi!: 8l = 0.9> O2 = 0. Go back to step 2a and assume O1 = 0.9, O 2 A= 0. Continue in the same way until IO1 - ell < t and IO2 < c . This finally leads, in seven iterations, to the solution O1 = O2 = 0 which is the solution of the problem. Note that if K = 0.1 the nonconvexity of the first connection constraint with respect to O1 does not disappear. The stronger version of the minimum principle developed in this paper does not succeed in giving the solution. Example 2. Consider the following example taken from Denn (1969), which is also a counterexample to Katz' strong minimum principle and is described by
e2
+
e2
Is1
u'.uj
subject to
236
+
+ x;) - 21( U ' ) *
+ U'
Ind. Eng. C h e m . , Fundam., Vol. 13, No. 3,1974
+
+- +
-
K x l ' i l l ( l x l ' ) - 2K;12x^1'(l x ? ' ) , . K2x1'(l x ~ ' ) ( u ~ )Kx,'(l + x^21)(u2)
+
+
4K^X1'X^Z2- 8Kx^1'~2'- 2K3CJ'Z; ~ K ; , ' x ? ~- 4Kx1'^u2+ 8 K x l ' u 2
+
+ 2K;2'u2 - 4K2,'u'
The necessary conditions to be satisfied are
aH1* - &I
-X1'(u1)
+
XI'
+
jl'
=
0
+ A t + K X ; + K - 2KxzL + p2 - K21'(I + F2') + 8KSl1 - 4Kx^,' = 0
- - K ( u ' ) ~dH,* au 2
A:(u2)
The adjoint variables satisfy the following equations
+ 2K(xI1)(1+ x21)2+ 32K(x1')- 16Kr ' - 2K;,'(1 + x ~ ' )+
A,' = A I L ( l + x I 1 ) 4Az2
K ( l x2'Xu2) - 8 K x ; - 8 K u 2 A?' = A12x11~ 2K(x,')'(1 + x 2 ' ) 8K(x2')16Kx1' - 2K;C1'x1' - Kxl'Z2 4KX; - 4Ku2
+ +
X12
minimize - x:
- 2x2'
^
and
2K(x1') - Xl']8'
= 4x1'
^
^
+ 6K(01)*+ [10K - 2Kx; +
X?l
A
K- + + ~ x ' L ( 1+ x z ' ) ( u 2 )K x l ' ( l + x 2 l ) ( i 2 )+ 4Kx^,';C2I - 8 K x l ' q 2 - 2Kx2'x,' + 4Kx2'x2 - 4KX^,'i2 + 8Kx11L2+ 2Kx,'u2 - 4Kx;u2
+ 2 K ( x 2 ' )- 2K[x; + 8'1
xll = x,o(1
- p1u* + K x i 2 x l ' ( l+ x , l ) -
2KxI2xl1(1 x 2 ' )
1 aH -K(81)3
88'
This is a constrained problem but it can be handled equally as well as the unconstrained problem by the developed method. The inequalities will be handled through Kuhn-Tucker multipliers. The above problem has a solution at the point ril = 1 and 02 = u*, but the Hamiltonian for the first stage does not have a minimum and thus Katz' strong minimum principle cannot be used. In fact the Hamiltonian of the first stage possesses one stationary point which is a maximum. Let us now apply the algorithm developed in the present work. Thus we take
+ (A; + jll)u' + X12x11(1+ x 2 ' ) + 4XI.2~1'- 2A2'~2' + K(x,')'(l + + 16K(xI')*+
The first necessary conditions yield
*
u* 5 u',u*
1 Hl* = - ~ X ~ ( U ' ) '
-
- 2 3 - +(iiiiZ] - 2 K ( x , ' X x 2 ' t 8'1 t 1-$8')']+ K(x,')[x,O+ B'] - X:x11-A;x21
with
A,'
t uz
xl@= -114
I Hl* = K
- 2x,'
U2)*
=
+ K + K W ) - 2Kxl'(l + i2') = 2K(x;) - 2Ku2 + 4K;C2'
2x12
A,' Note that in the above equations the variables p1 and g2 are Kuhn-Tucker multipliers through which we handle the inequalities u1 1 u* and u2 2 u*. For p 1 and p2 we have p1, p2 5 0.
0
KSMP
KTLM
-
Figure 3. Relationship of penalty constant K required for three Lagrangian based optimization algorithms.
Starting with K = 10 and initial assumptions u1 = 2 and u2 = 2, we apply the same algorithmic procedure as presented in the previous example, and after five iterations we find the solution zil = 1 and zi2 = u * , which is the solution for the problem. 6. Discussion and Conclusions
As mentioned before, the failure of the strong discrete minimum principle was caused by the fact that the stationary points of the stage Hamiltonians with respect to the controls are not always minima points. The method proposed in this paper turns every stationary point of the Hamiltonian into a minimum point for a large enough K . Thus we can find the solution of the original problem at a local minimum point of the stage Hamiltonians. This constitutes a stronger result than that currently available, where we must search for a stationary point of the stage Hamiltonians. The method used in this paper to establish a stronger version of the discrete minimum principle parallels in many respects the method used earlier by the authors to resolve the dual gaps in the two-level optimization procedure. The source of the shortcomings for both the methods (minimum principle and two-level method) is the nonconvexity of the objective function and/or the stage transformation equations. The two-level optimization method fails if either the objective function or transformation equations or both are nonconvex with respect to the control and/or the state variables. For the resolution of the dual gaps of the two-level method, the authors used the same penalty term multiplied by a positive constant K . It was required that K be large enough so that the stage sub-Lagrangians become lo-
cally convex with respect to the control and state variables. Since, in the present work, we have required that the K be large enough to turn the stage Hamiltonians convex with respect to the controls only, we conclude that the K required by the strengthened form of the minimum principle to solve a nonconvex problem is at most as large as the K required by the two-level optimization method to resolve dual gaps. Figure 3 compares the values of K required to solve a nonconvex problem for the three methods discussed in this paper, namely, weak discrete minimum principle, the strengthened form of discrete minimum principle developed in this work, and the two-level optimization method. Note, KTI.M2 K s ~ pI 0, where KTLMis the least value of K required for the two-level method and K h ~ isp the least value of K for the minimum principle developed here. Thus it should be clear that, although the methods are related, they do not have equivalent shortcomings; it is possible to find problems which can be solved by the weak minimum principle or even the method developed here, and not by the two-level method. However, given a K 2 K~LM all, three methods succeed.
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