ISSN 0012�5008, Doklady Chemistry, 2010, Vol. 435, Part 2, pp. 343–346. © Pleiades Publishing, Ltd., 2010.Original Russian Text © T.V. Lapteva,N.N. Ziyatdinov, G.M. Ostrovskii, I.M. Zaitsev, 2010, published in Doklady Akademii Nauk, 2010, Vol. 435, No. 4, pp. 497–500.

343

Problems of design, intensification, and control ofchemical engineering systems are generally solvedunder uncertainty of the initial physicochemical andtechnological information. Two formulations are typi�cally used in solving problems of optimal design ofchemical engineering systems, namely, a two�stepoptimization problem, in which the control variablescan be varied at the functioning stage, depending onthe state of the chemical engineering system; and aone�step optimization problem, in which the controlvariables are constant at the functioning stage. Thereare problems under hard constraints, in which all con�straints must necessarily be satisfied in all the states ofthe chemical engineering system, and problems undersoft constraints, in which constraints are met either onthe average, or with a certain probability.

Methods for solving two�step optimization prob�lems under hard constraints and one�step optimiza�tion problems under soft (probabilistic) constraints arewell�developed. A two�step optimization problemunder hard constraints has the form [1]

(1)

where

(2)

and f *(d, θ) is given by solving the problem

(3)

Eθ f* d θ,( )[ ],d D∈

min

Eθ f* d θ,( )[ ] f* d θ,( )ρ θ( ) θd

T

∫=

f * d θ,( ) f d z θ, ,( ),z H∈

min=

gj d z θ, ,( ) 0, j≤ 1 2 … m., , ,=

Here, d is the nd�vector of design variables; z is thenz�vector of control variables; θ is the nθ�vector ofuncertain parameters; f (d, z, θ) is an optimization cri�terion; z (H = {d, z: hl(d, z) ≤ 0}, l = 1, 2, …, p) is the

feasible domain of d and z; and T = {θi: ≤ θi ≤ ,

i = 1, 2, …, p} is the uncertainty region.

We assume that all the uncertain parameters θi areindependent random quantities characterized by nor�mal distribution N1(E[θi]; σi). A great contribution tothe development of methods for solving two�step opti�mization problems under hard constraints was madeby Grossmann and colleagues in the 1980s [1–3]. Fur�ther, these studies were continued (see, e.g., [4, 5]).

A one�step optimization problem under probabi�listic constraints has the form [6]

(4)

(5)

where Eθ{ f (d, z, θ)} is the mathematical expecta�tion of the function f (d, z, θ) and Pr{gj(x, θ) ≤ 0} isthe probabilistic measure of the region Ωj = {θ: gj(x,θ) ≤ 0}.

The main difficulty in solving problems of type (4)is the necessity of calculating the multidimensionalintegrals Pr{gj(d, z, θ) ≤ 0} and E[ f (d, z, θ)]. Usingstandard quadrature formulas [7] for this purposeinvolves very laborious procedures. This difficulty canbe overcome in three ways. The first way is to obtainspecial quadrature formulas. Three alternativequadrature formulas were derived [4]. A specialquadrature formula was also obtained [8], which sig�nificantly reduces the calculation time for the casewhere the parameters θ are normally distributed ran�dom quantities. The second way is to use Monte Carlomethods [6]. However, they also require rather labori�ous calculations. The third way is to transform proba�bilistic constraints into deterministic ones [9, 10].

θiL θi

U

E f d z θ, ,( )[ ],d z,

min

Pr gj d z θ, ,( ) 0≤{ } αj, j≥ 1 2 … m,, , ,=

CHEMICALTECHNOLOGY

Two�Step Problems of Optimizationof Chemical Engineering Processes

T. V. Laptevab, N. N. Ziyatdinovb, G. M. Ostrovskiia, and I. M. Zaitsevb

Presented by Academician Yu.D. Tret’yakov August 5, 2010

Received April 15, 2010

DOI: 10.1134/S0012500810120086

a Karpov Institute of Physical Chemistry, ul. Vorontsovo pole 10, Moscow, 103064 Russia

b Kazan State Technological University, ul. Karla Marksa 68, Kazan, 420015 Russia

344

DOKLADY CHEMISTRY Vol. 435 Part 2 2010

LAPTEVA et al.

Methods for solving two�step optimization problemsunder soft constraints are much less developed. Diffi�culties in formulating and solving such problems werediscussed previously [11]. A penalty�method�basedapproach to solving such problems was proposed [12],which unfortunately has the serious drawback that itdoes not guarantee a given probability of satisfaction ofconstraints. In this context, we here give a formulationof a two�step optimization problem under probabilis�tic constraints and propose an approximate methodfor solving it.

Let us initially consider two�step optimizationproblem under probabilistic constraints (1). Substitu�tion of the expression for Eθ[ f *(d, θ)] from Eq. (2)and the expression for f *(d, θ) from Eq. (3) intoEq. (1) gives

(6)

Let us assume that problem (3) has a solution ateach point of the region T. Then, because the solutionof problem (3) at a certain θ value does not affect thesolutions of problem (3) at the other θ values, we canchange the order of integration and minimization onthe right�hand side of expression (6):

(7)

In this formulation, the search variables z(θ) arefunctions of the parameters θ. The conditions deter�mining the domain H have the form hl(d, z(θ)) ≤ 0, l =1, 2, …, p. We will assume that these constraints arehard. Thus, the conditions determining the domain Hwill have the form

Let us now rewrite problem (7), explicitly takinginto account the conditions that the variables d andz(θ) belong to the domain H:

(8)

(9)

where

According to Eq. (9), the constraints should be met atall the points of the region T; therefore, they should be

f1

= f d z θ, ,( )/ g d z θ, ,( )θ T∈

max 0≤{ }ρ θ( )z H∈

min θ.d

T

∫d D∈

min

f1 f d z θ( ) θ, ,( )ρ θ( ) θ,d

T

∫d z θ( ) H∈,

min=

gj d z θ( ) θ, ,( ) 0, j≤ 1 2 … m, θ∀, , ,= T.∈

hl d z θ( ),( ) 0, l≤ 1 2 … p, θ∀, , , T.∈=

f1 f d z θ( ) θ, ,( )ρ θ( ) θ,d

T

∫d z θ( ),

min=

gj d z θ( ) θ, ,( ) 0, j≤ 1 2 … m p,+, , ,=

θ∀ T,∈

gj d z θ( ) θ, ,( ) hj m– d z θ( ),( ),≡

j m 1+ m 2+ … m p.+, , ,=

met with probability 1. Therefore, problem (8) can berewritten as

(10)

(11)

Let us consider the case of soft constraints, whereeach of the constraints should be satisfied with proba�bility αj (j is the number of a constraint). Since theconstraints in the problem under hard constraints arerepresented in form (11), it is natural to write the prob�lem under soft constraints in the form

(12)

(13)

At αj = 1, j = 1, 2, …, m + p, problem (12) reduces toproblem (10). Let us compare this problem with one�step optimization problem under soft constraints (4).These formulations differ in the fact that, in the one�step optimization problem, the target function andconstraints involve the usual search variables d and z,whereas, in two�step optimization problem (12), thesearch variables z are functions of the parameters θ.This reflects the fact that, at the functioning stage ofchemical engineering system, the control variables zmay be tuned, depending on the state of the chemicalengineering system (on the values of the parametersθ). Let us reformulate problem (12). Consider an indi�vidual constraint,

(14)

Let us assume that we have constructed, by one way oranother, the region such that the probability that

the point θ is in this region is more than or equal to αj,

(15)

Note that condition (15) determines not a singleregion but a set of regions. Let us call this set . Con�

sider the constraint

(16)

where is one of the regions of the set . It is clear

that this condition is equivalent to the condition

Thus, if constraint (16) is met, then the constraintgj(d, z(θ), θ) ≤ 0 is satisfied at each point of the region

. Consequently, if the probability that the point θ is

in is more than or equal to αj and condition (16) is

valid, then the probability of satisfaction of the ine�

f1 f d z θ( ) θ, ,( )ρ θ( ) θ,d

T

∫d z θ( ),

min=

Pr gj d z θ( ) θ, ,( ) 0≤{ } 1, j 1 2 … m p.+, , ,= =

f1 f d z θ( ) θ, ,( )ρ θ( ) θ,d

T

∫d z θ( ),

min=

Pr gj d z θ( ) θ, ,( ) 0≤{ } αj, j≥ 1 2 … m p.+, , ,=

Pr gj d z θ( ) θ, ,( ) 0≤{ } αj.≥

Tαj

Pr θ Tαj∈{ } αj.≥

Tαj

gj d z θ( ) θ, ,( )θ Tαj∈

max 0,≤

TαjTαj

gj d z θ( ) θ, ,( ) 0, θ∀ Tαj.∈≤

Tαj

Tαj

DOKLADY CHEMISTRY Vol. 435 Part 2 2010

TWO�STEP PROBLEMS 345

quality gj(d, z(θ), θ) ≤ 0 is more than or equal to αj.

Hence, for a fixed region , constraints (15) and

(16) are equivalent to constraint (14). Let us choose acertain region belonging to the set and replace

each constraint (13) in problem (12) by constraints(15) and (16). Then, we have

(17)

Because we chose an arbitrary region , the result

given by problem (17) is naturally worse than the resultgiven by problem (12). Therefore, problem (17) givesonly an upper estimate of the solution of problem (12).To find the exact solution of problem (12), it is neces�sary to seek not only for the optimal values of thesearch variables d and the vector functions z(θ), butalso for the optimal shapes and locations of the regions

(j = 1, 2, …, m). Hence, problem (12) can be

rewritten as

(18)

(19)

(20)

Unfortunately, the problem in form (18) is difficultto solve because it is difficult to arrange the search forthe optimal shape of the region . Therefore, fur�

ther, we will narrow the class of feasible shapes of theregion and will seek for regions in the shape of

multidimensional rectangles

(21)

Thus, the search for the optimal shape of regions

reduces to the search for the optimal values and

, i.e., for the optimal sizes and locations of multi�dimensional rectangles. Let us now transform con�straint (20). Because the parameter θi is an indepen�dent, normally distributed random quantity, the prob�

ability that the parameter θi is within the range ≤

θi ≤ is

(22)

Tαj

TαjTαj

E f d z θ( ) θ, ,( )[ ]d z θ( ),

min ,

Pr θ Tαj∈{ } αj, j≥ 1 2 … m p,+, , ,=

gj d z θ( ) θ, ,( )θ Tαj∈

max 0, j≤ 1 2 … m p.+, , ,=

Tαj

Tαj

f E f d z θ( ) θ, ,( )[ ],d z θ( ) Tαj

Tαj∈, ,

min=

gj d z θ( ) θ, ,( )θ Tαj∈

max 0, j≤ 1 2 … m p,+, , ,=

Pr θ Tαj∈{ } αj, j≥ 1 2 … m., , ,=

Tαj

TαjTαj

Tαjθi : θi

L j, θi θiU j,

, i 1 2 … n, , ,=≤ ≤{ }.=

Tαj

θiL j,

θiU j,

θiL j,

θiU j,

ρ θi( ) θid

θiL j,

θiU j,

∫ Φ θ̃iU j,

( ) Φ θ̃iL j,

( ),–=

where Φ(η) is the standard normal distribution and

Since all the parameters θi are independent, the

probability that the point θ is in is equal to the

product of the probabilities that the parameters θi are

within the ranges ≤ θi ≤ , i = 1, 2, …, n. Hence,condition (20) in this case takes the form

(23)

Taking into account the equivalence of conditions (23)and (20), let us transform problem (18) to the form

(24)

(25)

(26)

Let us seek for an approximate solution of problem(24) under the assumption that the control variablesz(θ) are approximated by linear functions (θ) of theparameters θ:

(27)

where bi is a vector of dimension dimz. Now, thesearch for the optimal functions z(θ) reduces to thesearch for the optimal values of the coefficients bi.Thus, the new search variables will be the componentsof the vectors bi. Since the uncertainty region is oftensmall, such an approximation may be good enough.Substitution of (θ) from Eq. (27) into problem (24)gives

(28)

(29)

where F(d, b, θ) ≡ f (d, b0 + b1θ1 + ··· , θ), Gj(d,

b, θ) ≡ gj(d, b0 + b1θ1 + ··· , θ).

Problem (28) is a generalization of a semi�infiniteprogramming problem in which the feasibility regions

θ̃iL j,

θiL j,

E θi[ ]–( ) σi( ) 1–,=

θ̃iU j,

θiU j,

E θi[ ]–( ) σi( ) 1–.=

Tαj

θiL j, θi

U j,

Φ θ̃1U j,

( ) Φ θ̃1L j,

( )–[ ] Φ θ̃2U j,

( ) Φ θ̃2L j,

( )–[ ]···

··· Φ θ̃nU j,

( ) Φ θ̃nL j,

( )–[ ] αj.≥

f E f d z θ( ) θ, ,( )[ ],d z θ( ) θi

L j,θi

U j,, , ,

min=

gj d z θ( ) θ, ,( )θ Tαj∈

max 0, j≤ 1 2 … m p,+, , ,=

Φ θ̃1U j,

( ) Φ θ̃1L j,

( )–[ ] Φ θ̃2U j,

( ) Φ θ̃2L j,

( )–[ ]···

··· Φ θ̃nU j,

( ) Φ θ̃nL j,

( )–[ ] αj, j≥ 1 2 … m p.+, , ,=

z̃

z̃ θ( ) b0 b1θ1 ···bnθθnθ

,+ +=

z̃

f E F d b θ, ,( )[ ],d b θi

L j,θi

U j,, , ,

min=

Gj d b θ, ,( )θ Tαj∈

max 0, j≤ 1 2 … m p,+, , ,=

Φ θ̃1U j,

( ) Φ θ̃1L j,

( )–[ ] Φ θ̃2U j,

( ) Φ θ̃2L j,

( )–[ ]···

··· Φ θ̃nU j,

( ) Φ θ̃nL j,

( )–[ ] αj, j≥ 1 2 … m p,+, , ,=

bnθθnθ

bnθθnθ

346

DOKLADY CHEMISTRY Vol. 435 Part 2 2010

LAPTEVA et al.

of maximization problems in constraints (29)

depend on the search variables and . There�fore, such problems can be solved by the externalapproximation method [13] modified for taking intoaccount this specific feature [14]. Because we nar�rowed the class of feasible shapes of regions and

restricted the form of the functions z(θ) to linear func�tions, the solution of problem (28) is not better than

the solution of problem (18) ( ≤ ). In solving prob�lem (28), the most laborious operation is to calculatethe mathematical expectation E[F(d, b, θ)]. At smalldimension of the vector θ, quadrature formulas can beused. However, at large dimension (dimθ > 3), the cal�culation of E[F(d, b, θ)] may be very laborious.In this case, an iterative approach to solving prob�lem (28) may be useful [14]. This approach is basedon partitioning the region T into subregions Tl. Byusing this approach, solving problem (28) reducesto an iterative procedure at each iteration of which

the region T is partitioned into Qk subregions ,

q = 1, …, Qk, where k is the number of an iteration.At each iteration, problem (28) is solved in whichthe value E[F(d, b, θ)] is replaced by its approxima�tion Eap[F(d, b, θ)]. The approximation Eap[F(d, b, θ)] iscalculated using a piecewise linear approximationof the function F(d, b, θ). Therefore, we replace thefunction F(d, b, θ) by its linear approximation in

each subregion :

(30)

where θq ∈ . One can show that the approximationEap[F(d, b, θ)] can be represented in the form

where

It is clear that the solution of problem (28) will notbe the solution of problem (18) (or problem (12)) forthe following reasons: (1) we restricted the class of fea�

sible shapes of regions to multidimensional rect�

angles and (2) the class of feasible functions z(θ) inproblem (28) is restricted to linear functions (27). Toexclude the error due to the restriction of feasibleshapes of regions , one can use an approach based

on partitioning the region T into subregions. Thisapproach was proposed [14] for solving one�step opti�mization problems under probabilistic constraints.However, this approach can also be used in the caseconsidered. To reduce the error due to the second rea�son, one can use not linear, but quadratic, approxima�tions of the functions z(θ):

(31)

where Ci is an nθ × nθ matrix. Surely, using quadraticapproximation (31) leads to a significant increase inthe number of search variables. Indeed, in this case,the search variables are the components of the vectorsbi and the elements of the matrices Ci. Their number is

m(nθ + ).

REFERENCES

1. Halemane, K.P. and Grossmann, I.E., AIChE J., 1983,vol. 29, pp. 425–433.

2. Swaney, R.E. and Grossmann, I.E., AIChE J., 1985,vol. 31, pp. 621–630.

3. Grossmann, I.E. and Floudas, C.A., Comp. Chem.Eng., 1987, vol. 11, pp. 675–693.

4. Bernardo, F.P., Pistikopoulos, E.N., and Saraiva, P.M.,Ind. Eng. Chem. Res., 1999, vol. 38, pp. 3056–3068.

5. Ostrovskii, G.M. and Volin, Yu.M., Dokl. Chem., 2001,vol. 376, nos. 1–3, pp. 38–41 [Dokl. Akad. Nauk, 2001,vol. 376, no. 2, pp. 215–218].

6. Diwaker, U.M. and Kalagnanam, J.R., AIChE J., 1997,vol. 43, pp. 440–447.

7. Bakhvalov, N.S., Zhidkov, N.P., and Kobel’kov, G.M.,Chislennye metody (Numerical Methods), Moscow:Labor. baz. znanii, 2000.

8. Bernardo, F.P., Pistikopoulos, E.N., and Saraiva, P.M.,Comp. Chem. Eng., 1999.

9. Call, P. and Wallace, S.W., in Stochastic Programming,New York: Wiley, 1994.

10. Wendt, M., Li, P., and Wozny, G., Ing. Chem. Res.,2002, vol. 41, pp. 3621–3629.

11. Ierapetritou, M.G. and Pistikopoulos, E.N., Ind. Eng.Chem. Res., 1994, vol. 33, pp. 1930–1942.

12. Wellons, H.S. and Reklaitis, G.V., Comp. Chem. Eng.,1989, vol. 13, pp. 115–126.

13. Hettich, R. and Kortanek, K.O., SIAM Rev., 1993,vol. 35, pp. 380–429.

14. Ostrovsky, G.M., Ziyatdinov, N.N., and Lapteva, T.V.,Chem. Eng. Sci., 2010, vol. 65, pp. 2373–2381.

Tαj

θiL j, θi

U j,

Tαj

f f

Tqk( )

Tqk( )

f x θ θ q( ), ,( ) f x θq,( ) ∂f x θq,( )∂θi

����������������� θi θiq–( ),

i 1=

p

∑+=

Tqk( )

Eap F d b θ, ,( )[ ] = aqF d b θq, ,( ) ∂F d b θq, ,( )∂θi

����������������������

i 1=

p

∑+⎝⎜⎛

q 1=

Qk

∑

∫ × E θi; Tqk( )[ ] aqθi

q–( )⎠⎟⎞

,

aq ρ θ( ) θ, E θi; Tqk( )[ ]d

Tqk( )

∫ θiρ θ( ) θ.d

Tqk( )

∫= =

Tαj

Tαj

zi θ( ) bi0 θTbi( ) θTCiθ( ),+ +=

nθ

2