CONSTRUCTION OF THE GRS - Application of the Generalized Reachable Sets Method to Water Resou

In t h i s Section, w e briefly discuss numerical methods f o r t h e construc- tion of t h e GRS. W e suppose t h a t t h e model of t h e system under study h a s t h e form

where A i s a given matrix, b is a given v e c t o r . The v e c t o r of c r i t e r i a

p

E Rm is connected with v a r i a b l e s by l i n e a r mapping

P =PX

^I (32)

where F i s a given matrix. The GRS which is defined implicitly as

should b e c o n s t r u c t e d in t h e form

= [ p

€ R m :

D f s d ] .

To c o n s t r u c t t h e GRS in t h e form (34), at t h e Computing Center of t h e USSR Academy of Sciences a group of numerical methods w a s developed.

These methods were combined into t h e software system POTENTIAL f o r t h e computer BESM-6. The f i r s t version of t h e system POTENTIAL w a s published in (Bushenkov and Lotov, 1980), t h e second one w a s described in (Bushenkov and Lotov, 1982 and 1984).

The methods included into t h e system POTENTIAL are based on t h e con- s t r u c t i o n of projections of finite dimensional polyhedral sets into sub- spaces. Suppose we have some polyhedral set M belonging t o

O,

+ q ) - dimensional l i n e a r s p a c e R P + Q . Suppose t h i s set is described in t h e form of solution of finite system of l i n e a r inequalities

where v E R P , w E R Q , t h e matrices A and B as w e l l as t h e v e c t o r c are given. The projection of t h e set

M

into t h e s p a c e RQ of v a r i a b l e s w i s defined as t h e set Mw of all points w E Rq f o r which t h e r e e x i s t s such a point v E RP t h a t [ v ,w

1

^ERP ^{+ Q} belongs t o M . An example of t h e two dimen- sional set and i t s projection into one dimensional s p a c e is p r e s e n t e d in Fig- u r e 1.

To c o n s t r u c t t h e GRS f o r t h e system (31)-(32) l e t s consider t h e set

The GRS i s t h e projection of t h e set Z into t h e s p a c e Rm of t h e c r i t e r i a f

.

The methods of t h e system POTENTIAL give t h e possibility t o c o n s t r u c t pro- jections of polyhedral sets in t h e form of solution of a system of l i n e a r ine- qualities. Therefore, using t h e system POTENTIAL i t i s possible t o c o n s t r u c t t h e GRS in t h e form (34).

The f i r s t method of t h e construction of projections of polyhedral sets described as solutions of systems of l i n e a r inequalities w a s introduced by F o u r i e r (1826). This method w a s based on exclusion of variables by combi- nation of inequalities. The application of t h e method given by F o u r i e r t o t h e construction of t h e projection of t h e set given in Figure 1 is described in (Lotov 1981a).

Figure 1

The idea suggested by F o u r i e r w a s used in more effective methods of construction of projections of polyhedral sets ( s e e Motzkin et al. 1953, as w e l l as Chernikov 1965) based on exclusion of v a r i a b l e s by combination of inequalitiess. The experimental study of t h e s e methods proved t h a t t h e methods of t h i s kind are effective for small systems only (n

-

10-30). For mathematical models (31) with hundreds of v a r i a b l e s new methods were sug- gested. The most effective of them a t t h e moment i s t h e method of

improvement of t h e approximations of t h e projection (IAP). The IAP method gives t h e possibility t o approximate t h e projection of t h e set 2 f o r all t h e models (31) f o r which l i n e a r optimization problems c a n b e solved. The IAP method i s described in s h o r t below. For more detailed information see Bushenkov and Lotov (1982) as well as Bushenkov (1985).

The g e n e r a l idea of t h e IAP method consists of combining methods based on exclusion of variables with t h e optimization methods of construc- 'tion of t h e GRS which are usually ineffective f o r m

>

2. The IAP method

consists of iterations having t h e following form.

Before t h e k-th iteration two polyhedral sets Pk and P' should b e given while i t holds

The set Pk which is t h e internal approximation of t h e set Gf should b e given in t h e two following forms simultaneously:

1 ) as t h e solution of a system of linear inequalities, i.e.

where cj a r e t h e v e c t o r s and dj a r e t h e numbers calculated on previous iterations,

2) as t h e convex combination of points (vertices) f ,

. . .

, f r k :

The description of t h e polyhedral set using both forms i s called double description (Motzkin et al. 1953). I t i s necessary t o note t h a t t h e conver- sion from one form to a n o t h e r is a v e r y difficult t a s k . Only if t h e number of inequalities sk in t h e f i r s t form o r t h e number of points r k in t h e second one a r e r a t h e r small c a n t h i s task b e solved numerically. This i s why on t h e z e r o iteration w e c o n s t r u c t a simple approximation P I f o r which t h e

conversion between forms c a n b e fulfilled easily. Subsequently, s t e p by s t e p t h e internal approximation i s improved: from t h e set P I w e obtain P 2 and s o on. On k -th i t e r a t i o n we c o n s t r u c t t h e set Pk f o r which

Each form of t h e presentation of t h e set Pk+l i s calculated on t h e basis of t h e same form f o r t h e set Pk

.

To obtain t h e set Pk+l on t h e basis of t h e set Pk w e add t o t h e s e t Pk a new v e r t e x FTk+l. This v e r t e x is chosen in t h e following manner. I t i s sup- posed t h a t f o r any v e c t o r c j in t h e f i r s t form of presentation t h e following optimiation problem w a s solved:

{ ( c j , f ) ^-B

7

max

=

Fz ,

Az

S b

Let {z; , f;

1

b e t h e optimal solution of t h i s problem. Let

where

lbjll

is t h e norm of cj. The values of Aj d e s c r i b e descrepancy between t h e i n t e r n a l approximation Pk and t h e e x t e r n a l approximation Pk which i s described as

Since

t h e value of max {Aj: j

=

¹,

. . .

, Sk

1

can b e used as estimation of descrepancy between t h e sets Pk and Gf

.

In Figure 2 f o r m

=

2 we have t h e internal approximation Pk (its v e r - t i c e s are A , B , C and D), t h e e x t e r n a l approximation Pk (its v e r t i c e s are

K ,

,

M and N) and t h e set Gf which i s unknown f o r t h e r e s e a r c h e r . The points E , F , G and H a r e t h e solutions of optimization problems (36) f o r ine- qualities describing t h e set Pk

.

Figure 2

The set Pk i s obtained by inclusion into t h e second form of descrip- tion of t h e i n t e r n a l approximation t h e new v e r t e x J'rk f o r which i s chosen one of t h e points

J',:

obtained in optimization problems (36). I t i s r a t h e r effective to u s e t h e point with maximal value of A; b u t d i f f e r e n t s t r a t e g i e s c a n b e applied as well. The description of t h e approximation of t h e set Pk in t h e second form i s found. Note t h a t to obtain J'rk t h e f i r s t form of t h e description of t h e set Pk was used. N o w w e will c o n s t r u c t t h e f i r s t form of description of t h e set Pk

In Figure 2 t h e point with maximal value of A, i s t h e point G . I t is included into t h e description of t h e s e t which t h e r e f o r e h a s v e r t i c e s A , B , C , D , and G. To obtain t h e description of t h e Pk in t h e f i r s t form in t w o dimensional case p r e s e n t e d in Figure 2 i t i s sufficient to exclude from t h e description t h e inequality corresponding to t h e line passing through t h e points D and C and to include in t h e description t w o new inequalities corresponding to t h e lines passing through points D and G as w e l l as C and G . In g e n e r a l (m >2) t h e problem of constructing t h e description of t h e set

Pk+l in t h e f i r s t form is not s o simple and c a n b e solved by using methods of excluding of variables in t h e system of l i n e a r inequalities.

F i r s t of all, we find t h e inequalities which a r e violated by t h e point

f r k + , . These inequalities a r e excluded from t h e system. Let

P I ,

f ^{2 ,}

. . .

^,^{f N k}be t h e v e r t i c e s belonging t o t h e excluded inequalities. Let us

consider t h e cone with t h e v e r t e x

Irk+,

and t h e edges f 1 -

P 2 -

^grk+l

. . . INk -

i.e. t h e cone

W e shall p r e s e n t t h i s cone in t h e form of solution of finite number of ine- qualities. For t h i s r e a s o n w e consider t h e set

and c o n s t r u c t i t s projection into t h e s p a c e

R m

of v a r i a b l e s f

.

This projec- tion coincides with t h e cone

K

and c a n b e constructed by means of methods of excluding of v a r i a b l e s in systems of l i n e a r inequalities. Note t h a t t h i s problem h a s small dimensionality and c a n b e solved easily. The obtained system of l i n e a r inequalities we include into t h e description of t h e approxi- mation. The set Pk in t h e f i r s t form i s constructed. Then we solve optimi- zation problems (36) f o r new inequalities. The e x t e r n a l approximation p k

i s constructed as well. The k-th i t e r a t i o n i s finished.

After a finite number of iterations t h e polyhedral set Gf could b e con- s t r u c t e d . Usually if t h e system (31) is l a r g e enough i t i s necessarily t o ful- fill millions of i t e r a t i o n s t o c o n s t r u c t Gf precisely. T h e r e f o r e in p r a c t i c a l problems i t i s reasonable t o find a good approximation of t h e set GI. The good approximation i s usually found a f t e r a small number of iterations. For example, f o r m

=

4 t h e set Gf i s approximated with 1% precision a f t e r 15-20 iterations. Note t h a t w e c o n s t r u c t both internal and e x t e r n a l approximations of t h e set G f ; so i t i s possible t o decide a f t e r e a c h iteration t o s t o p o r not t o stop t h e p r o c e s s on t h e basis of graphical presentation of both approximations of t h e set.

I t is n e c e s s a r y t o note t h a t t h e IAP method coincides in some details w a s studied in dialogue using presentation of two dimensional cross-sections (slices) of t h e s e t on t h e s c r e e n of t h e computer. Some of t h e c r o s s -

Im Dokument Application of the Generalized Reachable Sets Method to Water Resources Problems in the Southern Peel Region of the Netherlands (Seite 21-28)