Yugoslav Joural of Operatios Research 1 (00), Number 1, 49-60 ON WELLPOSEDNESS QUADRATIC FUNCTION MINIMIZATION PROBLEM ON INTERSECTION OF TWO ELLIPSOIDS M. JA]IMOVI], I. KRNI] Departmet of Mathematics Uiversity of Moteegro, Podgorica Abstract: This paper deals with the existece of solutios ad the coditios for the strog covergece of miimizig sequeces towards the set of solutios of the quadratic fuctio miimizatio problem o the itersectio of two ellipsoids i Hilbert space. Keywords: Quadratic fuctioal, miimizatio, wellposedess. 1. INTRODUCTION Suppose that H, F, G 1 ad G are Hilbert spaces; A: H F, H G 1 ad C: H G - bouded liear operators; f F a fixed elemet; r 1 > 0 ad r > 0 are give real umbers; U 1 ad U ellipsoids i the space H defied by operators B ad C : U = { u H: Bu r }, U = { u H: Cu r }. 1 1 This paper deals with the extremal problem: Ju ( ) = Au f if, u U= U U. (1) 1 We study the existece of solutios ad the wellposedess of the problem i the Tikhoov sese. As a example of the problem of this type, we ca quote the problem of miimizatio of the fuctio This research is supported by the Yugoslav Miistry of Scieces ad Ecology, Grat OSI63.
50 M. Ja}imovi}, I. Kri} / O Wellposedess Quadratic Fuctio Miimizatio Problem R Ju ( ) = xtu (, ) z, where z R ad xtu (, ) is a solutio of the system of differetial equatio x () t = B() t x() t + D() t u(), t t ( 0, T), x( 0) = 0 R, ut ( ) r : = ut ( ) dt r, xtu (, ) = xtut (, ( )) dt r L T T R 1 0 0 with give matrices B() = ( b ij()) ad D() = ( d ij()) r. These coditios guaratee the existece of the solutio xtu (, ) H 1 [ 0, T ] of the previous system for each r u L [ 0, T ]. The same problem with differet set of costraits U, was cosidered i [1], [] ad [3]. I [3], the set of costraits U was a ball. I [] the ecessary ad sufficiet coditios for the existece of a solutio have bee cosidered i the case whe U is a half-space, as have sufficiet coditios i the case whe U is a ellipsoid. Fially, the paper [1] cotais sufficiet ad ecessary coditios for these problems whe the set of costraits U is a polyhedro. It should be poited out that this article deals with the wellposedess problem with the exact iitial date which is also the case i the papers [1], [] ad [3]. Methods for approximate solvig of problem (1) with errors i the iitial data are cosidered, for example, i [3], [4], [5].. AUXILIARY RESULTS Let us itroduce the followig otios: RA ( ) = { Au: u H } - the set of operators values A, A( U) = { Au: u U}, Ker A = { u H: Au =0} - kerel of A ; M is the closure of the set M i the space H ; L is the orthogoal complemet of the subspace L H ; P is the operator orthogoally projectig the space H o the closed subspace AU ( ) ; RA ( ); Pr - operator projectig the space F o the closed ad covex set B A - restrictio o the operator B o the subspace Ker A ; C AB - restrictio of the operator C o the subspace Ker A Ker B. Geerally, every liear bouded operator A: H F geerates the decompositio H = R( A ) KerA. () Lemma 1. The operators A, B ad decompositios of the space H : C geerate the followig orthogoal A AB H = R( A ) R( B ) R( C ) (KerA KerB Ker C ), (3)
M. Ja}imovi}, I. Kri} / O Wellposedess Quadratic Fuctio Miimizatio Problem 51 B AB H = R( B ) R( A ) R( C ) (KerA KerB Ker C ). (4) Proof: By applyig () o operators B A :KerA G 1 ad CAB :KerA KerB G we get ad Ker A= R( B ) (Ker A Ker B ), AB Ker A (Ker B) H = R( C ) (Ker A Ker B Ker C ). Accordig to these relatios ad (), the relatio (3) follows. Relatio (4) ca be proved i the similar way. I order to formulate the ext statemets we eed the followig defiitio. It is said that the operator A is ormal solvable, if the coditio RA ( ) = RA ( ) is fulfilled. This coditio is equivalet to RA ( ) = RA ( ) ([4]). Lemma. [4] Liear bouded operator A : H F is ormal solvable if ad oly if m = if{ Au : u Ker A, u = 1} > 0, A ad tha we have ( x R( A )) m x Ax. (5) A This Lemma Immediately implies Lemma 3. If a liear bouded operator A : H F is ot ormal solvable the there exists a sequece ( p ) such that ( N) p R( A ), p = 1, p 0, Ap 0 ( ). 3. EXISTENCE OF SOLUTION It is obvious that for a give f F, the problem (1) has a solutio, if ad oly if Pr( f ) A( U ). Sice Pr( F) A( U ), we have that problem (1) has a solutio for every f F, if ad oly if AU ( ) = AU ( ). Fuctio J is weakly lower semicotiuous sice it is covex ad cotiuous. The set U is weakly closed sice it is covex ad closed i the orm of H. Suppose that ( u ) is miimizig sequece of problem (1), i.e. that u U, =1,,... ; ad lim Ju ( ) = J : = if{ Ju ( ) : u U }.
5 M. Ja}imovi}, I. Kri} / O Wellposedess Quadratic Fuctio Miimizatio Problem If for some f F the sequece ( u ) is bouded, the for such f problem (1) has a solutio. Namely, i that case there exists a subsequece ( u k ) of the sequece ( u ) ad a poit u U such that u u as k. k Sice the set U is weakly closed, it follows that u U. The fuctio J is weakly lower semicotiuous ad hece Ju ( ) limif Ju ( ) = J. k k Accordig to this we have that Ju ( )= J. It meas that u U : = U : = { u U : J( u) = J }. If U, the it is easy to prove that for each u U we have the equatio U = ( u + Ker A) U. (6) By usig the equatio Ju () = Jv () + J (), v u v + Au ( v), uv, U, ad the optimality coditios ( u U) J ( u ), u u 0 we get the iequality Au ( u) Ju ( ) J. From here we have that ( u ) is a miimizig sequece of problem (1) if ad oly if Au Au,. If A is a ormal solvable operator, the accordig to (5), we get that is Theorem 1. If m Pu Pu Au Au 0,, A Pu Pu,. (7) (i) A is a ormal solvable operator, (ii) B(Ker A ) - closed subspace of space G 1, (iii) C(Ker A Ker B ) - closed subspace of space G, the problem (1) has a solutio for each f F.
M. Ja}imovi}, I. Kri} / O Wellposedess Quadratic Fuctio Miimizatio Problem 53 Proof: Accordig to the theorem coditios, the equatio (3) may be writte dow as A AB H = R( A ) R( B ) R( C ) (KerA KerB Ker C ). Let ( u ) be a miimizig sequece. The A AB u = Pu + x + y + z, x R( B ), y R( C ), z Ker A Ker B KerC. Sequece ( v), v = Pu + x + y is also a miimizig sequece. Besides, Bv = P( Bu + x ), that is BPu ( + x ) r, CPu ( + x + y ) r. 1 Accordig to (7) we have that sequece ( Pu ) is bouded. By usig coditios ii) ad iii) ad applyig relatio (5) o the operators B A ad C AB, we coclude that the sequeces ( x ) ad ( y ) are bouded. O the basis of this, the sequece ( v ) is a bouded miimizig sequece. By usig the decompositio (4) ad a similar decompositio, we may prove that operators A, B ad C i Theorem 1 may mutually chage their places. Let us metio oe of these cases. Theorem. If (i) B is a ormal solvable operator, (ii) A(Ker B ) - closed subspace of space F, (iii) C(Ker A Ker B ) - closed subspace of space G, the problem (1) has a solutio for each f F. 4. WELLPOSEDNESS Let i the followig defiitio U H be a arbitrary closed ad covex set, ad J a arbitrary real fuctio defied o the set U. Defiitio. [1], [4], [5] We say that the extremal problem Ju ( ) if, u U is wellposed i the sese of Tikhoov if the followig coditios are satisfied: (i) J = if{ J( u) : u U } > ; (ii) U = if{ u U: J( u) = J } ; (iii) for each miimizig sequece ( u ) we have du (, U) = if{ u u : u U } 0 whe.
54 M. Ja}imovi}, I. Kri} / O Wellposedess Quadratic Fuctio Miimizatio Problem If at least oe coditio from this defiitio is ot valid, we will say that the problem is illposed. The followig example shows that coditios of Theorem 1, i geeral, do ot guaratee the wellposedess of the problem (1). Example. Let L = { x l: x = ( 0, x, x/ 3, x3, x 3/ 5,...)} ad A be operator of the orthogoal projectio o L. Operator A is ormal solvable. Let operators BC, : l l be defied as follows: Bx = ( 0, x, x,...,), Cx = ( x, 0, x, 0,...,), x = ( x, x, x,...) l. Here we have that 3 1 3 1 3 Ker A= L, Ker B= { x l : x = ( x, 00,,...)}, Ker C = { x l : x = ( 0, x, 0, x,...)}, 1 4 we ca see that B(Ker A) = B( L) = L ad Ker A Ker B = { 0}. It meas that for sets U = { u l : Bu 1}, U = { u l : Cu 1}, 1 ad for the elemet f = ( 100,,,...) the coditios of the Theorem 1 are fulfilled. Let us prove that i this case the problem (1) is ot wellposed. Sice f L, the Af = f. It is also Bf = 0 ad Cf = f. It meas that f U ad the u = f is a solutio to problem (1). Let us cosider the sequece u = α ( u + v ), where Sice v 1 = 00,,..., 01,,, 0,... + 1 v L = Ker A, we have that ad 1/ 1 α = 1+ 1,. ( + ) 1 Au = α u u = Au. Besides, we also have, Bu = αv ad α 1 Cu =,,...,,,,... 10 0 0. + 1 Therefore, Bu = 1 ad Cu = 1. Accordig to this, the sequece ( u ) is the miimizig sequece. Let us prove that u is the uique solutio of the problem (1). Let v U. The, accordig to relatio (6) we have that there exists z =,, z z,, 3 0 z z3,..., L= KerA, 3 5 such that
M. Ja}imovi}, I. Kri} / O Wellposedess Quadratic Fuctio Miimizatio Problem 55 z,, z v = u + z =,, 3 1 z z 3,...,. 3 5 From here we have z = + z Cv + 3 1 + > 1, 3 5 for z 0. I that way U = { u }. Ad ow, we have du (, U) = u u = α v ( α 1) u 1,. I the followig theorem we are provig that if we add the coditio U Γ 1, where Γ 1 is the boudary of the ellipsoid U 1, to the coditios from the previous theorem, the the problem (1) is wellposed. Theorem 3. If the coditios from the Theorem 1 are satisfied ad if U Γ= 1 { u H: Bu = r 1}, (8) the the problem (1) is wellposed. Proof: Let us suppose that ( u ) is the arbitrary miimizig sequece. We have proved i Theorem 1 that there are bouded sequeces ( x ), ( y ) ad ( z ), A AB x R( B ), y R( C ), z Ker A Ker B KerC such that u = Pu + x + y + z. Without a loss of geerality, we ca suppose that AB y y R( C ),. The 0 A x x R( B ) ad 0 Pu + x + y u = Pu + x + y U. (9) 0 0 Accordig to (8) we have Bu = r 1. Further, from (9) it follows that that is Further o r = Bu limif B( Pu + x ) limsup B( Pu + x ) r, 1 1 lim BPu ( + x ) = r 1. lim BPu ( + x ) Bu = r r + r = 0. (10) 1 1 1 get Operator B A is ormal solvable. By applyig relatio (5) o this operator we
56 M. Ja}imovi}, I. Kri} / O Wellposedess Quadratic Fuctio Miimizatio Problem m x x Bx ( x) = BPu ( + x ) Bu + BPu ( Pu ). B 0 0 O the basis of relatios (7) ad (10), we obtai the strog covergece x x, Pu + x Pu + x,. (11) 0 0 Let us cosider the sequece ( v), v = Pu + x0 + y + z ad let us otice that Av = Au, Bv = Bu ad v u,. (a) If Cv r, the v U ad i that case we have du (, U) u v = Pu Pu + x x,. 0 0 (b) We suppose here that Cv > r. The from Cv = C( Pu + x + y ) C( Pu Pu + x x ), ad from relatio (11), we ca coclude lim Cv r. = (b1) Let us first cosider the case Cu > r. By a argumet similar to the oe used i provig the first relatio i (11), the strog covergece may be proved: The It follows y y,. 0 Pu + x + y u = Pu + x + y U. 0 0 d( u, U ) u ( u + z ) = Pu Pu + x x + y y,, 0 0 0 so, i the case of (b1) the theorem is proved. (b) Let us ow suppose that Cu < r. Let us deote with = 0 AB g y y R( C ) ad let us defie the sequece ( α ) such that Cu ( + α g ) = r. (1) For α defied i this way, we have u + α g U. Cosiderig that Pu Pu, x x 0 ad g g 0 as, it is easy to prove that lim Cg r Cu. (13) = > 0 From (1) ad (13) it follows that lim α = 1. Ad fially,
M. Ja}imovi}, I. Kri} / O Wellposedess Quadratic Fuctio Miimizatio Problem 57 du (, U) u ( u + α g + z ) = Pu Pu + x x0 + ( 1 α ) g 0,, which proves the theorem. I the followig four theorems we will prove that if some of the coditios from the previous theorem are violated, the problem (1) does ot have to be wellposed. Theorem 4. If (i) RA ( ) RA ( ), (ii) U itu the problem (1) is illposed. Proof: From (i) ad Lemma 3 we have that there exists a sequece ( p ) such that ( ),, p R A p = 1 Ap 0 as. Accordig to (ii) we ca coclude that there are α > 0 ad the elemet u U itu such that ( N) v = u + α p U. The sequece ( v ) is miimizig, sice Av Au = α Ap 0 as. Let v U be a arbitrary elemet. Accordig to (6) we have that u v Ker A. The α α α v v = p + u v = + u v ad it meas that the sequece ( dv (, u )) does ot coverge to zero. Theorem 5. If (i) B(Ker A) B(Ker A ), (ii) U Γ= { u H: Bu = r }, (iii) U itu the problem (1) is illposed. 1 1 Proof: Let u U itu. Accordig to the coditio (ii) ad relatio (6), we have that U u + (Ker A Ker B ). (14) The set Ker A may be preseted as A Ker A= R( B ) (Ker A Ker B ). (15)
58 M. Ja}imovi}, I. Kri} / O Wellposedess Quadratic Fuctio Miimizatio Problem Accordig to the coditio (i) ad Lemma 3 we have that there exists a sequece ( q ) such that ( A),, q R B q = 1 Bq 0, as. (16) By takig ito accout the coditio (iii), there is ε > 0 such that ( N) C( u + εq ) < r. Let us cosider the sequece ( v), v = u + εq. Accordig (14)-(16), we have that Bv > r 1, ad lim Bv = r. If we take r, 1 u = αv = αu + αεq α =, Bv we ca see that α < 1 ad α 1 as. Ad ow Au = α Au Au as, Bu = r, Cu r. 1 Accordig to this, the sequece ( u ) is miimizig. O the basis of (14), we have that for each v U, there is xv ( ) KerA KerB such that v = u + x( v ). That is why the followig holds Theorem 6. If du (, U) = if{ u v : v U} = if{ ( α 1) u + α εq + xv ( ) : v U } ( 1 ) u as α ε α ε I a similar way, we ca prove the followig theorem. (i) C(Ker A) C(Ker A ), (ii) U Γ= : { u H: Cu = r }, (iii) U itu 1 the problem (1) is illposed. Theorem 7. If (i) C(Ker A Ker B) C(Ker A Ker B ), (ii) U Γ= : { u H: Bu = r, Cu = r }, the problem (1) is illposed.. 1 Proof: Accordig to the coditio (ii) ad relatio (6), we have that U = u + (Ker A Ker B Ker C ).
M. Ja}imovi}, I. Kri} / O Wellposedess Quadratic Fuctio Miimizatio Problem 59 where u U is a arbitrary elemet. The set Ker A Ker B may be preseted as AB Ker A Ker B= R( C ) (Ker A Ker B Ker C ). Accordig to the coditio (ii) ad Lemma 3 we ca coclude that there exists a sequece ( q ) whose elemets satisfy the followig coditios AB q R( C ), q = 1, Cq 0, as. The further argumet is similar to oe i the proof of Theorem 5. I the proof of Theorem 3, we used relatio (7). The coditios of Theorem, do ot guaratee this relatio. Theorem 8. If the coditios of Theorem are satisfied ad if U Γ= 1: { u H: Bu = r 1}, (17) the problem (1) is illposed. Proof: Let us suppose that ( u ) is a miimizig sequece. By usig the relatio (4) ad the coditios of the Theorem, the elemets of this sequece may be represeted as where The u = s + x + y + z, B AB s R( B ), x R( A ), y R( C ), z KerA KerB KerC. Bu = Bs, Au = A( s + x ), Cu = C( s + x + y ). By a argumet similar to the oe used i the proof of Theorem 3, we ca prove that the sequeces ( s), ( x ) ad ( y ) are bouded. Without a loss of geerality we ca suppose that The 0 0 B 0 AB s s R( B ), x x R( A ), y y R( C ), as. s + x + y u = s + x + y U as. 0 0 0 As i Theorem 3, usig relatio (17) ad weak covergece of ( s ) to s 0, strog covergece 0 s s as (18) is proved. Regardig the fact that the sequece ( u ) is miimizig we have that
60 M. Ja}imovi}, I. Kri} / O Wellposedess Quadratic Fuctio Miimizatio Problem Au = A( s + x ) Au = As + Ax, as. The from (18), we have that 0 Ax Ax, as. 0 0 If we suppose that A(Ker B) = A(Ker B ), applyig relatio (15) o operator A B strog covergece 0 x x, as., we get Cosideratio of the sequece ( y ) ad proof of the wellposedess of problem (1) are the same as i the poits a) ad b) i Theorem 3. Here we ca also prove that if ay of the coditios from the previous theorem is ot respected, the the problem (1) geerally is ot wellposed. REFERENCES [1] Ja}imovi}, M., Kri}, I., ad Potapov, M.M., "O well-posedess of quadratic miimizatio problem o ellipsoid ad polyhedro", Publicatios de l'istitute de Mathematique, 6 (1997) 105-11. [] Kri}, I., ad Potapov, M.M., "O coditios of wellposedess of quadratic miimizatio problem o ellipsoid ad halfspace", Mathematica Motisigri, 4 (1995) 7-41. (i Russia) [3] Vasilyiev, F.P., Ishmuhametov, A.E., ad Potapov, M.M., Geeralized Momet Method i Optimal Cotrol Problem, Moscow State Uiversity, Moscow, 1989. (i Russia) [4] Vaiikko, G.M., ad Vereteikov, A.Yu., Iterative Procedures i Ill-posed Problems, Nauka, Moscow, 1986. (i Russia). [5] Vasilyev, F.P., The Numerical Solutio of Extremal Problems, Nauka, Moscow, 1988. (i Russia) [6] Ja}imovi}, M., ad Krai}, I., "O some classes of regularizatio methods for miimizatio problem of quadratic fuctioal o halfspaces", Hokaido mathematical Joural, 8 (1999) 57-69. [7] Zolezzi, T., "Wellposed optimal cotrol problems", VINITI, 60 (1998) 89-106. (i Russia) [8] Zolezzi, T., "Well-posedess ad coditioig of optimizatio problems of optimal", Pliska Stud. Math. Bulgar., 1 (1998) 1001-1018. [9] Dochev, A., ad Zolezzi, T., Well-posed Optimizatio Problems, Lect. Notes Math., 1993.