Parametric Linear Complementarity Problems. Klaus Tammer. Humboldt-Universitat Berlin, FB Mathematik, Unter den Linden 6, D Berlin, Germany

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1 Parametri Linear Complementarity Problems Klaus Tammer Humboldt-Universitat Berlin, FB Mathematik, Unter den Linden 6, D Berlin, Germany Abstrat We study linear omplementarity problems depending on parameters in the right-hand side and (or) in the matrix. For the ase that all elements of the right-hand side are independent parameters we give a new proof for the equivalene of three dierent important loal properties of the orresponding solution set map in a neighbourhood of an element of its graph. For oneand multiparametri problems this equivalene does not hold and the orresponding graph may have a rather ompliate struture. But we are able to show that for a generi lass of linear omplementarity problems depending linearly on only one real parameter the situation is muh more easier. 1 Introdution Linear omplementarity problems with parameters in the right-hand side and in the matrix have been extensively studied by many authors (e.g. [1], [3], [5], [9], [17], [21]). Further interesting papers onerning more general problems ontain essential onsequenes also for the speial ase of parametri linear omplementarity problems (f. [4], [15], [16], [18], [23], [24], [25]). In our paper we onsider parametri linear omplementarity problems of the form P 0 () Compute all x 2 R n satisfying q()+k()x 0 x 0 x 0 (q()+k()x) =0 for whih generally both the vetor q 2 R n and the (nn)-matrix K depend on a parameter vetor 2 R d. Conerning the kind of the parameter dependene we onsider two ases, denoted by A 0 and A 1 : A 0 : q() :R d! R n and K() :R d! R nn are loally Lipshitz. A 1 : q(t) =q 0 + t 1 q 1 + ::: + t d q d and K is onstant. Intheasethatwe assume A 1 we denote the parameter vetor by t and the orresponding parametri linear omplementarity problem by P 1 (t). Only few results will be devoted to the general problem P 0 () under assumption A 0. Our partiular interest onerns its speial ases P 01 (q ) Compute all x 2 R n satisfying q + K()x 0 x 0 x 0 (q + K()x) =0 where all omponents of q together with the omponents of are independent parameters, P 02 (q) with q as parameter, P 03 (q K) with q and K as parameters and the one-dimensional speial ase P 11 (t) ofp 1 (t) P 11 (t) Compute all x 2 R n satisfying (q 0 + tq 1 )+Kx 0 x 0 x 0 (q 0 + tq 1 + Kx)=0: This work was supported by the Deutshe Forshungsgemeinshaft under grant Gu 304/1-4.

2 Let us denote the set of all solutions x of P 0 () (P 1 (t) P 01 (q ) :::) for the orresponding parameter value by () ( (t) (q ) :::). In setion 2 of our paper we summarize some essentially known global results on the solution set maps of the onsidered parametri problems and give ertain supplements onerning their polyhedral struture. Motivated by a reent equivalene statement of Donthev and Rokafellar [4] onerning three dierent loal properties of the solution set map of a more general lass of parameter-depending problems (lower semiontinuity, pseudo-lipshitz ontinuity and strong regularity) we present in setion 3 for problem P 01 (q ) another proof whih gives a better insight into the situation. Espeially, it will be lear, for whih reason lower semiontinuity around a given point of the graph of the solution set map does not hold, if this map is not strongly regular at this point. Counterexamples in the ase of problem P 11 (t) show, that in general lower semiontinuity around agiven point of the graph and strong regularity at this point are not equivalent. In setion 4 we show that generially the graph of the solution set map of P 11 (t) hasamuh more easier struture as in the general ase and that only six types of solutions may appear. Throughout the whole paper we use the symbols "gph" for the graph of a set-valued map, "sgn" for the sign of a real number, "rg" for the rank of a matrix, "dim" for the dimension, "lin" for the linear hull, "int" for the interior, "bd" for the boundary and "ri" for the relative interior of a onvex set. The symbol B stands for the losed unit ball in R n. 2 Some global results on the solution set map In the following theorem we summarize some known properties (f. [3], [24]) of the solution set map of P 0 () andp 1 (t), respetively. Theorem For the problem P 0 () the map is losed (i.e., for eah sequene f g, onverging to any 0,andeah sequene fx g with x 2 ( ),onverging to any x 0,we have x 0 2 ( 0 )). 2. For the problem P 1 (t) the map is polyhedral (i.e., its graph is a union of a nite number of onvex polyhedra). 3. For the problem P 1 (t) the map is loally upper Lipshitz with a uniform modulus (i.e., there isaonstant >0 and for eah t 0 2 R d there exists a neighbourhood Voft 0 suh that for all t 2 V we have (t) (t 0 )+ k t ; t 0 kb. Obviously, Statement 3 of Theorem 2.1 has the following onsequenes. Corollary 2.1 For the problem P 1 (t) the sets Q b = ft = (t) is boundedg as well as Q e = ft = (t) = g are open. The following statements on the ardinality j (q) j of the set of solutions of problem P 02 (q) have been proven in [19] and [25]. Theorem 2.2 For the problem P 02 (q) it holds: 1. j (q) j 1 8q 2 R n () K is a Q-matrix (for the denition f. [19]). 2. j (q) j< 18q 2 R n () K is a N-matrix (i.e., all its prinipal subminors are nonzero). 3. j (q) j= 18q 2 R n () K is a P-matrix (i.e., all its prinipal subminors are positive). Remark 2.1 If the matrix K belongs to the lass of Q-matries (N-matries, P-matries, resp.) then for the problem P 1 (t) we have j (t) j 1 8t 2 R d (j (t) j< 18t 2 R d j (t) j= 18t 2 R d, resp.). But the reverse statements are not true in general. 2

3 For any 2 R d the set P () =fx 2 R n =K()x + q() 0 x 0g is a onvex polyhedron assoiated with the problem P 0 (). For any pair (I,J) of index sets I J f1 ::: ng let be P I J () = fx 2 P () = (K()x + q()) i =0 i2 I x j =0 j2 Jg ~P I J () = fx 2 P I J () = (K()x + q()) i > 0 i2 I xj > 0 j2 Jg A I J = f 2 R d =P I J () 6= g and ~ A I J = f 2 R d = ~ P I J () 6= g where I = f1 ::: ngni. Further let be S := f(i J)=I [ J = f1 ::: ngg. For the speial ase that J = I we write shortly P I () ~ P I () P I A I ~ A I instead of P I I () ~ P I I () P I I A I I ~ A I I. Moreover, we introdue the set A = f 2 R d = () 6= g, the matrix V (I J ) formed by therows of ;K() with the indies i 2 I and the rows of the (n n)-unit matrix with the indies j 2 J and the vetor p(i J ) formed orrespondingly by the omponents q i () i2 I and n; ji j zero omponents otherwise. The following two theorems summarize and supplement orresponding results ontained in [1], [3] and [25]. The proof of the rst one is obvious. Theorem 2.3 For the problem P 0 () it holds: 1. For 2A I J the set P I J () is a losed faet of the onvex polyhedron P () and for 2 ~ A I J the set ~ P I J () is the orresponding open faet. For (I 0 J 0 ) 6= (I 00 J 00 ) and any 2 R d it holds ~ P I 0 J 0 () \ ~ P I 00 J 00 () = and gph ~ P I 0 J 0 \ gph ~ P I 00 J 00 =. 2. The sets A gph and () for any 2 R d may be deomposed in the form: [ A = A I = [ [ ~A I J gph = gph P I = [ gph P ~ I J If1 ::: ng (I J)2S If1 ::: ng (I J)2S and () = [ P I () = [ ~P I J (): If1 ::: ng (I J)2S 3. If for any 2 R d apoint x 2 R n is an isolated solution of P 0 () then x orresponds to a vertex x(i J ) :=V ;1 (I J ) p(i J ) of the onvex polyhedron P (), i.e., there isapair (I,J) of index sets I J f1 ::: ng with j I j + j J j= n suh that the system of (in x) linear equations V (I J ) x = p(i J ) has a unique solution x(i J ) and this solution is also a solution of P 0 (). Eah vertex x(i J ) of the onvex polyhedron P () depends loally Lipshitz ontinuous on the parameter. We note that for the problem P 1 (t) the set gphp is a onvex polyhedron and the sets gphp I respetively gph ~ P I J are losed respetively open faets of gphp. Now we use the submatrix K IJ of K formed by the elements k ij of K with i 2 I and j 2 J. Theorem 2.4 For the problem P 02 (q) and eah pair (I J) 2S it holds: 1. The set A I J is a nonempty polyhedral one and ~ A I J = ria I J. 2. dimp I J (q) =dim ~ P I J (q) =d(i J) 8q 2 ~ A I J, where d(i J) =j J j;rg(ki J ) does not depend on q. 3. dima I J + d(i J)+ j I j + j J j =2n and dima I J + d(i J)+ j I \ J j= n. 4. dima I J = n () J = I ^ d(i J)=0 () J = I ^ KII is regular. 5. dima I J = n ; 1 () a) (J = I ^ d(i J)=1) _ b) (j I \ J j= 1 ^ d(i J)=0). 3

4 Proof: In our proof we apply the ideas already used in the proof of Theorem of [1]. To prove Statement 1we write A I J and A ~ I J in the form A I J = fq 2 R n =q= ;Kx + y x 0 y 0 x j = 0 j 2 J y i = 0 i 2 Ig and A ~ I J = fq 2 A I J =x j > 0 j 2 J y i > 0 i 2 Ig. Thus, the set A I J is the image of a losed faet of the polyhedral one R 2n + under the linear map whih is dened by the matrix (-K,E) and A ~ I J the image of the orresponding open faet of R 2n +. But this implies Statement 1. Beause of P ~ I J (q) 6= for all q 2 A ~ I J it follows ~P I J (q) =rip I J (q) anddim P ~ I J (q) =dimp I J (q) =diml I J =2n; ji j;jj j;rgb(i J)= 2n; j I j ; j J j ; j I j ;rg(ki J ) = n; j J j ;rg(k I J ) = j J j ;rg(ki J ), where L I J = f(x y) 2 R 2n = ; Kx + y = 0 y i = 0 i 2 I x j = 0 j 2 Jg and B(I J) is the (n (j I j + j J j))-matrix formed by the olumns of -K with the numbers i 2 I and by the olumns of the (n n) unit matrix with the numbers j 2 J. This implies Statement 2 and Statement 3,ifwe use the fat that dima I J = rg B(I J) holds. Statements4and5follow diretly from Statement 3. q.e.d. As an immediate onsequene of Theorem 2.4 we mention the following fat. Corollary 2.2 For the problem P 02 (q) and any q 2 R n we have j (q) j= 1() q 2 with S := f(i J) 2S=d(I J) 1g, where for all (I J) 2S it holds dim A ~ I J <n. S (I J)2S ~ A I J For the appliation in the following setions we give now some additional properties of those sets A I J orresponding to problem P 02 (q) whih have the maximal dimension n. Note that the number of these sets is not zero, sine for I = and J = f1 ::: ng we have A I J = R n +. All results follow from the Theorems 2.3 and 2.4 and from generally known fats on basi solutions in linear optimization. Remark 2.2 For eah set A I J with the dimension n orresponding to problem P 02 (q) it holds J = I and the orresponding matrix K II must be regular (where for the index set I = this regularity ondition is satised per denition). The orresponding set P I I (q) =P I (q) is formed xi (I q) by a single point, whih is a vertex x(i q) = of P x I(I q) I (q), where x I (I q) =;K ;1 II q I and x I(I q) = 0. Moreover, it holds A I = f q = K ;1 II q I 0 qi + K IIK ;1 II q I 0g and ~A I = f q=k ;1 II q I < 0 qi + K IIK ;1 II q I > 0g. After introduing slak variables y we an write the onvex polyhedron P(q) equivalently in the form P 0 (q) =f(x y) 2 R 2n = ; Kx + y = q x 0 y 0g and the omplementarity slakness ondition x(i q) an be expressed by x'y=0. The y-part of the vertex of the onvex polyhedron P'(q), y(i q) whih orresponds to the vertex x(i,q) of P(q) is given by y I =0and y I = q I + K IIK ;1 II q I. The orresponding simplex table to this vertex of P' (q) with the vetors of basi variables x I and y I and the vetors of non-basi variables y I and x I is given by y I x I x I ;K ;1 II K ;1 II K II ;K ;1 II q I y I K IIK ;1 II K IIK ;1 II K II ; K I I q I + K IIK ;1 II q I : (1) This simplex table ontains all oeients whih will be obtained ifwetransform the system of equations ;KII 0 xi ;KI I E x I qi ;Kx + y = q or + = ;K II E ;K I I 0 y I y I q I 4

5 in the equivalent form xi = ; y I ;K ;1 II K ;1 II K I I K IIK ;1 II K IIK ;1 II K I I ; K I I yi x I + ;K ;1 II q I q I + K IIK ;1 II q I : 3 Loal properties Besides global properties of the solution set maps of the onsidered parametri problems studied in the previous setion also loal properties are of interest. This means properties of the intersetion of the orresponding graph with a suiently small neighbourhood of one of its elements. Beause of the fat that we do not restrit our onsiderations to the ase that the matrix K has only nonnegative prinipal subminors, the set of solutions must not be onneted or even onvex, suh that loal properties are not entirely determined by the global ones. Of ourse, if the set of all solutions for a xed value of the parameter is nite, then neessarily eah solution must be isolated, and, on the other hand, if loally the set of solutions is not nite, then this also must hold globally. But these trivial statements are already almost all relations between loal and global properties. As already done in the paper [4] we are interested to apply the following denitions for general set-valued maps ; : R m! 2 Rn to the solution set maps of our parametri linear omplementarity problems. Denition 3.1 Let ; be a set-valued map and (u 0 v 0 ) 2 gph;. Then; is alled (*) lower semiontinuous around (u 0 v 0 ),ifthereare neighbourhoods U of u 0 and V of v 0 suh that ; is lower semiontinuous at every point (u v) 2 (U V )\gph; (i.e., for every sequene fu g onverging to u there isasequene fv g with v 2 ;(u ) for suiently hight, onverging to v). (**) pseudo-lipshitz at (u 0 v 0 ) with the onstant L>0, if there are neighbourhoods U of u 0 and Vofv 0 suh that ;(u 1 ) \ V ;(u 2 )+L k u 1 ; u 2 kb 8u 1 u 2 2 U. (***) strongly regular at (u 0 v 0 ), if there are neighbourhoods U of u 0 and V of v 0 suh that the map u! ;(u) \ V is single-valued and Lipshitz-ontinuous relative to U. The following lemma is the main basis to study parametri linear omplementarity problems P 0 () loally. Suppose that ( 0 x 0 )isany element of the graph of the solution set map of P 0 () and we denote I 1 = fi =(q( 0 )+K( 0 )x 0 ) i =0 x 0 i > 0g I 2 = fi =(q( 0 )+K( 0 )x 0 ) i =0 x 0 i =0g and I 3 = fi =(q( 0 )+K( 0 )x 0 ) i > 0 x 0 i =0g: Lemma 3.1 For eah suiently small neighbourhood Wof( 0 x 0 ) 2 gph we have [ [ W \ gph = W \ gph P I = W \ gph P ~ I J I2T ( 0 x 0 ) (I J)2S( 0 x 0 ) where T ( 0 x 0 ):=fi =I 1 I I 1 [ I 2 g and S( 0 x 0 ):=f(i J) 2S=I 1 I I 1 [ I 2 I 3 J I 2 [ I 3 g. S Proof: Statement 2 of Theorem 2.3 implies W \ gph W \ gphp I and W \ gph I2T ( 0 x 0 ) S W \ gph P ~ I J.For eah index set I=2T( 0 x 0 ) and for eah pair(i J) =2 S( 0 x 0 ) (I J)2S( 0 x 0 ) it holds ( 0 x 0 ) =2 gphp I and ( 0 x 0 ) =2 gphp I J. Hene, sine the set gphp I respetively gphp I J is losed, we get W \ gphp I = respetively W \ gph P ~ I J =, ifwehoose W suiently small (f. also [23]). q.e.d. 5

6 Remark 3.1 For eah ( 0 x 0 ) 2 gph there is a minimal subsystem Z( 0 x 0 )=fp 1 ::: P k g with k = k( 0 x 0 ) of the system of onvex polyhedra P I J ( 0 ) for (I J) 2S( 0 x 0 ) suh that ks S P i = P I J ( 0 ) and rip i1 \ rip i2 = for i 1 6= i 2.For i =1 ::: k let be S i ( 0 x 0 )= i=1 (I J)2S( 0 x 0 ) f(i J) 2S( 0 x 0 )=P I J ( 0 x 0 )=P i g. The sets S i ( 0 x 0 ) i=1 ::: k are pairwise disjoint and S for eah (I J) 2S( 0 x 0 ) n k S i ( 0 x 0 ) the onvex polyhedron P I J ( 0 ) is a losed faet of at i=1 least one of the onvex polyhedra P i i=1 ::: k. In the following proposition we summarize some immediate observations with respet to the appliation of Denition 3.1 to parametri linear omplementarity problems. Proposition For eah set-valued map; we have () =) () =) (). 2. For any element ( 0 x 0 ) of the graph of the solution set map of problem P 0 () ondition (*) is equivalent with the existene of neighbourhoods U of 0 and V of x 0 satisfying: (+) For eah ( ~ ~x) 2 (U V ) \ gph there isaneighbourhood ~ U of ~ suh that for eah 2 ~ U and eah onvex polyhedron Pi 2Z( ~ ~x) there isatleast one pair (I J) 2S i ( ~ ~x) with 2A I J and dimp i dimp I J (). 3. For the solution set maps of the problems P 01 (q ) as well as P 1 (t) the onditions (*) and (**) are equivalent. 4. For any element ( 0 x 0 ) of the graph of the solution set map of problem P 0 () ondition (***) is equivalent with the property that there are neighbourhoods U of 0 and V of x 0 suh that the map! () \ V is single-valued and ontinuous on U. Proof: Statement 1 follows diretly from Denition 3.1 (f. [4]). To prove the rst diretion of Statement 2we assume that there are neighbourhoods U of 0 and Vofx 0 having the property (+).Now let ( ~ ~x) be an arbitrary element of(u V ) \ gph f g any sequene onverging to ~ and P i 2Z( ~ ~x). Obviously, it holds ~x 2 P i. Aording to (+) for eah suiently hight there exists a pair (I J) 2S i ( ~ ~x) (depending on ) with 2A I J and dimp i dimp I J ( ). As in the proof of Theorem in [1] the sequene fx g, where x minimizes the Eulidean distane between ~x and P I J ( ), onverges to ~x. But this means that is lower semiontinuous at ( ~ ~x) and, hene, ondition (*) is fullled at ( 0 x 0 ). To prove the seond diretion of Statement 2 let us suppose that there do not exist any neighbourhoods U of 0 andvofx 0 with the property (+). This means that for eah neighbourhoods Uof 0 and V of x 0 there are an element ( ~ ~x) 2 (U V ) \ gph, a sequene f g onverging to ~ and a onvex polyhedron P i 2Z( ~ ~x) (depending on ) suh for all =1 2 ::: it holds either =2 A I J 8 (I J) 2S i (~ ~x) or for all pairs (I J) 2Si (~ ~x) with 2 A I J wehave dimp i >dimp I J ( ). Hene, there must be an innite subsequene of the sequene f g (for simpliity we denote it again by f g)suh that one of the following ases holds true. The rst ase is that there exists a onvex polyhedron P i 2Z( ~ ~x) suh that for all pairs (I J) 2S i ( ~ ~x) it holds =2 A I J. Lemma 3.1 and Remark 3.1 imply that for any element ( ~ x ) with x 2 rip i suf- iently near to ~x there an not be any sequene fx g with x 2 ( )onverging to x suh that annotbelower semiontinuous at ( ~ x ) and, onsequently, (*) is not satised at ( 0 x 0 ). The seond ase is that there exists a onvex polyhedron P i 2Z( ~ ~x) and a nonempty system S ~ S i ( ~ ~x) suh that for = 1 2 ::: it holds S ~ = f(i J) 2 S( ~ ~x)= 2 A I J g and dimp I J ( ) <dimp I J ( ) ~ 8(I J) 2 S. ~ For eah pair (I J) 2 S ~ let be Q I J = fx=9fx g x 2 P I J ( ) x! xg. This set is obviously onvex, ontained in P i and an only have a dimension less or equal to the minimal dimension of the sets P I J ( ). Toprove this last ondition let us suppose the opposite. Then there must be an innite subsequene of the sequene f g (for simpliity 6

7 we denote it again by f g)suhthatd = dimq I J >dimp I J ( )for =1 2 :::. Thus, there must be d+1 linearly independent points z l l=0 1 ::: din Q I J, eah of them limit of a sequene fx l g with x l 2 P I J ( ). Beause of our supposition dimp I J ( ) <dfor eah there must be a normed vetor 2 R d satisfying d P l=1 l (x0 ; x l )=0 =1 2 :::. The sequene f g must have an (again normed) aumulation point andwe obtain (using an innite subsequene of the P sequene f g onverging to ) the relation d l (z 0 ; z l )=0whihontradits our supposition l=1 that the points z l are linearly independent. Hene, it holds dimq I J dimp I J ( ) <dimp i and, onsequently, Q I J P i for eah pair(i J) 2 S. ~ Using Lemma 3.1 and Remark 3.1 this relation implies that in eah suiently small neighbourhood of ~x there are elements of the onvex polyhedron P i whih may not be a limit of any sequene fx g with x 2 ( ). But this ontradits (*). For the problem P 01 (q ) Statement 3 follows from Theorem 1 of [4]. Now let us prove Statement 3 for the problem P 1 (t). Aording to Statement 1wehave onlytoshow() =) (). We assume (*) at any element (t 0 x 0 ) 2 gph andhoose polyhedral neighbourhoods U 0 U of t 0 and V 0 V of x 0 small enough suh that for W = S U 0 V 0 Lemma 3.1 an be used. Consider the map 0 dened for t 2 U 0 by 0 (t) =V 0 \ P I J (t), whih must be lower semiontinuous (I J)2S(t 0 x 0 ) on int U'. The graph of 0 is a union of a nite number of onvex polyhedra. Consider those edges of these onvex polyhedra, whih belong to the boundary of the graph but not to the set bdu 0 V 0. Beause of the lower semiontinuity of 0 all these edges an not be perpendiular to the parameter spae R d.for eah suhedgewe onsider its angel to the parameter spae R d.if we nowhoose L as the maximal absolute value of the tangent of all these angels we nd that for arbitrary t 1 t 2 2 U it holds 0 (t 1 ) 0 (t 2 )+L k t 1 ; t 2 kband, hene, ondition (**). The rst diretion of Statement 4 is trivial, sine Lipshitz ontinuity implies ontinuity. Onthe other hand, Statement 3 of Theorem 2.3 implies that the vetor funtion x() is a ontinuous seletion of a nite number of vetor funtions x(i J ), whih are loally Lipshitz, and is, thereby, loally Lipshitz itself. q.e.d. Unlike the fat that properties (*) and (**) are equivalent for the problem P 1 (t), the properties (*) and (***) dier generally. The following three examples of the type P 11 (t) illustrate dierent possibilities whih may appear although (*) is fullled. Example 1: We dene 1(t) =fx 2 R 2 +=t ; x 1 +2x 2 0 3t +2x 1 + x 2 0 x 1 (t ; x 1 +2x 2 )+x 2 (3t +2x 1 + x 2 )=0g: f(0 ;3t) An easy omputation shows 1 (t) = 0 (;t ;t) 0 g for t 0 f(0 0) 0 (t 0) 0 suh that g for t 0 1 satises ondition (*) but not (***) at the solution (0,0)' for t=0. Loally (and in this ase even globally) this solution for t=0 is unique but in eah neighbourhood of (0,0)' and for eah t 6= 0 suiently near to zero we have more than one element x (namely exatly two) with (t x) 2 gph 1. Example 2: We dene 2(t) =fx 2 R 3 + = ; 2x 2 +2x 3 0 2t ; 1+x 1 +2x 2 +2x 3 0 ;t +1; x 1 ; x 2 0 x 1 (;2x 2 +2x 3 )+x 2 (2t ; 1+x 1 +2x 2 +2x 3 )+x 3 (;t +1; x 1 ; x 2 )=0g: f(;0:5t +1 ;0:5t ;0:5t) An easy omputation shows 2 (t) = 0 g for t 0 f(x 1 0 0) 0 = 1 ; 2t x 1 1 ; tg for 0 t 0:5 suh that 2 satises ondition (*) but not (***) at the solution (1,0,0)' for t=0. Loally (and again 7

8 globally) this solution for t=0 is unique but in eah neighbourhood of (1,0,0)' and for eah suiently small t>0wehave an innite number of points x with (t x) 2 gph 2. But globally eah onneted omponent of 2 (t) having a nonempty intersetion with a suiently small neighbourhood of (1,0,0)' is bounded. Example 3: We dene 3(t) =fx 2 R 3 + = 2t ; 2x 2 +2x 3 0 ;4t + x 1 +2x 2 +2x 3 0 ;x 1 ; x 2 0 x 1 (2t ; 2x 2 +2x 3 )+x 2 (;4t + x 1 +2x 2 +2x 3 )+x 3 (;x 1 ; x 2 )=0g: f(0 0 x3 ) An easy omputation shows 3 (t) = 0 =x 3 ;tg for t 0 f(0 0 x 3 ) 0 suh that also in this =x 3 2tg for t 0 ase 3 satises ondition (*) but not (***) at eah solution (0 0 x 3 ) 0 with x 3 0 for t=0. Here we have the situation that the intersetion of gph 3 with any neighbourhood of an arbitrary element (0 0 0 x 3 ) of this graph onsists of innitely manypoints. But here one omponent of 3 (t) having a nonempty intersetion with a suiently small neighbourhood of a solution (0 0 x 3 ) for t=0 is unbounded. Reently, Donthev and Rokafellar [4] have shown a general equivalene statement for parametri variational inequalities over polyhedral onvex sets, whih we formulate here for problem P 01 (q ). Theorem 3.1 For the problem P 01 (q ) the properties (*), (**) and (***) are equivalent. Note that this assertion is valid also for the speial ases P 02 (q) and P 03 (q K) ofp 01 (q ). The essential assumption is only that at least all omponents of q are independent parameters. The proof given in [4] is rather abstrat and uses a redution approah, known general properties of projetions and normal as well as pieewise linear maps. However, it is not seen immediately, whih requirements of (*) would be violated if (***) does not hold. Moreover, it will not intelligible why this proof an not be extended, for instane, to the problem P 1 (t). For this reason we will give another proof at the end of this setion after some preparations. A reent paper of Kummer [14] is devoted to a orresponding aim, however for the Karush-Kuhn-Tuker onditions for nonlinear and quadrati optimization problems. Aording to the deomposition of the whole 0 index set f1 ::: ng into the disjoint subsets I 1 I 2 and I 3 we also deompose K in the form K K 1 11 K 12 K 13 K 21 K 22 K 23 A. K 31 K 32 K 33 The following neessary and suient ondition for strong regularity isshown in [23] and [3] (if we use, additionally, Statement 4 of Proposition 3.1.) Theorem 3.2 For the problem P 01 (q ) and eah element (q 0 0 x 0 ) 2 gph ondition (***) is equivalent with K 11 is regular and the Shur-omplement N = K 22 ; K 21 K ;1 11 K 12 is a P-matrix. (2) In the following for eah index set I 2T(q 0 0 x 0 )we onsider the Jaobian M I of the linear system, whih desribes the set P I KII K (q), namely M I = I I.Obviously, it holds detm 0 E I = detk II. Nowwe are able to give another equivalent ondition for strong stability in problems of the type P 01 (q ), whih is already known from [11] for the Karush-Kuhn-Tuker onditions of nonlinear parametri optimization problems. For this ase the assertion of the following theorem is shown in [10]. 8

9 Theorem 3.3 For any element (q 0 0 x 0 ) 2 gph ondition (***) is equivalent to sgn detm I = onst 6= 0 8I 2T(q 0 0 x 0 ): (3) Proof: Aording to Theorem 3.2 we onlyhave toshow, that (2) and (3) are equivalent. Let (2) be satised. Then for I = I 1 wehave detm I1 = detk I1I1 = detk 11 6=0. For any index set I with K11 K I 1 I I 1 [ I 2 we an write K II = I1I 0, where I K I 0 I 1 K 0 = I n I 1. Moreover, using a I 0 I 0 known determinant rule for Shur omplements (f. [20]), we have detk II = detk 11 detn 0, where N 0 = K I 0 I 0 ;K I 0 I 1 K ;1 K 11 I 1I0 is a prinipal submatrix of N having aording to (2) a positive determinant. Hene, sgn detm I = sgn detk II = sgn detk 11 = onst 6= 0 as required in (3). The other diretion of the proof is similar. We only mention the fat that any prinipal submatrix of N an be expressed in the form K I 0 I 0 ;K I 0 I 1 K ;1 K 11 I 1I 0 with an index set I satisfying I 1 I I 1 [I 2. q.e.d. Corollary 3.1 If the solution set map of P 01 (q ) does not satisfy ondition (***) at any element (q 0 0 x 0 ) 2 gph then one of the following two ases a) or b) holds true. Case a) There is an index set I 2T(q 0 0 x 0 ) satisfying rgk I I <j I j. Case b) There are two index sets I 0 I 00 2T(q 0 0 x 0 ) and one index i 0 =2 I 0 suh that I 00 = I 0 [fi 0 g and it holds sgn detm I 0 = ;sgn detm I 00 6= 0. Proof: This assertion follows by negation of (3) taking into aount that the ondition in ase a) is only a reformulation of the equation detm I = 0 and that (if ase a) does not ome true) the existene of two dierent index sets I 0 I 00 2T(q 0 0 x 0 ) with sgn detm I 0 = ;sgn detm I 00 6= 0 implies that there are also two index sets I' and I" and an index i' with the properties given in ase b). q.e.d. Remark As shown in [4] even parametri nonlinear omplementarity problems satisfying ertain dierentiability properties an be haraterized loally (espeially onerning the property of strong regularity) in the same way as it was done here and in former papers for linear problems, namely by analyzing its orresponding linearization. Hene, many results of this setion may be used, for instane, to study the Karush-Kuhn-Tuker onditions of nonlinear (and not only quadrati) optimization problems depending on parameters. 2. Aording to [3] problem P 01 (q ) may be written equivalently as a Lipshitz ontinuous equation of the form F (z ) := K()z ; + z + = q (4) where z + = max(0 z) and z ; = min(0 z) omponentwise. The neessary and suient onditions (2), (3) (as well as all other equivalent onditions of other papers as [3] and [4]) are equivalent with the nondegeneray of the projetion of the generalized Jaobian ) (in the sense of Clarke [2]) onto the subspae of the z-variables. This follows from a result of [8] and from the fat that the verties of this projetion are losely related to the matries K II onsidered inourpaper. As we know from [12] this nondegeneray is in general only a suient (but not neessary) ondition for the so-alled Lipshitz invertibility of systems of the form (4). Only beause of a speial rank property of the verties of the mentioned projetion (whih has been applied already in [8] for the Karush-Kuhn-Tuker ondition for nonlinear parametri optimization problems desribed by C 2 -funtions) this nondegeneray ondition of Clarke is also neessary for Lipshitz invertibility and, hene, equivalent to a neessary and suient ondition of Kummer [13] for Lipshitz invertibility of Lipshitz systems and (for the speial ase of problem P 1 (t)) toa orresponding neessary and suient ondition of Sholtes [26] for pieewise linear systems. 9

10 Proof of Theorem 3.1: Beause of Statement 1 of Proposition 3.1 we onlyhave to show :( ) ):(). Consider an arbitrary element (q 0 0 x 0 ) 2 gph and suppose, that (***) is not satised there. Using Corollary 3.1 we have to study now more preisely the two ases a) and b) desribed there. In the following we delete the dependene on the parameter by xing = 0 and study the orresponding problem P 02 (q). If we an show, that ondition (*) is not satised at the point (q 0 x 0 ) of the graph of the solution set map of P 02 (q), then, obviously, ondition (*) is also not fullled at the point (q 0 0 x 0 ) of the graph of the solution set map of P 01 (q ). Consider at rst ase a). Aording to our assumptions we have (q 0 x 0 ) 2 gphp I and thus q 0 2A I. Statements 2 and 3 of Theorem 2.4 applied to I = I and J = I implies d(i I ) 1 and dima I <n.nowwehave to distinguish two subases. Subase a 1 ): If q 0 2 A ~ I then dimp I (q 0 )= d(i I ) 1 and, hene, j (q 0 ) j= 1. But aording to Corollary 2.2 in eah neighbourhood U of q 0 there must exist a parameter value q with j (q) j< 1 and, hene, dimp I (q) < dimp I (q 0 ), where beause of q 0 2 A ~ I wehave T (q 0 x 0 )=fi g Z(q 0 x 0 )=fp 1 g P 1 = P I (q 0 ) and S 1 = f(i I )g. But aording to Statement 2 of Proposition 3.1 this ontradits ondition (*). Subase a 2 ): If q 0 =2 A ~ I then q 0 belongs to the relative boundary of A I and for any neighbourhoods U of q 0 and V of x 0 there are points ~q 2 U \ A ~ I and (beause of the fat that the map P I is ontinuous relative toa I )~x 2 V \ P I (~q), for whih our argumentation of subase a 1 )anbe repeated. Now we onsider ase b). Aording to our assumptions we have (q 0 x 0 ) 2 gphp I00 I 0 and thus q 0 2A I00 I 0. The ondition sgn detm I 0 = ;sgn detm I 00 6= 0 is equivalent with the ondition sgn detk I 0 I 0 = ;sgn detk I 00 I 00 6= 0. Beause of I00 = I 0 [fi 0 g with i 0 =2 I 0 the matrix K I 00 I 0 is formed by K I 0 I 0 relled by one additional row. Hene, it holds rg(k I 00 I 0)=j I0 j. Applying Statements 2 and 3 of Theorem 2.4 we getd(i 00 I 0 ) = 0 and dima I00 I 0 = n ; 1. As in the ase a) we want to distinguish two subases. Subase b 1 ): If q 0 2 A ~ I00 I 0 then T (q 0 x 0 ) = fi 0 I 00 g Z(q 0 x 0 ) = fp 1 g P 1 = fx 0 g and S 1 (q 0 x 0 )=f(i 0 I0 ) (I 00 I 00 ) (I 00 I0 )g. Let us onsider the simplex tableaus (1') of the vertex x(i',t) as wellas(1")ofthevertex x(i",t) in the form desribed in (1). Beause of I 00 = I 0 [fi 0 g tableau (1") an be generated from tableau (1') by exatly one pivot step with the element d 0 i 0 i 0 of the i'-th row and i'-th olumn in tableau (1') as pivot element. This pivot element is loated in the main diagonal of the submatrix K I 0 I 0K;1 I 0 I 0K I 0 I 0 ; K I 0 I 0. Let us denote the linear funtions of q in the last olumn in (1') (whih is formed aording to Remark 2.2 by the elements of the vetors ;K I ;1 0 I 0q I 0 and (q I 0 + K I 0 I 0K;1 I 0 I 0q I 0)) by d0 i0 (q) and the orresponding funtions in (1") by d00 (q). i0 Aording to the rules of the pivot tehnique it holds d 0 i 0 (q) 0 =d0 i 0 i 0d00 i 0 0 (q). Using the already mentioned determinant rule for Shur omplements one an show that detk I 00 I 00 = ;d0 i 0 i 0detK I 0 I 0 suh that beause of sgn detk I 00 I 00 = ;sgn detk I 0 I 0 neessarily d0 i 0 i0 > 0follows. Aording to Remark 2.2 it holds A I0 = f q= d 0 (q) 0 i=1 ::: ng and i0 Ai00 = f q= d 00 i0 (q) 0 i=1 ::: ng. Hene, both sets A I0 and A I00 are ontained in the same halfspae H i 0 = fq=d 0 i 0 0 (q) 0g and A I00 I 0 belongs to the orresponding hyperspae. But due to Statement 2 of Proposition 3.1 this ontradits (*). Subase b 2 ): If q 0 =2 A ~ I00 I 0 then q 0 belongs to the relative boundaryofa I00 I 0 and for any neighbourhoods U of q 0 and V of x 0 there are points ~q 2 U \ A ~ I00 I 0 and (beause of the fat that the map P I00 I 0 is ontinuous relative toa I00 I 0 )~x 2 V \ P I00 I 0 (~q), for whih our argumentation of subase b 1 ) an be repeated. q.e.d. With other words the proof of Theorem 3.1 says: If at an element (q 0 0 x 0 ) 2 gph ondition (***) is violated, then in eah neighbourhood of this element there is another element (~q 0 ~x) of this graph, at whih the solution set map is not lower semiontinuous. In (q 0 0 x 0 ) itself lower semiontinuitymay hold or not. The violation of lower semiontinuityat(~q 0 ~x) may happen for 10

11 two dierent reasons. One possibility is that for all suiently small neighbourhoods U of ~q and Vof~x there exists a value q 2 U suh that there does not exist any solution x of P(q 0 )inv. The other possibility is that for all suiently small neighbourhoods U of ~q and V of ~x there exists avalue q 2 U suh that the number of solutions x of P(q 0 )invisnite,whereasthenumber of solutions of P(~q 0 ) in V is innite. Corollary 2.2 shows that this seond possibility only leads to a ontradition to ondition (*) if all omponents of q may be perturbed independently of eah other. This would be not the ase, for instane, in the problem P 1 (t) ifd<n. 4 Generi properties of one-parametri linear omplementarity problems Let us study in this setion the one-parametri linear omplementarity problem P 11 (t). The examples given in the foregoing setion show thateven for problems of small size the solution set map of this problem may have a rather ompliate struture. As the result of this setion we will see that generially the graph of the solution set map has a very easy struture. We an show this with help of the results on the problem P 02 (q) given in Theorem 2.4. Lemma 4.1 There isanopen and dense subset Q R 2n suh that for all (q 0 q 1 ) 2 Q the set g = fq 2 R n =q = q 0 + tq 1 t2rg has the following two properties: 1. For all (I J) 2S with dima I J = n ; 1 it holds g 6 lin A I J. 2. For all (I J) 2S with dim A I J n ; 2 it holds g \A I J =. Proof: We show that those values (q 0 q 1 ), for whih 1 or 2 is violated, is ontained in the union of a nite number of nondegenerated smooth manifold with dimension less or equal to 2n If g lin A I J with dim A I J = n ; 1, then neessarily it follows that (q 0 q 1 ) belongs to the linear subspae lin A I J lin A I J of R 2n, whih has the dimension 2n If g \A I J 6= with dim A I J n ; 2, then also g \ lin A I J 6=. Aording to our assumption on the dimension of A I J there must are two linear independent vetors a b 2 R n suh that lin A I J L n;2 with L n;2 = fq 2 R n =a 0 q =0 b 0 q =0g. This means that the two linear equations for one variable t, namely a 0 (q 0 + tq 1 ) = 0 and b 0 (q 0 + tq 1 )=0must have a solution. But this implies that either (q 0 q 1 ) is an element of the linear subspae L n;2 L n;2, whih has the dimension 2n-4, or (q 0 q 1 ) belongs to the set desribed by a 0 q 0 b 0 q 1 ; b 0 q 0 a 0 q 1 =0,whihis outside of L n;2 L n;2 a nondegenerated quadrati manifold of dimension 2n-1. q.e.d. The given proof shows that it sues to disturb slightly only one of the both vetors q 0 or q 1 to reah the set Q, ifagiven pair (q 0 q 1 ) originally would not belong to Q. Only beause of the possibility that q i 2 L n;2 may ome true we an not restrit our disturbations on q i0 (i =0 1 i 0 =1 0). Using the set Q desribed in Lemma 4.1 we are now abletoprove in the next theorem an essential generial property (in the sense that this property holds true for all vetors (q 0 q 1 )fromanopen and dense subset Q of R 2n ) for the graph of the solution set map of P 11 (t). For a given element (t x) 2 gph we use here the notation I(t x) =fi=(kx+q 0 +tq 1 ) i =0g and J(t x) =fj=x j =0g. Theorem 4.1 For all vetors (q 0 q 1 ) 2 Q we have: 1. Eah onneted omponent of gph for P 11 (t) is a runode-free edge polygon, whih may be either a) homeomorphi to the real line or b) homeomorphi to a irle or ) an isolated point of gph. 2. Eah element (t x) 2 gph belongs to exatly one of the following six types: 11

12 Type 1: I(t x) \ J(t x) = (strit omplementarity) ^ rgk II =j I j, where I = I(t x). Type 2: I(t x) \ J(t x) = (strit omplementarity) ^ rgk II =j I j;1, where I = I(t x). Type 3: I(t x) \ J(t x) =fi 0 g ^ sgn detk I 0 I 0 = sgn detk I 00 I 00 6= 0,where I0 = I(t x) J 0 = J(t x) nfi 0 g I 00 = I(t x) nfi 0 g J 00 = J(t x). Type 4: I(t x) \ J(t x) =fi 0 g^sgn detk I 0 I 0 = ;sgn detk I 00 I00 6= 0, where I', J', I"and J" are dened asabove. Type 5: I(t x) \ J(t x) =fi 0 g ^ sgn detk I 0 I 0 6= 0 ^ sgn detk I 00 I00 =0(or vie versa), where again I', J', I"and J" are denedasabove. Type 6: I(t x) \ J(t x) =fi 0 g^sgn detk I 0 I 0 = sgn detk I 00 I00 =0, where again I', J', I"and J" are dened asabove. 3. For almost all values of t only Type 1oours. Almost all elements (t x) 2 gph are ofthe types 1 and 2. Proof: Aording to Lemma 4.1 for (q 0 q 1 ) 2 Q the line g mayinterset only those sets A ~ I J with dimension n or n-1, where the seond ase may only aour for a nite number of values t. Hene, taking Theorem 2.4 and Remark 2.2 into aount, the graph of will be formed by all points (t x) satisfying a) (q 0 + tq 1 ) 2 A ~ I for any index set I 2f1 ::: ng suh thatdima I = n and K II is regular and xi (I t) x = x(i t)= with x x I(I t) I (I t)=;k ;1 II (q0 I + tq1 I ) and x I(I t)=0 or b) (q 0 + tq 1 ) 2 A ~ I J for any pair(i J) 2S suh thatdima I J = n ; 1andx2P I J (q 0 + tq 1 ). Aording to Theorem 2.4 and Remark 2.2 the points (t x) 2 gph satisfying a) are just those elements of gph of the type 1 and form, together with their boundary points, a rst nite system of edges of the set gph. These boundary points also must belong to gph, sine this set is losed. At these boundary points neessarily b) must be satised. For the nite number of values t, for whih there is a point x suh that(t x) 2 gph satises b) we an use Statements 2 and 5 of Theorem 2.4 and Lemma 3.1. Aording to Statement 5of Theorem 2.4 we must study two subases of ase b). In the subase b 1 )wehave I \ J = and d(i J) = 1. Hene, the points (t x) 2 gph satisfying b 1 ) are just the elements of gph ofthetype 2 and form together with their boundary points a seond nite system of edges of the set gph. Also these boundary points must belong to gph and are in this ase just the elements of gph ofthetype 5. In the subase b 2 )wehave the situation I \ J = fi 0 g and d(i J) = 0 suh that the set f(t x)=x 2 P I J (q 0 + tq 1 )g is a singleton. Obviously, with the types 3-6 all possibilities for sgn detk I 0 I 0 and sgn detk I 00 I 00 under the assumption I \ J = fi0 g are exhausted suh thatthepoints (t x) 2 gph satisfying b 2 ) are just the elements of gph ofthetypes 3-6. The points of the types 3 and 4 are ommon boundary points of exatly two adjaent edges of the rst system, one of them given by the vertex x(i 0 t) of the onvex polyhedron P(t), the other one by the vertex x(i 00 t). This follows by Lemma 3.1. Again by Lemma 3.1 we see that the points of the type 5 are the ommon boundary points of exatly one edge of the rst system and exatly one of the seond system. Finally, the points of the type 6 are isolated points of gph. Both systems of edges together with the isolated points of type 6 form the whole graph of. This ompletes the proof. q.e.d. Remark For all elements (t x) 2 gph of the types 1 and 3-6 the x-part is a vertex of the onvex polyhedron P(t). For all elements (t x) 2 gph of the type 2 the x-part is an inner point of an edge of the onvex polyhedron P(t). 12

13 2. For all elements (t x) 2 gph of the type 6 the orresponding verties (x y) of the onvex polyhedron P'(t) with y = q 0 + tq 1 + K x are exatly those verties of P'(t) whih atually satisfy the omplementarity ondition x 0 y =0, but for whih there does not exist any simplex tableau of the form (1), i.e., there does not exist any basis solution with the property that for eah i=1,...,n exatly one of the varibles x i and y i is a basi variable and the other one a non-basi variable. Under dierent assumptions on the matrix K suh verties and, hene, elements (t x) 2 gph of the type 6an not exist (f. [1]). 3. For an arbitrary element (t x) 2 gph of one of the types1or3-5onsider a orresponding simplex tableau (1). As in the proof of Theorem 3.1 let us denote the elements of (1) by d ij and the linear funtions of t in the last olumn of (1) by d i0 (t). Aording to Remark 2.2 we assume d i0 (t) 0 i=1 ::: n. With help of the data of (1) we an haraterize uniquely the type of this point as follows: a) (t x) is of the type 1 () d i0 (t) > 0 i=1 ::: n. b) (t x) is of the type 3 () there is exatly one index i 0 2f1 ::: ng suh that d i 0 0 (t) =0 and it holds d i 0 i 0 < 0. ) (t x) is of the type 4 () there is exatly one index i 0 2f1 ::: ng suh that d i 0 0 (t) =0 and it holds d i 0 i 0 > 0. d) (t x) is of the type 5 () there is exatly one index i 0 2f1 ::: ng suh that d i 0 0 (t) =0 and it holds d i 0 i 0 =0. 4. The unique open edge of the seond lass formed by solutions (t x) 2 gph of the type 2 with an element (t x) 2 gph of the type 5 (whih satises d) of Statement 3) as one boundary point an be onstruted with help of the orresponding simplex table (1) to (t x) as follows: We put z Bi 0 =0 z Bi = d i0 (t) ; d ii 0s i 6= i 0 z Ni 0 = s 0 <s<s z Ni =0 i6= i 0, where z B stands for the vetor of basi variables in (1), z N for the vetor of non-basi variables and s = supfs=d i0 (t) ; d ii 0s 0 i6= i 0 g. 5. Conerning the edges of gph of the seond lass there are three dierent ases to distinguish: Case a) If all elements d ii 0 i6= i 0,are nonpositive then this edge is unbounded (s = 1) and is, hene, the rst or last edge of the orresponding edge polygon, to whih it belongs. Otherwise the edgeisbounded and must have a seond boundary point (t 0 x 0 ). Let be i 00 2 dj0(t) d ji0 g and s = d i00 0 (t) d. For (q 0 q 1 ) 2 Q the index i" is uniquely i00i0 fi 6= i 0 = di0( t) d = min j:d ii0 ji0 >0 determined and it holds d i 0 i 00 6= 0. Hene, we an obtain a simplex tableau (1') to ( t 0 x 0 ) di from (1) by one (2 2)-pivot step with the (2 2)-matrix 0 i 0 d i 0 i 00 d i 00 i 0 d as pivot matrix. i 00 i 00 Beause of d i 0 i 0 =0, d i 00 i 0 > 0 and d i 0 i 00 6= 0this matrix is regular, if (q0 q 1 ) 2 Q. The index set I', whih orresponds to (1'), is formed by I, i' and i" in the following way. First we put I = I nfi 0 g if i 0 2 I respetively I = I [fi 0 g otherwise. Analoguesly, we set I 0 = I nfi 00 g if i 00 2 I respetively I 0 = I [fi 00 g otherwise. The orresponding matrix K I 0 I0 will always be regular. With respet to suiently small neighbourhoods W of (t x) and W' of (t 0 x 0 ) the following two possibilities b) and ) may appear. Case b) If d i 0 i 00 < 0 then sgn detk II = sgn detk I 0 I 0 and for t< t the intersetion of W with gph onsists of all points (t x(i t)) and is empty for t>t and the intersetion of W' with gph is empty for t<t and onsists for t>t of all points (t x(i 0 t)) (or vie versa). Case ) If d i 0 i 00 > 0 then sgn detk II = ;sgn detk I 0 I 0 and for t< t the intersetion of W with gph onsists of all points (t x(i t)) and is empty for t>t and the intersetion of W' with gph is empty for t>t and onsists for t<t of all points (t x(i 0 t)) (or vie versa). 13

14 6. At all elements (t x) 2 gph of the types 1 and 3 ondition (***) is satised, whereas at all other types 2 and 4-6 even ondition (*) is not fullled. 7. With respet to an open and dense subset of the (n n)-dimensional Eulidean spae of all elements k ij the matrix K is an N-matrix. Hene, for the problem P 11 (t) only the types 1, 3 and 4 remains generi, if we permit to disturb beside the vetors q 0 and q 1 also the elements of the matrix K. 8. If all prinipal minors of the matrix K are nonnegative, then the types4and6aswellasthe ase ) desribed in Statement 5 an not appear, the graph of the solution set map of P 11 (t) onsists of exatly one edge polygon and is always homeomorphi to the real line (f. [1]). If the matrix K is even a P-matrix, then also the types2and5an not appear, all elements (t x) 2 gph satisfy (***) and gph is formed only by edges of the rst lass. In the following theorem we haraterize the loal struture of the graph of the solution set map ofp 11 (t) for the six dierent types given above. We use the notations I', I" and x(i,t) as in Theorem 4.1. Theorem 4.2 Let be (t x) 2 gph and W a suiently small neighbourhood of(t x). 1. If (t x) is of the type 1 then we have W \ gph = f(t x) 2 W = x = x(i t)g, where I = I(t x). 2. If (t x) is of the type 2 then we have W \ gph =f(t x) 2 W = t= t x 2 P I (t)g, where I = I(t x). 3. If (t x) is of the type 3 then we have W \gph =f(t x) 2 W=t t x = x(i 0 t)g[f(t x) 2 W=t t x = x(i 00 t)g or W \gph =f(t x) 2 W=t t x = x(i 0 t)g[f(t x) 2 W=t t x = x(i 00 t)g. For t = t it holds x(i 0 t)=x(i 00 t). 4. If (t x) is of the type 4 then we have W \gph =f(t x) 2 W=t t x = x(i 0 t)g[f(t x) 2 W=t t x = x(i 00 t)g or W \gph =f(t x) 2 W=t t x = x(i 0 t)g[f(t x) 2 W=t t x = x(i 00 t)g. For t = t it holds x(i 0 t)=x(i 00 t). 5. If (t x) is of the type 5 then we have W \gph =f(t x) 2 W=t t x = x(i 0 t)g[f(t x) 2 W=t= t x 2 P I00 (t)g or W \ gph =f(t x) 2 W=t t x = x(i 0 t)g[f(t x) 2 W=t= t x 2 P I00 (t)g or W \ gph =f(t x) 2 W=t t x = x(i 00 t)g[f(t x) 2 W=t= t x 2 P I0 (t)g or W \ gph =f(t x) 2 W=t t x = x(i 00 t)g[f(t x) 2 W=t= t x 2 P I0 (t)g. 6. If (t x) is of the type 6 then we have W \ gph =f(t x)g. Proof: The proof follows by Theorem 4.1 and Remark 4.1. q.e.d. 14

15 The following piture illustrates the given six types of solutions and the possible situations onerning the struture of the graph of the solution set map. In this example this graph has fore onneted omponents, namely two isolated points, one edge polygon whih is homeomorphi to the real line and one edge polygon whih is homeomorphi to a irle. Moreover, there are three edges of the seond lass, one of them orresponds to ase a) desribed in Statement 5 of Remark 4.1, one to ase b) and one to ase ) hhhhhhhhhhhhhhhhh hhhhhhhhhhhhhhhhh 5 2b t 5 1 2a - Generi properties of the Karush-Kuhn-Tuker onditions for one-parametri quadrati optimization problems are the ommon subjet of Setion 4 of our paper with the papers [7] of Jongen et al and [6] of Henn et al. For the speial ase of one-parametri linear optimization problems we refer also to the relevant paper [22] of Patewa. The results are partially similar but not idential beause of the following essential dierenes in the assumptions. Firstly, in the papers [7] and [6] the dependene on the parameter t is assumed to be more general (of the type C 3 respetively C 1 ), whereas we restrit our onsiderations to the ase that only the vetor q depends on t and this dependene is linear. This point is onneted with the seond dierene, namely with the fat, that our notion "generi" is based only on small pertubations of the problem in the nite dimensional 15

16 spae R 2n of the data q 0 and q 1 with the orresponding topology, whereas in [7] and [6] small perturbations of all underlying funtions in the strong C 3 -topology respetively strong C 1 -topology are allowed. Finally, an exat omparison of the three papers would require, on the one hand, to inlude the Lagrange multipliers and their dependene on the parameter into the onsiderations of [7] and [6] and, on the other hand, to inlude all aspets of [7] whih are only relevant forthe Karush-Kuhn-Tuker onditions of an optimization problem into our onsiderations. Remember that in [7] ve types of (generalized) ritial points of one-parametri nonlinear optimization problems desribed by C 3 -funtions have been identied to be generi, whereas the paper [6] shows that in the orresponding speial ase of quadrati respetively linear optimization only the types 1, 2 and 5 respetively 1 and 5 from [7] remain generi. Taking into onsideration all dierenes between the approahes in [7] and in our Setion 4 mentioned above, we an see, that the types 1 of both papers are idential and that the two subases of type 2 of [7] orrespond to our types 3 and 4. Type 3 of [7] has some ommon properties as our type 2 but both are not idential. Finally, type 5 of [7] is related to our types 2 and 5 but is not idential. All other types of both papers dier essentially of eah other. Aknowledgement The author is indepted to Bernd Kummer and Diethard Klatte for helpful disussions on the subjet of this paper. Referenes [1] B. Bank, J. Guddat, D. Klatte, B. Kummer, K. Tammer: Non-linear Parametri Optimization. Akademie-Verlag, Berlin, [2] F. H. Clarke: Optimization andnonsmooth Analysis. Wiley, NewYork, [3] R. W. Cottle, J.-S. Pang, R. E. Stone: The Linear ComplementarityProblem. Aademi Press In., Boston et., [4] A. L. Donthev, R. T. Rokafellar: Charaterizationofstrong regularityforvariational inequalities overpolyhedral onvexsets. submitted 1995 to SIAM J. Optimization. [5] M. S. Gowda: On the ontinuity of the solution set map in linear omplementarity problems. SIAM J. Optimization 2, 1992, [6] M. Henn, P. Jonker, F. Twilt: On the ritial sets of one-parameter quadrati optimization problems. In: R. Durier, Chr. Mihelot (Eds.), Reent Developments in Optimization, Seventh Frenh-German Conferene on Optimization, Springer-Verlag, Berlin, Heidelberg, 1995, [7] H. Th. Jongen, P. Jonker, F. Twilt: Critialsetsinparametrioptimization. Math. Progr. 34, 1986, [8] H. Th. Jongen, D. Klatte, K. Tammer: Impliitfuntionsandsensitivityofstationarypoints. Math. Progr. 49, 1990, [9] D. Klatte: On the Lipshitz behavior of optimal solutions in parametri problems of quadrati optimizationandlinear omplementarity. Optimization 16, 1985, [10] D. Klatte, K. Tammer: Strong stability ofstationary solutions andkarush-kuhn-tuker points in nonlinearoptimization. Annals Oper. Res. 27, 1990, [11] M. Kojima: Strongly stable stationary solutions in nonlinear programs. In: S. M. Robinson (Ed.), Analysis and Computation of Fixed Points, Aademi Press, New York, 1980, [12] B. Kummer: The inverse ofalipshitz funtion inr n : Complete haraterization bydiretional derivatives. Preprint Nr. 195 (1988), Humboldt-Universitt zu Berlin, Sektion Mathematik. [13] B. Kummer: Lipshitzian inverse funtions, diretional derivatives andappliations inc optimization. JOTA 70, 1991,