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Sensitivity Analysis of Solutions to a Class of Quasi-variational Inequalities
Samir Adly, Mohamed Ait-Mansour, Laura Scrimali
To cite this version:
Samir Adly, Mohamed Ait-Mansour, Laura Scrimali. Sensitivity Analysis of Solutions to a Class of Quasi-variational Inequalities. Bulletino Della Unione Mathematica Italiana, 2006, 8, pp.767-772.
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B OLLETTINO
U NIONE M ATEMATICA I TALIANA
Samir Adly, Mohamed Ait Mansour, Laura Scrimali
Sensitivity analysis of solutions to a class of quasi-variational inequalities
Bollettino dell’Unione Matematica Italiana, Serie 8, Vol. 8-B (2005), n.3, p. 767–771.
Unione Matematica Italiana
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Bollettino dell’Unione Matematica Italiana, Unione Matematica Italiana, 2005.
Bollettino U. M. I.
(8)8-B (2005), 767-771
Sensitivity analysis of solutions to a class of quasi-variational inequalities
SAMIRADLY- MOHAMEDAITMANSOUR- LAURASCRIMALI
Sunto.-- Si propone un risultato di sensitivitaÁ delle soluzioni di disequazioni quasi- variazionali finito-dimensionali del tipo:
(QVI) u2K(u);hC(u);v ui 0; 8v2K(u);
in presenza di perturbazioni dell'operatoreCe dell'insieme convessoK. In partico- lare, si prova la continuitaÁ HoÈlderiana degli insiemi delle soluzioni dei problemi perturbati intorno al problema iniziale. I risultati illustrati possono essere estesi anche al caso infinito-dimensionale.
Summary.--We provide a sensitivity result for the solutions to the following finite-di- mensional quasi-variational inequality
(QVI) u2K(u);hC(u);v ui 0; 8v2K(u);
when both the operatorCand the convexKare perturbed. In particular, we prove the HoÈlder continuity of the solution sets of the problems perturbed around the original problem. All the results may be extended to infinite-dimensional(QVI).
Sensitivity analysis plays a central role in variational inequality theory, since it arises in many different applied problems, which range from transportation theory to physics, from economics to finance.
For this reason, we were motivated to study the HoÈlder continuity of the solution sets of the parametric finite-dimensional (QVI), measuring the t- Hausdorff distance (see Attouch and Wets [2]) between the solution sets of the problems perturbed around the original problem.
Let E be a convex and compact subset of Rm, C:E!Rm, r:E!Rl, Kr:E!!Rm a set-valued map with convex and closed values and letAdenote an opportunelm matrix (landmare two given integers, m>l). We consider the following quasi-variational inequality:
(QVI) u2Kr(u);hC(u);v ui 0; 8v2Kr(u);
where
Kr(u): fv2E:Avr(u)g (see [4] for a survey on (QVI)).
In order to state the parametric (QVI), we assume that the operator C is subject to change, which can be seen in a general perturbation form by involving a parameterm;wherembelongs to a subset of a finite-dimensional spaceL, whose norm is denoted byk k. Thus, we consider the family of operatorsC(;m)mde- fined fromEintoRm:
The perturbation of constraints will be done with respect to the map r;
whereas the matrix A will be fixed. Specifically, we consider a kind of small perturbation of the map r as follows: r will be perturbed by a parameter l, element of a subsetMof a given Euclidean subspace, whose norm is also denoted byk k. For anymandl,V(m) andV(l) will denote a neighborhood ofmandl, respectively. The reference value of the parameters are given bymandl.
We also suppose thatrsatisfies a HoÈlder continuity assumption:
(h0) ris HoÈlder continuous, i.e., for someL1;L2>0 andj;j02]0;1[;
jrl(u) rl0(v)j L1kl l0kj0L2ju vjj 8u;v2E;8l;l02 V(l):
The family frlgl2V is considered as a perturbation of the initial r:
Hereinafter, for the sake of simplicity, the constraintsKrl(u) will be denoted by Kl(u), for every l2 V(l) and u2E. Therefore, the corresponding perturbed problem can be stated as follows
(QVIm;l) 9u(m;l)2Kl(u(m;l)): hC(u(m;l);m);v u(m;l)i 0;
8v2Kl(u(m;l)):
We suppose that the solution set of (QVIm;l) is nonempty and focus our attention on the local behavior of the set-valued mapS:LM!!Ewhich associates the setS(m;l) with each pair (m;l). Our aim is to prove the HoÈlder continuity of the solution mapSaround the reference value of the parameters (m;l).
First, we apply the result of Walkup and Wets (1969) [7], which can be seen as a particular case of famous Hoffman's Lemma, to derive the following:
LEMMA1. -- Let A be an lm-matrix, d1and d2be given vectors inRl:Let Si denote the solution set of the linear equality Axdi;i1;2:Then, there exists uu(A)>0such that for each x12S1there exists x22S2satisfying
jx1 x2j ujd1 d2j:
In the subsequent proposition we show that for HoÈlder continuous mapr;we 768 SAMIR ADLY - MOHAMED AITMANSOUR - LAURA SCRIMALI
can dispose of a HoÈlder-type behavior of the map (u;r)7!Kr(u); which is a stronger property than the well-known Aubin Lipschitz property, or the Pseudo- Lipschitz property as referred in the literature (see [3]).
PROPOSITION1. --Let us assume that(h0)holds. Then, there existk1; k2>0 such that8l;l02 V(l)and8u; v2E one has:
Kl(u)Kl0(v)(k1kl l0kj0k2ju vjj)Bm; 1
whereBmstands for the closed unit ball ofRm:
REMARK1. --We suppose that v are nontrivial and norm-bounded from be- low, namely for some f0>0and for all v2E:Enf0g;it results thatkvk f0:
Now, we are able to state our main result.
THEOREM1. -- Let us assume that(h0)holds and the following conditions are satisfied:
(h1) for some m>0anda2;C is uniformly(a;m)-strongly monotone, i.e., hC(u;m) C(v;m);u vi mju vja; 8u;v2E;8m2 V(m);
(h2) for some b0>0, C is uniformly b0-bounded, i.e., for allm2 V(m)and all u2E one hasjC(u;m)j b0;
(h3) for some g2]0;1[ and c>0; m7!C(:;m) is uniformly (in u) (g;c)- HoÈlder, i.e., for all u2E and allm;m02 V(m);
jC(u;m) C(u;m0)j ckm m0kg; (h4) m>(2k2)bb; wherebaj andbfbb01
0 ,
where f0 is the minimal value of v as in Remark 1 and k2 is given in Proposition1. Then, the solution map S is HoÈlder continuous around(m;l), i.e., there exist c1;c2>0;d1;d22]0;1[such that for somet>0
haust(S(m;l);S(m0;l0))c1km m0kd1c2kl l0kd2; 2
for allm;m02 V(m);l;l02 V(l):
It is worth noting that:
1. estimate (2) means that:
S(m;l)t S(m0;l0)(c1km m0kd1c2kl l0kd2)Bm: 3
SENSITIVITY ANALYSIS OF SOLUTIONS TO A CLASS ETC. 769
2. For any sequencesfmngn andflngn converging tom andl;respectively, denotingS(mn;ln) bySn andS(m;l) byS; (2) implies that:
n!1lim haust(Sn;S) 0:
4
Equivalently, for anyt>t0(t0>0 arbitrarily chosen) and anye>0;fornlarge enough the following conditions hold:
St SneBm and (Sn)tSeBm: 5
3. IfEis a finite-dimensional space, (4) forces fSngn to converge toS (the strong stability result) in the sense of PainleveÂ-Kuratowski, i.e.,
lim sup
n!1 SnS and Slim inf
n!1 Sn: 6
In other words, (6) implies:
upper stability, i.e.,whenever a sequence fungn of solutions to (QVI)(mn;ln) converges to some point, sayu; thenuis necessarily a solution to (QVI)m;l; lower stability, i.e., every solution u to (QVI)m;l can be approximated by a
sequence, sayfungn;of solutions to (QVI)(mn;ln):
The above results, which improve and extend some existing results in the literature (see [1, 3, 5, 6]), may be generalized to infinite-dimensional quasi- variational inequalities assigned in a time-dependent setting.
REMARK2. --As a practical application of the sensitivity result, we may refer to the traffic network equilibrium problem. In this case, the operator C represents the path cost operator depending on the distribution of flows through the network, andrdenotes the travel demand affected by the equilibrium flow u.
REFERENCES
[1] M. AITMANSOUR- H. RIAHI,Sensitivity analysis for abstract equilibrium problems, J. Math. Anal. App.,306No. 2 (2005), 684-691.
[2] H. ATTOUCH - R. WETS, Quantitative stability of variational systems: I. The epigraphical distance, Trans. Am. math. Soc.,328No. 2, (1991), 695-729.
[3] J.-P. AUBIN, Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res.,9(1984), 87-111.
[4] L. SCRIMALI, Quasi-Variational inequalities in Transportation networks, Math.
Models. Meth. Appl. Sci,14, No. 10 (2004), 1541-1560.
[5] A. SHAPIRO,Sensitivity analysis of generalized equations, Journal of Mathematical Sciences,115(2003), 2554-2565.
770 SAMIR ADLY - MOHAMED AITMANSOUR - LAURA SCRIMALI
[6] A. SHAPIRO,Sensitivity analysis of parameterized variational inequalities, Mathe- matics of Operations Research,30(2005), 109-126.
[7] D. W. WALKUP- R. J-B. WETS,A Lipschitzian of convex polyhedral, Proc. Amer.
Math. Society,23(1969), 167-178.
Laco, University of Limoges, 123, Avenue A. Thomas, 87000 Cedex France e-mail: samir.adly@unilim.fr mohamed.ait-mansour@unilim.fr DMI, UniversitaÁ di Catania, Viale A. Doria 6, 95125 Catania, Italy
e-mail: scrimali@dmi.unict.it
________
Pervenuta in Redazione il 20 maggio 2005
SENSITIVITY ANALYSIS OF SOLUTIONS TO A CLASS ETC. 771
