Elsevier

Nonlinear Analysis: Theory, Methods & Applications

Volume 63, Issues 5–7, 30 November–15 December 2005, Pages e1655-e1664
Nonlinear Analysis: Theory, Methods & Applications

Duality theory and optimality conditions for Generalized Complementarity Problems

https://doi.org/10.1016/j.na.2004.12.019Get rights and content

Abstract

In this paper Generalized Complementarity Problems are expressed in terms of suitable optimization problems and some necessary optimality conditions are given. The infinite dimensional Lagrangean and Duality Theories play an important role in order to achieve the main results.

Introduction

Let S be a nonempty subset of a real linear space X. Let Y be a partially ordered real normed space with the ordering cone C. Let Z be the set of nonnegative measurable functions and letL:SZ,B:SZbe two operators. Let g:SY be a given constraint mapping and let us setK={vS:g(v)-C}.Let us suppose that L(v)0B(v)0vSand let us observe that the Generalized Complementarity ProblemB(u)L(u)=0,uKexpresses many economic and physical equilibrium problems. In fact, starting from the classical Signorini problem, it has been observed that the Obstacle problem, the Elastic–Plastic Torsion problem, the Traffic Equilibrium problem both in the discrete and continuous cases, the Spatial Price Equilibrium problem, the Financial Equilibrium problem and many others (see [5], [9], [8], [15]) satisfy the Generalized Complementarity Problem (2). For example, the continuous traffic equilibrium problem fits very well with the above scheme assumingX=Ldiv2(Ω)={uL2(Ω,R2):divuL2(Ω)},Y=L2(Ω),S=Ldiv2(Ω),Bv=v,Lv=ci(x,v(x))-μxii=1,2.Problem (2) becomesci(x,u(x))-μ(x)xiui(x)=0uKi=1,2,a.e.inΩ,where μH1(Ω) is a given function (potential) and K is given by K={uLdiv2(Ω):ui(x)0,ui(x)|Ω=ϕi(x),divu+t(x)=0},with Ω a simply connected bounded domain in R2 with Lipschitz boundary Ω.

In this model, ui(x)i=1,2 represent the traffic density through a neighbourhood of x in the direction of the increasing axis xi. ui(x) has nonnegative fixed trace ϕi(x) on Ω which represents the entering flow. If we associate to each point xΩ a scalar field t(x)L2(Ω), which measures the density of the flow originating or terminating at x, the flow u(x) satisfies the conservation law: divu(x)+t(x)=0 a.e. in Ω. The function ci(x,v(x)) represents the travel cost density along the axis xi(i=1,2).

The equilibrium condition is the following one:

Definition 1

u(x)K is an equilibrium distribution flow if there exists a potential μH1(Ω) such that ci(x,u(x))-μ(x)xiui(x)=0,ci(x,u(x))-μ(x)xi0,i=1,2a.e.inΩ.

The potential μ measures the cost occurred when a user travels from the point x to the boundary Ω using the cheapest possible path (see [4], [6], [11]).

The same happens for the Elastic–Plastic Torsion Problem. In this case we have X=H2(Ω) (or H1(Ω) if we consider the weak formulation); Y=L2(Ω);Bv=1-i=1nvxi2Lv an elliptic operator. The equilibrium condition is; 1-i=1nuxi2Lu=0and K=vH01(Ω):v0,1-i=1nvxi20(see [12], [13]).

The Evolutionary Financial Time Equilibrium has been considered very recently and also it perfectly agrees with the above scheme. In this case a vector of sector assets, liabilities and instrument prices (x*(t),y*(t),r*(t))i=1mPi×L2([0,T],R+n), wherePi=(xi(t),yi(t))L2([0,T],R2n):j=1nxij(t)=si(t),j=1nyij(t)=si(t),xij(t),yij(t)0a.e.in[0,T],with si(t) the total financial volume held by sector i at the time t is an equilibrium of the evolutionary financial model if and only if it satisfies the system of equalities:xij*(t)[2[Q11i(t)]jTxi*(t)+2[Q21i(t)]jTyi*(t)-rj*(t)-μi(1)(t)]=0,yij*(t)[2[Q12i(t)]jTxi*(t)+2[Q22i(t)]jTyi*(t)+rj*(t)-μi(2)(t)]=0,i=1m(xij*(t)-yij*(t))rj*(t)=0with all the functions xij*(t), yij*(t), 2[Q11i(t)]jTxi*(t)+2[Q21i(t)]jTyi*(t)-rj*(t)-μi(1)(t), 2[Q12i(t)]jTxi*(t)+2[Q22i(t)]jTyi*(t)+rj*(t)-μi(2)(t), i=1m(xij*(t)-yij*(t)) nonnegative.

The meaning of this definition is the following one:

To each financial volume si(t) invested by sector i there are associated two functions μi(1)(t) and μi(2)(t) related to the assets and to the liabilities which represent the “Equilibrium Utilities” per unit of the sector i, respectively; 2[Q11i(t)]jTxi*(t)+2[Q21i(t)]jyi*(t)-rj*(t) is the personal utility of the investor in the instrument j as an asset. Then if this personal utility equals the equilibrium utility μi(1)(t), it results xij*(t)0, whereas if the personal utility is greater than the equilibrium utility μi(1)(t), it results xij*(t)=0. The meaning of the second condition is analogous, whereas the third one i=1m(xij*(t)-yij*(t))rj*(t)=0 states that if the price rj* of the instrument j is positive, then the amount of the assets is equal to the amount of liabilities, on the contrary if there is an excess supply of an instrument in the economy: i=1mxij*(t)>i=1myij*(t), then rj*(t)=0 (see [7]).

In this paper we observe that the Generalized Complementarity Problem (2) can be written as the Optimization ProblemminB(v)L(v)=0,vKand we investigate how we can associate to Problem (3), by means of Lagrangean and Duality Theories, some optimality conditions. For the sake of simplicity, we confine ourselves to a less general case. Let us suppose that X and Y are real Hilbert spaces with the usual inclusion XYX*; let it be C the ordering convex cone of Y and let L, B, g three functions defined on X with values in Y. Let us suppose that the set K={vX:g(v)-C} is nonempty and let us assume that the Generalized Complementarity Problem (3) holds in the sense of the scalar product on Y and that Lv,Bv0 vX. Then Problem (3) becomesminvKLv,Bv=0.The main result of this paper is the following:

Theorem 1

Let the function (L(v),B(v),g(v)) be convex-like. Let us assume that qri[g(X)+C] and cone(qri(g(X)+C))¯ is not a linear subspace of Y. In addition suppose that C is closed, C-C¯=Y and there exists v¯X such that g(v¯)-qriC. Then if the functions L, B, g are Fréchet differentiable and Problem (4) admits a solution uK, then there exists an element l¯C* such that Lu(u)v,B(u)+L(u),Bu(u)v+l¯,gu(u)v=0vXand l,g(u)0,lC*,l¯,g(u)=0.

Note that, in virtue of Proposition 6 in Section 2, qriC. Further, it is worth remarking that, taking into account that Lu(u)v,B(u), L(u),Bu(u)v, l¯,gu(u)v define three continuous linear mappings on the Hilbert space X, there exist three elements of X*, that we denote by B(u)Lu(u), L(u)Bu(u), l¯gu(u), such thatLu(u)v,B(u)+L(u),Bu(u)v+l¯,gu(u)v=B(u)Lu(u)+L(u)Bu(u)+l¯gu(u),v=0vX.Hence we derive the equivalent condition B(u)Lu(u)+L(u)Bu(u)+l¯gu(u)=0X*.We can generalize the result obtained in Theorem 1 assuming that the set of the constraints is given by K={vS:g(v)-C}, where S is a nonempty convex subset of X and that L(v), B(v) and g(v) are defined on S. In this case the following result holds:

Theorem 2

Let the function (L(v),B(v),g(v)) be convex-like. Let us assume that qri[g(S)+C] and cone(qri(g(S)+C))¯ is not a linear subspace of Y. In addition suppose that C is closed, C-C¯=Y and there exists v¯S such that g(v¯)-qriC. Then if the functions L, B, g are Fréchet differentiable and Problem (4) admits a solution uK, then there exists an element l¯C* such that L(u),Bu(u)(v-u)+Lu(u)(v-u),B(u)+l¯,gu(u)(v-u)0vSand l,g(u)0,lC*,l¯,g(u)=0.

Moreover, we observe that the above technique is complementary to the study of the generalized complementarity problems by means of variational inequalities.

Finally we would like to mention that the results of Theorem 1 have been presented at the International Conference “Variational Analysis and Applications” and an abridged version of these results has been published in [10].

Section snippets

The Lagrangean and duality theory

Let us introduce the dual cone C* C*={lY*:l(v)0vC}that, in virtue of the usual identification Y=Y*, can be rewritten C*={lY:l,v0,vC}.Then, using the same technique used by Jhan in [14], it is possible to show the following result:

Theorem 3

Let the ordering cone C be closed. Then u is a minimal solution of (4) if and only if u is a solution of the problemminvXsuplC*{Lv,Bv+l,g(v)}.and the extremal values of the two problems are equal.

Now let us introduce the Dual ProblemmaxlC*infvX{Lv,Bv

Proof of Theorem 1

Let us consider the Lagrangean functional L:X×C*R L(v,l)=L(v),B(v)+l,g(v).Using the preceding theorems we are able to state the following.

Theorem 7

Let the assumptions of Theorem 6 be fulfilled, with C closed. Then a point (u,l¯)X×C* is a saddle point of L, namelyL(u,l)L(u,l¯)L(v,l¯),vX,lC*if and only if u is a solution of Problem (4) (or (5)), l¯ is a solution of Problems (6) and (7) holds, namelyminvXsuplC*{Lv,Bv+l,g(v)}=maxlC*infvX{Lv,Bv+l,g(v)}=Lu,Bu+l¯,g(u)=0.

(see for

Proof of Theorem 2

Assume that L, B and g are defined on S and that K={vS:g(v)-C}. The following version of Theorem 6 holds.

Theorem 8

Let the function (L(v),B(v),g(v)) be convex-like with respect to the product cone R+×C in R×Y. Let qri[g(S)+C] and cone(qri(g(S)+C))¯ be not a linear subspace of Y. In addition suppose that qriC and C-C¯=Y. If Problem (4) is solvable and there exists v¯S with g(v¯)-qriC, then also Problem (6), with X replaced by S, is solvable and the extremal values of the two problems are equal.

References (15)

  • S. Dafermos

    Continuum modeling of transportation networks

    Transportation Res.

    (1980)
  • J.M. Borwein et al.

    Notions of relative interior in Banach space. Optimization and related topics

    J. Math. Sci. (N.Y.)

    (2003)
  • J.M. Borwein, A.S. Lewis, Practical conditions for Fenchel duality in infinite dimensions, in: J.B. Baillon, M. Thera...
  • F. Cammaroto et al.

    Separation theorem based on the quasirelative interior and application to duality theory

    J. Optim. Theory Appl.

    (2005)
  • P. Daniele, F. Giannessi, A. Maugeri (Eds.), Equilibrium Problems and Variational Models, Kluwer Academic Publishers,...
  • P. Daniele et al.

    Variational inequalities and the continuum model of transportation problems

    Int. J. Nonlinear Sci. Numer. Simul.

    (2003)
  • P. Daniele, Variational inequalities for evolutionary financial equilibrium, in: A. Nagurney (Ed.), Innovations in...
There are more references available in the full text version of this article.

Cited by (13)

View all citing articles on Scopus
View full text