Nonlinear Analysis: Theory, Methods & Applications
Duality theory and optimality conditions for Generalized Complementarity Problems
Introduction
Let S be a nonempty subset of a real linear space . Let Y be a partially ordered real normed space with the ordering cone . Let Z be the set of nonnegative measurable functions and letbe two operators. Let be a given constraint mapping and let us setLet us suppose that and let us observe that the Generalized Complementarity Problemexpresses 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 assumingProblem (2) becomeswhere is a given function (potential) and is given by with a simply connected bounded domain in with Lipschitz boundary .
In this model, represent the traffic density through a neighbourhood of x in the direction of the increasing axis . has nonnegative fixed trace on which represents the entering flow. If we associate to each point a scalar field , which measures the density of the flow originating or terminating at , the flow satisfies the conservation law: a.e. in . The function represents the travel cost density along the axis .
The equilibrium condition is the following one: Definition 1 is an equilibrium distribution flow if there exists a potential such that
The same happens for the Elastic–Plastic Torsion Problem. In this case we have (or if we consider the weak formulation); an elliptic operator. The equilibrium condition is; and (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 , wherewith 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:with all the functions , , , , nonnegative.
The meaning of this definition is the following one:
To each financial volume invested by sector i there are associated two functions and related to the assets and to the liabilities which represent the “Equilibrium Utilities” per unit of the sector i, respectively; is the personal utility of the investor in the instrument j as an asset. Then if this personal utility equals the equilibrium utility , it results , whereas if the personal utility is greater than the equilibrium utility , it results . The meaning of the second condition is analogous, whereas the third one states that if the price 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: , then (see [7]).
In this paper we observe that the Generalized Complementarity Problem (2) can be written as the Optimization Problemand 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 ; let it be C the ordering convex cone of Y and let , , g three functions defined on X with values in . Let us suppose that the set is nonempty and let us assume that the Generalized Complementarity Problem (3) holds in the sense of the scalar product on Y and that . Then Problem (3) becomesThe main result of this paper is the following: Theorem 1 Let the function be convex-like. Let us assume that and is not a linear subspace of Y. In addition suppose that C is closed, and there exists such that . Then if the functions , , g are Fréchet differentiable and Problem (4) admits a solution , then there exists an element such that and
Note that, in virtue of Proposition 6 in Section 2, . Further, it is worth remarking that, taking into account that , , define three continuous linear mappings on the Hilbert space X, there exist three elements of , that we denote by , , , such thatHence we derive the equivalent condition We can generalize the result obtained in Theorem 1 assuming that the set of the constraints is given by , where S is a nonempty convex subset of X and that , and are defined on S. In this case the following result holds: Theorem 2 Let the function be convex-like. Let us assume that and is not a linear subspace of Y. In addition suppose that C is closed, and there exists such that . Then if the functions , , g are Fréchet differentiable and Problem (4) admits a solution , then there exists an element such that and
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 that, in virtue of the usual identification , can be rewritten 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 problemand the extremal values of the two problems are equal.
Proof of Theorem 1
Let us consider the Lagrangean functional 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 is a saddle point of , namelyif and only if u is a solution of Problem (4) (or (5)), is a solution of Problems (6) and (7) holds, namely
Proof of Theorem 2
Assume that , and g are defined on S and that . The following version of Theorem 6 holds. Theorem 8 Let the function be convex-like with respect to the product cone in . Let and be not a linear subspace of Y. In addition suppose that and . If Problem (4) is solvable and there exists with , then also Problem (6), with X replaced by S, is solvable and the extremal values of the two problems are equal.
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