Time-Dependent Equilibrium Problems - CiteSeerX

Time-Dependent Equilibrium Problems Antonino Maugeri1 and Carmela Vitanza2 1

2

Department of Mathematics and Computer Sciences, University of Catania, Catania, Italy [email protected] Department of Mathematics, University of Messina, Messina, Italy [email protected]

Abstract The paper presents variational models for dynamic traffic, dynamic market, and evolutionary financial equilibrium problems taking into account that the equilibria are not fixed and move with time. The authors provide a review of the history of the variational inequality approach to problems in physics, traffic networks, and others, then they model the dynamic equilibrium problems as time-dependent variational inequalities and give existence results. Moreover, they present an infinite dimensional Lagrangean duality and apply this theory to the above time-dependent variational inequalities.

Key words: time-dependent variational inequalities, dynamic traffic, dynamic market and evolutionary financial equilibrium problems, Lagrangean duality.

1 Introduction The scientific life of the theory of Variational Inequalities has revealed itself full of events and surprises. This theory arose in the 1970s as an innovative and effective method to solve a group of equilibrium problems originated from mathematical physics as the Signorini problem, the obstacle problem, and the elastic-plastic torsion problem, and it is still an open question to decide who must be considered the founder between G. Fichera and G. Stampacchia, who first dealt with Variational Inequalities (see [10] and [15]). The critical point for which the other theories, available in the literature, have revealed themselves unable to solve the above-mentioned problems is that these problems request a condition of complementarity type on the boundary or on a part of the set where the problems are defined, and, in general, it is not possible to express them as an optimization problem. After an intense period of successes and of fundamental results obtained by means of the Variational Inequality theory, which someone defines as the Italian way of mathematics, maybe in consequence of the untimely death of

250

A. Maugeri, C. Vitanza

G. Stampacchia in 1979, the interest for Variational Inequalities declined and it seemed that the theory had no more to say. On the contrary, in the beginning of the 1980s, it was proved by M.J. Smith (see [22]) and S. Dafermos (see [2]) that the traffic network equilibrium problem can be formulated in terms of a finite-dimensional Variational Inequality and, hence, it is possible to study in this way existence, uniqueness, stability of traffic equilibria, and to compute the solutions. In consequence of this fact, the past decades have witnessed an exceptional interest for Variational Inequalities, and an enormous amount of papers and books have been devoted to this topic. As a relentless river, more and more problems arising from the economic world, as the spatial price equilibrium problem, the oligopolistic market equilibrium problem, the migration problem, and many others (see [19]), are formulated in terms of a finite dimensional Variational Inequality and, by means of this theory, solved. The last event goes back to the end of the 1990s: the traffic network equilibrium problem with feasible path flows that have to satisfy time-dependent capacity constraints and demands has been formulated in [3] and [4] (see also [11]) as an evolutionary Variational Inequality, for which existence theorems and computational procedures are given. Starting from this first result, many other problems with time-dependent data have been formulated in the same terms. In [5] and [6], the authors consider the spatial price equilibrium problem when the prices and the commodity shipment bounds vary over the time. [8] addresses the time-dependent spatial price equilibrium problem in which the variables are commodity shipments. In [7] and [9], the authors consider a time-depending financial network model consisting of multiple sectors, each of which seeks to determine its optimal portfolio given time-dependent supplies of the financial holdings. Although in the theory of Variational Inequalities an important chapter is constituted by parabolic or hyperbolic Variational Inequalities, the models that formulate the above problem are different from the previous ones and then they request an appropriate study and an improvement of some aspects of Variational Analysis. All these problems have a common element: their equilibrium conditions can be handled as generalized complementarity problems and moreover the evolutionary Variational Inequality formulation can be expressed in a unified way (see [1]). The aim of this paper is to present the essential aspects of the problems considered and to focus on the new questions that the evolutionary framework provides.

2 Time-Dependent Equilibrium Conditions and Evolutionary Variational Inequalities The driving forces of the problems that we examine are considered timedependent on a fixed time interval [0, T ]. Consequently, the response of the system is time-dependent, too. Here the system is assumed to respond to

Time-Dependent Equilibrium Problems

251

changes of the driving forces so gradually in the considered timescale that, at each instant, equilibrium conditions prevail. However, we can consider models with presence of delay effects on the response (see [21]), but in this paper we will just mention this subject. We start considering a model of a traffic network on a finite directed graph (see [3] and [4]). There is given a set W of origin-destination pairs and a set R of routes. Each route r ∈ R links some origin-destination pair w ∈ R. This leads to the set R(w) of all w ∈ W. The topology of the network is described by the pair-route incidence matrix Φ = {Φw,r } with w ∈ W, r ∈ R, where 1 if the route r connects the pair w Φw,r = 0 otherwise. Because the feasible flows have to satisfy time-dependent capacity constraints and demand requirements, the flow vectors are time-dependent flow vectors f (t) ∈ RR , where t varies in the fixed time interval T = [0, T ], while the topology remains fixed. Each component fr (t) of f (t) gives the flow trajectory f : T → RR , which have to satisfy almost everywhere on T the capacity constraints λ(t) ≤ f (t) ≤ µ(t) and the so-called “traffic conservation law”: Φ f (t) = ρ(t), where the bounds λ ≤ µ and the demand ρ = (ρw )w∈W ≥ 0 are given. Considering a Lp setting with p ∈ (1, ∞), we assume that λ and µ ∈ Lp (T , RR ) and that ρ lies in Lp (T , RW ). Assuming in addition that Φ λ(t) ≤ ρ(t) ≤ Φ µ(t) a.e. on T , we obtain that the set of feasible flows K = {f ∈ E : λ(t) ≤ f (t) ≤ µ(t), Φ f (t) = ρ(t) a.e. on T }

(1)

is nonempty (see [13]). Clearly K is convex and weakly compact. The cost trajectory C, which assigns to each flow trajectory

f ∈ K the 1 1 ∗ q R + =1 cost trajectory C(f ), is a mapping C : K → E = L (T , R ) p q and it results C(f ), g != C(f (t)), g(t) dt = Cs (f ) gs (t) dt. T

T s∈R

The equilibrium condition is given by a generalized version of Wardrop’s condition, namely: Definition 1. h ∈ K is an equilibrium flow if and only if, for all w ∈ W and r, s ∈ R(w) and a.e. on T there holds: Cr (h)(t) < Cs (h)(t) =⇒ hr (t) = µr (t) or hs (t) = λs (t).

(2)

252


We remark that the kind of equilibrium defined by condition (2) is different from the one obtained considering a minimization of an objective, like total cost set by society or some authority. The equilibrium approach defined by (2) is called user-oriented traffic equilibrium and has the meaning that every agent in traffic strives for his individual cost and it, when abandoning artificial assumptions of symmetry and thus abandoning the existence of a potential, cannot be formulated as simple optimization problems. The overall flow pattern obtained according to condition (2) fits very well in the framework of the theory of Variational Inequality. In fact in [3] and [4], the following result is shown: Theorem 1. h ∈ K is an equilibrium solution according to Definition 1 if and only if h is a solution to the following Variational Inequality C(h), f − h !=

T

“Find h ∈ K : C(h(t)), f (t) − h(t) dt ≥ 0

∀f ∈ K.”

(3)

The next equilibrium conditions that we present are those of the spatial price equilibrium problem in the case of the price formulation. In this case, we have n supply markets P1 , P2 , . . . , Pn and m demand markets Q1 , Q2 , . . . , Qm of a commodity m, whose geometry remains fixed during the interval of time T = [0, T ]. For each t ∈ T we have: the the the the the the

supply price vector p(t) ∈ Rn ; total supply vector g(t) ∈ Rn ; demand price vector q(t) ∈ Rm ; total demand vector f (t) ∈ Rm ; flow vector x(t) ∈ Rnm ; unit cost vector c(t) ∈ Rnm .

The feasible vectors u(t) = (p(t), q(t), x(t)) have to satisfy the time-dependent constraints on prices and transportation flows, namely u(t) ∈

n $ n $ n / m % / % / / ) * pi (t), pi (t) × q j (t), q j (t) × xij (t), xij (t) i=1

j=1

i=1 j=1

where pi (t), pi (t), q j (t), q j (t), xij (t), xij (t) are given. The functional setting for the trajectories u(t) is the Hilbert space L = L2 (T , Rn ) × L2 (T , Rm ) × L2 (T , Rnm ) and, hence, the set of feasible vectors u(t) is given by K = K1 × K2 × K3 = {p ∈ L2 (T , Rn ) : 0 ≤ p(t) ≤ p(t) ≤ p(t) a.e. on T } × {q ∈ L2 (T , Rm ) : 0 ≤ q(t) ≤ q(t) ≤ q(t) a.e. on T } × {x ∈ L2 (T , Rnm ) : 0 ≤ x(t) ≤ x(t) ≤ x(t) a.e. on T },


253

where p(t), p(t) ∈ L( T , Rn ), q(t), q(t) ∈ L( T , Rm ), x(t), x(t) ∈ L( T , Rnm ). K is a convex, closed, weakly compact set. Furthermore, we are giving the mappings: g = g(t, p(t)) : T × K1 → L2 (T , Rn ) f = f (t, q(t)) : T × K2 → L2 (T , Rm ) c = c(t, x(t)) : T × K3 → L2 (T , Rnm ) which, at time t, assign to each price trajectory p ∈ K1 and q ∈ K2 the supply g ∈ L2 (T , Rn ) and the demand f ∈ L2 (T , Rm ), respectively, and to the flow trajectory x ∈ K3 the cost c ∈ L2 (T , Rnm ). Now, we allow that, during the activities of the market in the time interval [0, T ], supply and demand excesses can occur, namely that there exists n non-negative functions si (t) i = 1, 2 . . . , n and m non-negative functions tj (t) j = 1, . . . , m such that gi (t, p(t)) =

m

xij (t) + si (t) i = 1, 2, . . . , n

(4)

xij (t) + tj (t)

(5)

j=1

fj (t, q(t)) =

n

j = 1, 2, . . . , m.

i=1

The equilibrium conditions of this evolutionary market take the following form: Definition 2. u(t) = (p(t), q(t), x(t)) ∈ L is a dynamic market equilibrium if and only if for each i = 1, 2, . . . , n and j = 1, 2, . . . , m and a.e. in T there hold: =⇒ pi (t) = pi (t) si (t) > 0 i = 1, 2, . . . , n; (6) pi (t) < pi (t) < pi (t) =⇒ si (t) = 0 tj (t) > 0

=⇒ qj (t) = q j (t)

j = 1, 2, . . . , m; q j (t) < qj (t) < q j (t) =⇒ tj (t) = 0 ⎧ > qj (t) if xij (t) = xij (t) ⎪ ⎪ ⎪ ⎪ ⎨ pi (t) + cij (t, x(t)) = qj (t) if xij (t) ≤ xij (t) ≤ xij (t) ⎪ ⎪ ⎪ ⎪ ⎩ < qj (t) if xij (t) = xij (t).

(7)

(8)

Conditions (6) and (7), in a reasonable way, are satisfied when the excesses vanish in dependence of the prices; conditions (8) control the amounts of commodity shipments between the supply and the demand markets according to the equilibrium condition that the supply price plus the transportation cost is greater, equal, or less than the demand price.

254


Denoting by v : T × K → L the operator defined setting ⎛⎛ = ⎝⎝gi (t, p(t)) −

m j=1

⎞

v = v(t, u(t))

xij (t)⎠

, fj (t, q(t)) −

n

xij (t)

i=1

i=1,...,n

(pi (t) + cij (t, x(t)) − qj (t)) i=1,...,n

, j=1,...,m

,

(9)

j=1,...,m

also here the following characterization in terms of Variational Inequalities holds (see [5] and [6]): Theorem 2. u(t) = (p(t), q(t), x(t)) ∈ K is a dynamic market equilibrium if and only if u(t) is a solution to T v(t, u(t)), u ˜(t) − u(t) dt v(u), u ˜ − u != 0 ⎧ ⎛ ⎞ T ⎨ n m ⎝gi (t, p(t)) − xij (t)⎠ (˜ pi (t) − pi (t)) = 0 ⎩ i=1 j=1 m n − xij (t) (˜ qj (t) − qj (t)) fj (t, q(t)) − j=1

+

n m

i=1

(pi (t) + cij (t, x(t)) − qj (t)) (˜ xij (t) − xij (t))

dt ≥ 0

i=1 j=1

∀˜ u = (˜ p, q˜, x ˜) ∈ K.

(10)

For what concerns the quantity formulation of the spatial price equilibrium problem, in this case the only change is that the supply prices pi and the demand prices qj are considered as functions of the supply g and the demand f and the equilibrium conditions are related to a vector w(t) = (g(t), f (t), x(t), s(t), t(t)), which represents the variables of the model. More precisely, we are giving two mappings p = p(t, g(t)) : T × L2 ([0, T ], Rn+ ) → 2 m L2 ([0, T ], Rn+ ) and q = q(t, f (t)) : T × L2 ([0, T ], Rm + ) → L ([0, T ], R+ ), which assign to each supply g(t) the supply price p(t, g(t)) and to each demand f (t) the demand price q(t, f (t)). We assume that capacity constraints on p, q and the transportation cost c(t, x(t)) are fixed in such a way that: p(t) ≤ p(t, g(t)) ≤ p(t),

q(t) ≤ q(t, f (t)) ≤ q(t),

c(t) ≤ c(t, x(t)) ≤ c(t). The set of feasible vectors w(t) is given by


255

K=

w(t) = (g(t), f (t), x(t), s(t), t(t)) ∈

L2 ([0, T ], Rn ) × L2 ([0, T ], Rm ) × L2 ([0, T ], Rn ) × L2 ([0, T ], Rm ) : w(t) ≥ 0 a.e. in [0, T ]; m gi (t) = xij (t) + si (t), i = 1, . . . , n; j=1

fj (t) =

n

xij (t) + tj (t),

j = 1, . . . , m a.e. in [0, T ]

(11)

j=1

and the dynamic market equilibrium conditions in the case of the quantity formulation take the following form: Definition 3. w∗ (t) ∈ K is a dynamic market equilibrium if and only if for each i = 1, . . . , n and j = 1, . . . , m and a.e. in [0, T ] there hold: if s∗i (t) > 0,

then pi (t, g ∗ (t)) = pi (t);

if pi (t) < pi (t, g ∗ (t)), then s∗i (t) = 0;

(12)

then qj (t, f ∗ (t)) = q j (t);

if tj (t) > 0,

if qj (t, f ∗ (t)) < q j (t), then t∗j (t) = 0;

(13)

if x∗ij (t) > 0, then pi (t, g ∗ (t)) + cij (t, x∗ (t)) = qj (t, f ∗ (t)); if pi (t, g ∗ (t)) + cij (t, x∗ (t)) > qj (t, f ∗ (t)), then x∗ij (t) = 0.

(14)

Then in [8], [16], [17] the following result is shown: Theorem 3. w∗ ∈ K is a dynamic market equilibrium if and only if w∗ is a solution to the Variational Inequality Find w∗ ∈ K such that T ∗ ∗ v(t, w∗ (t)), w(t) − w∗ (t) dt v(w ), w − w != = 0

0 T

p(t, g ∗ (t)), g(t) − g ∗ (t) − q(t, f ∗ (t)), f (t) − f ∗ (t)

+c∗ (t, x∗ (t)), x(t) − x∗ (t) − p(t), s(t) − s∗ (t) +q(t), t(t) − t∗ (t) dt ≥ 0 ∀w ∈ K.

(15)

Here v denotes the operator v(t, w) = (p(t, g(t)), −q(t, f (t)), c(t, x(t)), −p(t), q(t)).

256


Now we pass to present the evolutionary financial equilibrium conditions and the equivalent variational inequality formulation. We consider a multisector, multiinstrument financial equilibrium problem with a general utility function and including policy interventions in form of taxes and price controls. Then we have m sectors, with a typical sector denoted by i, and n instruments, with a typical financial instrument denoted by j, in the period [0, T ]. Let si (t) be the total financial volume held by sector i at the time t. xij denotes the amount of instrument j held as an asset in sector i’s portfolio, yij the amount of instrument j held as liability in sector i’s portfolio. The assets xij in sector i’s portfolio are grouped into the column vector xi (t) and the sector asset vectors into the matrix x(t); similarly, yi (t) denotes the column vector of the liabilities in sector i’s portfolio and y(t) the matrix of the sector liability vectors. The instrument prices rj (t) are variables of the problem but are fixed instrument floor prices rj (t) and instrument ceiling prices rj (t), which represent the form of policy interventions; r(t), r(t), r(t) denote the column vectors of the prices, of the floor prices, and of the ceiling prices, respectively. Moreover, the policy interventions act by imposing a tax rate τij (t) on sector i’s net yield on financial instrument j. We assume that the tax rates in this model have the flexibility of adjusting the tax rate following the evolution of the system. Then, assuming as the functional setting the Lebesgue space L2 ([0, T ], Rp ), the set of feasible assets and liabilities for the sector i becomes

9 : xi (t) ∈ L2 ([0, T ], R2n ) : Pi = yi (t) n

xij (t) = si (t),

j=1

n

yij (t) = si (t) a.e. in [0, T ],

j=1

xij (t) ≥ 0, yij (t) ≥ 0 a.e. in [0, T ] and the set of feasible instrument prices is R = r(t) ∈ L2 ([0, T ], Rn ) : rj (t) ≤ rj (t) ≤ rj (t), j = 1, . . . , n a.e. in [0, T ] , where r(t) and r(t) are assumed to belong to L2 ([0, T ], Rn ). We introduce for each sector i a utility function Ui (t, xi (t), yi (t), r(t)), which is constituted by two terms (see [9] and [19]): Ui (t, xi (t), yi (t), r(t)) n rj (t) − rj (t) (1 − τij (t)) (xij (t) − yij (t)) (16) = ui (t, xi (t), yi (t)) + j=1


257

The first term is connected with the opposite of the risk-aversion and an example of this type of function is the well-known one used in the quadratic model (see [19] and [7]): 9 ui (t, xi (t), yi (t)) = −

xi (t) yi (t)

:T

9 Qi (t)

xi (t) yi (t)

:

where Qi (t) is a 2n × 2n matrix, which, following the concept that assessment of risk is based on a variance-covariance matrix denoting the sector’s assessment of the standard deviation of prices for each instrument, represents a measure of this aversion. In the general case, we require a lot of qualitative assumptions on ui (t, xi (t), yi (t)). Precisely, we require that ui (t, xi (t), yi (t)) is defined and concave on [0, T ] × Rn × Rn , is measurable in t, and continuous with respect ∂ui ∂ui to xi and yi . Moreover, we assume that and exist and they are ∂xij ∂yij measurable in t and continuous with respect to xi and yi . Further, we require that the following growth conditions hold: |ui (t, x, y)| ≤ αi (t) x y,

∀x, y ∈ Rn , (17)

a.e. in [0, T ], i = 1, . . . , m; 0 0 0 0 0 0 0 ∂ui (t, x, y) 0 0 ≤ βij y, 0 ∂ui (t, x, y) 0 ≤ γij x, 0 0 0 0 0 ∂xij ∂yij

(18)

i = 1, . . . , m; j = 1, . . . , n, where αi , βij , γij are non-negative functions of L∞ ([0, T ]). The second term expresses the request to maximize the value of the asset holding and to minimize the value of the liabilities. Moreover, the second term incorporates the the presence of the (1 − τij (t)) term premultiplying the tax rate through rj (t) − rj (t) (xij (t) − yij (t)) . We can provide the following definition of an evolutionary financial equilibrium. Definition 4. A vector of sector assets, liabilities, and instrument prices m / Pi × R is an equilibrium of the evolutionary financial (x∗ (t), y ∗ (t), r∗ (t)) ∈ i=1

model if and only if it satisfies the system of inequalities: −

∂ui (t, x∗i (t), yi∗ (t)) (1) − (1 − τij (t)) rj∗ (t) − rj (t) − µi (t) ≥ 0 ∂xij

∂ui (t, x∗i (t), yi∗ (t)) (2) − − (1 − τij (t)) rj∗ (t) − rj (t) − µi (t) ≥ 0, ∂yij

(19)

258


and equalities: 9 : ∂ui (t, x∗i (t), yi∗ (t)) (1) − (1 − τij (t)) rj∗ (t) − rj (t) − µi (t) = 0 x∗ij (t) − ∂xij 9 : ∂ui (t, x∗i (t), yi∗ (t)) (2) ∗ yij (t) − − (1 − τij (t)) rj∗ (t) − rj (t) − µi (t) = 0, ∂yij (20) (1) (2) where µi (t), µi (t) ∈ L2 ([0, T ]) are Lagrangean functions, for all sectors i : i = 1, . . . , m and for all instruments j : j = 1, . . . , n and verifies the condition: ⎧ m ⎨ ≤ 0 if rj∗ (t) = rj (t) ∗ ∗ = 0 if rj (t) < rj∗ (t) < rj (t) (1 − τij (t)) xij (t) − yij (t) (21) ⎩ i=1 ≥ 0 if rj∗ (t) = rj (t). The meaning of Definition 4 is that to each financial volume si (t) held by the (1) (2) sector i, we associate the functions µi (t), µi (t), related, respectively, to the assets and to the liabilities and that represent the “equilibrium disutilities” for unit of the sector i. The financial volume invested in the instrument j as assets x∗ij (t) is greater or equal to zero if the j-th component −

∂ui (t, x∗i (t), yi∗ (t)) − (1 − τij (t)) rj∗ (t) − rj (t) ∂xij (1)

of the disutility is equal to µi (t), whereas if −

∂ui (t, x∗i (t), yi∗ (t)) (1) − (1 − τij (t)) rj∗ (t) − rj (t) > µi (t), ∂xij

then x∗ij (t) = 0. The same occurs for the liabilities. (1)

(2)

The functions µi (t) and µi (t) are Lagrangean functions associated, respectively, with the constraints n j=1

(xij (t) − si (t)) = 0 and

n

(yij (t) − si (t)) = 0.

j=1

They are not known a priori, but this has not influence, as Definition 4 is (1) (2) equivalent to a Variationsl Inequality in which µi (t) and µi (t) do not appear, as the following theorem shows: Theorem 4. A vector (x∗ (t), y ∗ (t), r∗ (t)) ∈

m /

Pi × R is an evolutionary

i=1

financial equilibrium if and only if it satisfies the following Variational Inequality:


Find (x∗ (t), y ∗ (t), r∗ (t)) ∈

Pi × R such that

i=1

: n 9 ∗ ∂ui (t, x∗i (t), yi∗ (t)) − (1 − τij (t)) rj (t) − rj (t) − ∂xij 0 j=1 * ) × xij (t) − x∗ij (t) : n 9 ∂ui (t, x∗i (t), yi∗ (t)) + − (1 − τij (t)) rj∗ (t) − rj (t) − ∂yij j=1 * ) ∗ (t) × yij (t) − yij n ∗ ) * ∗ ∗ xij (t) − yij (t) × rj (t) − rj (t) dt ≥ 0 +

m i=1

m /

259

T

j=1

∀(x, y, r) ∈

m /

Pi × R.

(22)

i=1

Now, if we give a look to the variational inequalities and to the underlying constraint sets that express the above equilibrium problems, we are led to conclude that all these problems can be formulated in a unified way. In fact, let us consider the nonempty, convex, closed, bounded subset of L2 ([0, T ], Rq ) given by: K = u ∈ L2 ([0, T ], Rq ) : λ(t) ≤ u(t) ≤ µ(t) a.e. in [0, T ]; q

ξi ui (t) = ρ(t) a.e. in [0, T ], ξi ∈ {−1, 0, 1} , i ∈ {1, . . . , q} .

(23)

i=1

For chosen values of the scalars ξi , of the dimension q, and of the boundaries λ, µ, we obtain each of the previous above cited constraint sets (see [1] for details). Therefore, we obtain the following standard form for the above cited problems: Find u ∈ K such that T F (t, u(t)), v(t) − u(t) dt ≥ 0 ∀v ∈ K, F (u), v − u !=

(24)

0

where K is given by (23) and F is a mapping from [0, T ]×K onto L2 ([0, T ], Rq ). Further, it directly derives from the proofs of Theroems 1–4 that problem (24) is also equivalent to the following one: Find u ∈ K such that F (t, u(t)), v(t) − u(t)

∀v ∈ K, a.e. in [0, T ],

which can be useful for computational purpose.

(25)

260


3 Qualitative Results As observed in [3] and [4], there are two standard approaches to the existence of equilibria, namely, with and without a monotonicity requirement. We shall employ the following definitions. F : [0, T ] × K → L2 ([0, T ], Rq ) is said to be •

pseudomonotone if and only if for all u, v ∈ K F (u), v − u !≥ 0 =⇒

• •

F (v), v − u !≥ 0;

hemicontinuous if and only if for all v ∈ K, the function u → F (u), v − u ! is upper semicontinuous on K; hemicontinuous along line segments if and only if for all u, v ∈ K, the function w → F (w), v−u ! is upper semicontinuous on the line segment [x, y].

The following general result holds: Theorem 5. Let F : [0, T ]×K → L2 ([0, T ], Rq ) and K ⊆ L2 ([0, T ], Rq ) convex and nonempty. Assume that (a) there exists A ⊆ K nonempty, compact and B ⊆ K compact, convex such that, for every u ∈ K \ A, there exists v ∈ B with F (u), v − u !< 0; and that either (b) or (c) below holds: (b) F is hemicontinuous; (c) F is pseudomonotone and hemicontinuous along line segments. Then there exists u ∈ K such that

F (u), v − u !≥ 0, ∀v ∈ K.

We may apply this result with K given by (23). Then K is convex, closed, and bounded, hence weakly compact. So, if we endow L2 ([0, T ], Rq ) with the weak topology, then K is compact and condition (a) in Theorem 5 is automatically satisfied by choosing A = K and B = ∅. If we endow the space L2 ([0, T ], Rq ) with the strong topology, condition (a) must be used (we can avoid the request of convexity of K, as observed in [4]) as well as (b). Finally, because weak and strong topology coincide on line segments, condition (c) is enough to ensure the existence of a solution. Now let us suppose that F is a Carathéodory function, namely that F (t, u) is measurable in t and continuous with respect to u and that the following condition holds: (26) F (t, u)Rq ≤ f (t) + α(t) uRq with f (t) ∈ L2 ([0, T ]), α(t) ∈ L∞ ([0, T ]). Then it is possible to show (see [6], Theorem 3) that F is hemicontinuous. In consequence of this fact, we get the following existence result.


261

Theorem 6. Assume that condition (26) holds. Then each of the following conditions is sufficient for the existence of a solution to the Variational Inequality (24): 1. There exist A ⊆ K nonempty, compact and B ⊆ K compact such that, for every u ∈ K \ A, there exists v ∈ B with F (u), v − u !< 0; 2. F is pseudomonotone; 3. F is hemicontinuous with respect to the weak topology. Interesting problems concerning the qualitative study of solutions to the Variational Inequality (24) are the stability and sensitivity analysis and the so-called regularization theory of solutions. The sensitivity analysis tries to clarify the behavior of solutions when some changes in the data occur, and the aim of the stability analysis is to check if a small change in the mapping F produces a small change in the solution. Some results in these fields can be found in [8, 17, 21]. The regularization theory deals with the problem to see if, imposing that the data fulfill some regularity assumptions, as H¨ older-continuity, differentiability, and so on, the solutions to (24) verify in turn these major properties. For example, in [13], Section 2.1, the author asks whether the solution to (24) (or (25)) can be in C([0, T ], Rq ). Even if some partial results are available, the question is still open.

4 Lagrangean and Duality Theory It is worth remarking that the Lagrangean theory provides interesting contributions, absolutely necessary for the better understanding and handling of the equilibrium problems considered. In fact, not only do the Lagrangean variables have a meaning intrinsic to the nature of the problems considered, but also the Lagrangean theory is essential in order to obtain the equivalence between the equilibrium conditions and a Variational Inequality. However, in our infinite dimensional setting, new problems arise with respect to the finite dimensional Lagrangean theory. The crucial difference with respect to the finite dimensional setting is that the interior of the cone (27) C = v ∈ L2 ([0, T ], Rq ) : v(t) ≥ 0 a.e. in [0, T ] is empty and, as a consequence, the separation theorems as well as the socalled Slater regularity assumption do not hold. Then one can try to overcome this difficulty either introducing the new concept of quasi-relative interior and proving separation theorems by means of this new concept (see [13] for details and applications) or using a more general regularity assumption that does not require any condition on the interior of C. We will follow this second way and, to this end, let us consider the Variational Inequality (24) and let us introduce the following function:

262


L(v, l1 , l2 , m) = Ψ (v) − l1 (t), v(t) − λ(t) dt Ω + l2 (t), v(t) − µ(t) dt + m(t), Φv(t) − ρ(t) dt Ω

Ω

∀v ∈ L ([0, T ], R ), 2

q

∀l1 , l2 ∈ C, ∀m ∈ L2 ([0, T ], Rl )

(28)

which is called Lagrangean functional. In (28) we denote by Ψ : L2 ([0, T ], Rq ) → R the mapping Ψ (v) = F (u), v − u with u solution to the Variaional Inequality (24). It results min Ψ (v) = Ψ (u) = 0. v∈K

By the term Φ v(t) − ρ(t), we denote the term

q

ξi vi = ρ(t), which appears

i=1 2

in the convex set K given by (23); here ρ ∈ L ([0, T ], Rl ) and Φ is a l × q matrix whose entries are −1, 0, 1. Our aim is to prove the following characterization: Theorem 7. u ∈ K is a solution to Variational Inequality (24) if and only if there exist l1 , l2 ∈ C and m ∈ L2 ([0, T ], Rl ) such that (u, l1 , l2 , m) is a saddle point of the Lagrange functional (28), namely L(u, l1 , l2 , m) ≤ L(u, l1 , l2 , m) ≤ L(v, l1 , l2 , m) (29) ∀v ∈ L ([0, T ], R ), 2

q

∀l1 , l2 ∈ C and ∀m ∈ L ([0, T ], R )

and in addition T l1 (t), u(t) − λ(t) dt = 0, 0

2

l

T

l2 (t), u(t) − µ(t) dt = 0.

(30)

0

Proof. Let (u, l1 , l2 , m) be a saddle point of the Lagrange functional L. Taking into account that L(u, l1 , l2 , m) = 0, from the right-hand part of (29) we get: L(v, l1 , l2 , m) ≥ 0,

∀v ∈ L( [0, T ], Rq ).

(31)

Considering (31) for each v ∈ K, namely for λ(t) ≤ v(t) ≤ m(t) and Φv(t) = ρ(t), we obtain: Φ(v) = F (u), v − u ≥ L(v, l1 , l2 , m) ≥ 0, ∀v ∈ K and therefore u is a solution to (29).

(32)


263

Vice versa, let u be a solution to (24) and, first, let us prove that there exist l1 , l2 ∈ C and m ∈ L([0, T ], Rl ) such that L (u, l1 , l2 , m)(v − u) ≥ 0,

∀v ∈ L2 ([0, T ], Rq )

(33)

and

T

l1 (t), u(t) − λ(t) dt = 0, 0

T

l2 (t), u(t) − m(t) dt = 0, 0

where L (u, l1 , l2 , m) denotes the Fréchet derivative of L(u, l1 , l2 , m) at u. We derive the estimate (33) using Theorem 5.3 of [14], provided that the Kurcyusz–Robinson–Zowe condition (5.2) of [14] (see also [23] and [20]): ⎞ ⎛ g (u) ⎠ cone L2 ([0, T ], Rq ) − {u} ⎝ h (u) ⎞ ⎛ ⎞ ⎛ 2 (34) L ([0, T ], Rq ) c + {g(u)} ⎠ ⎠=⎝ + cone ⎝ 2 l 0 L ([0, T ], R ) is fulfilled (see also the remark at the end of page 120 of [14]). In order to verify this condition, let us set g(v) = (λ − v, v − m) and h(v) = Φ v − ρ. It results g (v)(w) = (−w, w), h (v)(w) = Φ w, and the condition (34) is fulfilled because it results: − cone L2 ([0, T ], Rq ) − {u} + cone (C + {λ − u}) = −L2 ([0, T ], Rq ) + cone {u} + C + cone {λ − u} = L2 ([0, T ], Rq ), (35) cone L2 ([0, T ], Rq ) − {u} + cone (C + {u − m}) = L2 ([0, T ], Rq ), (36) and Φ cone L2 ([0, T ], Rq ) − {u} = Φ L2 ([0, T ], Rq ) − cone {u} = Φ L2 ([0, T ], Rq ) = L2 ([0, T ], Rl ).

(37)

Then, the other assumption of Theorem 5.3 of [14] being fulfilled, (33) holds, and, in virtue of the linearity of L(v, l1 , l2 , m) with respect to v, it follows that u is a minimal point for L, namely 0 = L(u, l1 , l2 , m) ≤ L(v, l1 , l2 , m),

∀v ∈ L2 ([0, T ]).

So the right-hand part of (29) is proved. Now, taking into account that L(u, l1 , l2 , m) is reduced to T T L(u, l1 , l2 , m) = − l1 (t), λ(t)−u(t) dt+ l2 (t), u(t)−µ(t) dt ≤ 0 (38) 0

0

for each l1 , l2 ∈ C, ∀m ∈ L2 ([0, T ], Rl ), our result is achieved.

264


Some interesting consequences can be derived from Theorem 7 and from the estimate (29). The first consequence concerns the meaning of the Lagrangean variables. Taking into account that

T

T

l1 (t), u(t) dt =

l1 (t), λ(t) dt = 0,

0

0

T

T

l2 (t), u(t) dt = 0

l2 (t), µ(t) dt = 0, 0

and that Φ u(t) = ρ(t) form the right-hand part of (29), we get:

T

T

F (t, u(t)), v(t) − u(t) dt −

l1 (t), v(t) − u(t) dt

0

0

T

T

l2 (t), v(t) − u(t) dt +

+ 0

m, Φ(v − u) dt ≥ 0, 0

∀v ∈ L2 ([0, T ], Rq ) and hence F (t, u(t)) − l1 (t) + l2 (t) + ΦT m(t) = 0.

(39)

It is possible to derive from (30) and (39) interesting information about the meaning of the Lagrangean variables l1 , l2 , and m. In fact, taking into account that (30) can be rewritten as i

i

l1 (t) (ui (t) − λi (t)) = 0,

l2 (t) (ui (t) − mi (t)) = 0 a.e. in [0, T ],

i

i

it follows that when li (t) > 0, then ui (t) = λi (t), namely the variables li (t) give information about the point for which the vector attains the minimal i value; a similar remark holds also for l2 (t). Moreover, from (39) we deduce that m gives information about the equilibrium value of the functional F − li + l2 , which represents a generalized “cost” functional. Many other consequences about the meaning of the Lagrangean variables could be derived (we refer for this to [3–5, 7, 9, 16]). Another group of consequences concerns the duality theory. In fact, from estimate (29), we immediately deduce that the so-called duality gap cannot arise, namely that it results max

l1 , l2 ∈C m∈L2 ([0,T ],Rl )

=

min

v∈L2 ([0,T ],Rq )

inf

v∈L2 ([0,T ],Rq )

sup l1 , l2 ∈C m∈L2 ([0,T ],Rl )

L(v, l1 , l2 , m)

L(v, l1 , l2 , m) = L(u, l1 , l2 , m).

(40)

Moreover, taking into account (38) and (30), we can introduce a Dual Variational Inequality in the following way:


265

˜ : find u, l1 , l2 , m ∈ L2 ([0, T ], Rq ) × K T T λ(t) − u(t), l1 (t) − l1 (t) dt + u(t) − µ(t), l2 (t) − l2 (t) dt ≥ 0 0

0

˜ ∀(u, l1 , l2 , m) ∈ K, where

(41)

˜ = K

3 (u, l1 , l2 , m) ∈ L2 ([0, T ], Rq ) × L2 ([0, T ], Rl ) :

l1 (t), l2 (t) ≥ 0, a.e. in [0, T ]; Φ u(t) − ρ(t) = 0, a.e. in [0, T ]; F (t, u(t)) − l1 (t) + l2 (t) + ΦT m(t) = 0, a.e. in [0, T ] .

(42)

So the Dual Variational Inequality associated with our problem is a QuasiVariational Inequality.

5 Conclusion We conclude the current paper remarking that the time-dependent theory of equilibrium problems have received from the related Variational Inequality formulation a very fruitful setting and that Variational Inequalities seem to be the key to solve some of the principal challenges of our time. In fact, they allow us to manage the market and financial equilibria, following their evolution in time and achieving a light on the next future. New future research directions deal with the study of evolutionary equilibria by means of projected dynamic systems theory, the introduction to the elastic model for which the data depend also on the expected equilibrium solutions.

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