VARIATIONAL INEQUALITIES, INFINITE-DIMENSIONAL DUALITY, INVERSE - - PowerPoint PPT Presentation

VARIATIONAL INEQUALITIES, INFINITE-DIMENSIONAL DUALITY, INVERSE PROBLEM AND APPLICATIONS TO OLIGOPOLISTIC MARKET EQUILIBRIUM PROBLEM

Annamaria Barbagallo Joint work with Antonino Maugeri

Department of Mathematics and Computer Science University of Catania

Conference on Applied Inverse Problems 2009 Wien, July 20th, 2009

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 1 / 36

Outlines

Outline

1 Introduction

Some contributions on variational inequalities, infinite-dimensional duality and oligopolistic market equilibrium problem Dynamic oligopolistic market equilibrium problem

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 2 / 36

Outlines

Outline

1 Introduction

Some contributions on variational inequalities, infinite-dimensional duality and oligopolistic market equilibrium problem Dynamic oligopolistic market equilibrium problem

2 Existence results

Existence theorem

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 2 / 36

Outlines

Outline

1 Introduction

Some contributions on variational inequalities, infinite-dimensional duality and oligopolistic market equilibrium problem Dynamic oligopolistic market equilibrium problem

2 Existence results

Existence theorem

3 Regularity results

Mosco’s convergence Regularity results

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 2 / 36

Outlines

Outline

1 Introduction

Some contributions on variational inequalities, infinite-dimensional duality and oligopolistic market equilibrium problem Dynamic oligopolistic market equilibrium problem

2 Existence results

Existence theorem

3 Regularity results

Mosco’s convergence Regularity results

4 Duality theorem and inverse problem

Duality theory Inverse problem

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 2 / 36

Introduction Some contributions on VI, IDD and OMEP

Motivation Problem

An oligopolistic market equilibrium problem is the problem of finding a trade equilibrium in a supply-demand market between a finite number of spatially separated firms.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 3 / 36

Introduction Some contributions on VI, IDD and OMEP

Motivation Problem

An oligopolistic market equilibrium problem is the problem of finding a trade equilibrium in a supply-demand market between a finite number of spatially separated firms.

Motivation

The reason for which dynamic oligopolistic market problem and evolutionary variational inequality which expresses dynamic equilibrium condition are studied is that necessary to consider the dynamics of network adjustment process in which a lag response is operating.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 3 / 36

Introduction Some contributions on VI, IDD and OMEP

Some contributions on variational inequalities, infinite-dimensional duality and oligopolistic market equilibrium problem

• Cournot (Researches into the Mathematical Principles of the Theory
• f Wealth, 1838): consider a two-firm competitive oligopoly problem

in which firms sought to determine their profit-maximizing production quantities;

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 4 / 36

Introduction Some contributions on VI, IDD and OMEP

Some contributions on variational inequalities, infinite-dimensional duality and oligopolistic market equilibrium problem

• Cournot (Researches into the Mathematical Principles of the Theory
• f Wealth, 1838): consider a two-firm competitive oligopoly problem

in which firms sought to determine their profit-maximizing production quantities;

• Lions-Stampacchia (Compt. Rend. Acad. Sci. Paris, 1965): study of

variational inequalities;

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 4 / 36

Introduction Some contributions on VI, IDD and OMEP

Some contributions on variational inequalities, infinite-dimensional duality and oligopolistic market equilibrium problem

• Cournot (Researches into the Mathematical Principles of the Theory
• f Wealth, 1838): consider a two-firm competitive oligopoly problem

in which firms sought to determine their profit-maximizing production quantities;

• Lions-Stampacchia (Compt. Rend. Acad. Sci. Paris, 1965): study of

variational inequalities;

• Brezis (Compt. Rend. Acad. Sci., 1967): introduction of variational

inequalities in infinite-dimensional spaces;

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 4 / 36

Introduction Some contributions on VI, IDD and OMEP

Some contributions on variational inequalities, infinite-dimensional duality and oligopolistic market equilibrium problem

• Cournot (Researches into the Mathematical Principles of the Theory
• f Wealth, 1838): consider a two-firm competitive oligopoly problem

in which firms sought to determine their profit-maximizing production quantities;

• Lions-Stampacchia (Compt. Rend. Acad. Sci. Paris, 1965): study of

variational inequalities;

• Brezis (Compt. Rend. Acad. Sci., 1967): introduction of variational

inequalities in infinite-dimensional spaces;

• Mosco (Adv. Math., 1969): introduction of sets convergence;
• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 4 / 36

Introduction Some contributions on VI, IDD and OMEP

Some contributions on variational inequalities, infinite-dimensional duality and oligopolistic market equilibrium problem

• Cournot (Researches into the Mathematical Principles of the Theory
• f Wealth, 1838): consider a two-firm competitive oligopoly problem

in which firms sought to determine their profit-maximizing production quantities;

• Lions-Stampacchia (Compt. Rend. Acad. Sci. Paris, 1965): study of

variational inequalities;

• Brezis (Compt. Rend. Acad. Sci., 1967): introduction of variational

inequalities in infinite-dimensional spaces;

• Mosco (Adv. Math., 1969): introduction of sets convergence;
• Dafermos and Nagurney (Regional Science and Urban Economics,

1987): study on oligopolistic and competitive behavior of spatially separated markets in the static case;

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 4 / 36

Introduction Some contributions on VI, IDD and OMEP

Some contributions on variational inequalities, infinite-dimensional duality and oligopolistic market equilibrium problem

• Cournot (Researches into the Mathematical Principles of the Theory
• f Wealth, 1838): consider a two-firm competitive oligopoly problem

in which firms sought to determine their profit-maximizing production quantities;

• Lions-Stampacchia (Compt. Rend. Acad. Sci. Paris, 1965): study of

variational inequalities;

• Brezis (Compt. Rend. Acad. Sci., 1967): introduction of variational

inequalities in infinite-dimensional spaces;

• Mosco (Adv. Math., 1969): introduction of sets convergence;
• Dafermos and Nagurney (Regional Science and Urban Economics,

1987): study on oligopolistic and competitive behavior of spatially separated markets in the static case;

• Daniele-Idone-Giuffr´

e-Maugeri (Math. Ann., 2007): study the infinite-dimensional duality by means of quasi-relative interior.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 4 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let us consider:

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 5 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let us consider:

• m firms Pi, i = 1, 2, . . . , m;
• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 5 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let us consider:

• m firms Pi, i = 1, 2, . . . , m;
• n demand markets Qj, j = 1, 2, . . . , n;
• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 5 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let us consider:

• m firms Pi, i = 1, 2, . . . , m;
• n demand markets Qj, j = 1, 2, . . . , n;
• are generally spatially separated.
• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 5 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let us consider:

• m firms Pi, i = 1, 2, . . . , m;
• n demand markets Qj, j = 1, 2, . . . , n;
• are generally spatially separated.

Let us assume that:

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 5 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let us consider:

• m firms Pi, i = 1, 2, . . . , m;
• n demand markets Qj, j = 1, 2, . . . , n;
• are generally spatially separated.

Let us assume that:

• homogeneous commodity, produced by the m firms and consumed at

the n markets, is involved during a period of time [0, T], T > 0.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 5 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let us denote:

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 6 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let us denote:

• pi(t), t ∈ [0, T], i = 1, 2, . . . , m, the nonnegative commodity output

produced by firm Pi at the time t ∈ [0, T];

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 6 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let us denote:

• pi(t), t ∈ [0, T], i = 1, 2, . . . , m, the nonnegative commodity output

produced by firm Pi at the time t ∈ [0, T];

• qj(t), t ∈ [0, T], j = 1, 2, . . . , n, the demand for the commodity at

demand market Qj at the same time t ∈ [0, T];

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 6 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let us denote:

• pi(t), t ∈ [0, T], i = 1, 2, . . . , m, the nonnegative commodity output

produced by firm Pi at the time t ∈ [0, T];

• qj(t), t ∈ [0, T], j = 1, 2, . . . , n, the demand for the commodity at

demand market Qj at the same time t ∈ [0, T];

• xij(t), t ∈ [0, T], i = 1, 2, . . . , m, j = 1, 2, . . . , n, the nonnegative

commodity shipment between the supply market Pi and the demand market Qj at the time t ∈ [0, T].

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 6 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let us denote:

• pi(t), t ∈ [0, T], i = 1, 2, . . . , m, the nonnegative commodity output

produced by firm Pi at the time t ∈ [0, T];

• qj(t), t ∈ [0, T], j = 1, 2, . . . , n, the demand for the commodity at

demand market Qj at the same time t ∈ [0, T];

• xij(t), t ∈ [0, T], i = 1, 2, . . . , m, j = 1, 2, . . . , n, the nonnegative

commodity shipment between the supply market Pi and the demand market Qj at the time t ∈ [0, T]. Let us group:

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 6 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let us denote:

• pi(t), t ∈ [0, T], i = 1, 2, . . . , m, the nonnegative commodity output

produced by firm Pi at the time t ∈ [0, T];

• qj(t), t ∈ [0, T], j = 1, 2, . . . , n, the demand for the commodity at

demand market Qj at the same time t ∈ [0, T];

• xij(t), t ∈ [0, T], i = 1, 2, . . . , m, j = 1, 2, . . . , n, the nonnegative

commodity shipment between the supply market Pi and the demand market Qj at the time t ∈ [0, T]. Let us group:

• the production output into a vector-function p : [0, T] → Rm

+;

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 6 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let us denote:

• pi(t), t ∈ [0, T], i = 1, 2, . . . , m, the nonnegative commodity output

produced by firm Pi at the time t ∈ [0, T];

• qj(t), t ∈ [0, T], j = 1, 2, . . . , n, the demand for the commodity at

demand market Qj at the same time t ∈ [0, T];

• xij(t), t ∈ [0, T], i = 1, 2, . . . , m, j = 1, 2, . . . , n, the nonnegative

commodity shipment between the supply market Pi and the demand market Qj at the time t ∈ [0, T]. Let us group:

• the production output into a vector-function p : [0, T] → Rm

+;

• the demands into a vector-function d : [0, T] → Rn

+;

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 6 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let us denote:

• pi(t), t ∈ [0, T], i = 1, 2, . . . , m, the nonnegative commodity output

produced by firm Pi at the time t ∈ [0, T];

• qj(t), t ∈ [0, T], j = 1, 2, . . . , n, the demand for the commodity at

demand market Qj at the same time t ∈ [0, T];

• xij(t), t ∈ [0, T], i = 1, 2, . . . , m, j = 1, 2, . . . , n, the nonnegative

commodity shipment between the supply market Pi and the demand market Qj at the time t ∈ [0, T]. Let us group:

• the production output into a vector-function p : [0, T] → Rm

+;

• the demands into a vector-function d : [0, T] → Rn

+;

• the commodity shipments into a matrix-function x : [0, T] → Rmn

+ .

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 6 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

We have that for every i and j and a.e. in [0, T]:

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 7 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

We have that for every i and j and a.e. in [0, T]:

• the quantity produced by a firm, at the time t ∈ [0, T], must be equal

to the sum of the commodity shipments from that firm to all the demand markets, at the same t ∈ [0, T]: pi(t) =

n

• j=1

xij(t),

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 7 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

We have that for every i and j and a.e. in [0, T]:

• the quantity produced by a firm, at the time t ∈ [0, T], must be equal

to the sum of the commodity shipments from that firm to all the demand markets, at the same t ∈ [0, T]: pi(t) =

n

• j=1

xij(t),

• the demand for the commodity at a demand market, at the time

t ∈ [0, T], must be equal to the sum of all the commodity shipments to that demand market, at the same t ∈ [0, T]: qj(t) =

m

• i=1

xij(t).

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 7 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let us assume that the nonnegative commodity shipment between the supply market Pi and the demand market Qj has to satisfy time-dependent constrains: xij(t) ≤ xij(t) ≤ xij(t), ∀i = 1, 2, . . . , m, j = 1, 2, . . . , n, a.e. in [0,T], where xij(t) and xij(t) are given.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 8 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let us assume that the nonnegative commodity shipment between the supply market Pi and the demand market Qj has to satisfy time-dependent constrains: xij(t) ≤ xij(t) ≤ xij(t), ∀i = 1, 2, . . . , m, j = 1, 2, . . . , n, a.e. in [0,T], where xij(t) and xij(t) are given. Let us assume that the matrix-function x(t) belongs to L2([0, T], Rmn).

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 8 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let us assume that the nonnegative commodity shipment between the supply market Pi and the demand market Qj has to satisfy time-dependent constrains: xij(t) ≤ xij(t) ≤ xij(t), ∀i = 1, 2, . . . , m, j = 1, 2, . . . , n, a.e. in [0,T], where xij(t) and xij(t) are given. Let us assume that the matrix-function x(t) belongs to L2([0, T], Rmn). The set of feasible vectors x(t) is K =

• x ∈ L2([0, T], Rmn) : x(t) ≤ x(t) ≤ x(t), a.e. in [0, T]
• .

It is easily seen that K is a convex, closed, bounded subset of the Hilbert space L2([0, T], Rmn).

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 8 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let us associate

• with each firm Pi a production cost fi(t), t ∈ [0, T], i = 1, 2, . . . , m,

and fi = fi(p(t));

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 9 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let us associate

• with each firm Pi a production cost fi(t), t ∈ [0, T], i = 1, 2, . . . , m,

and fi = fi(p(t));

• with each demand market Qj a demand price dj(t), t ∈ [0, T],

j = 1, 2, . . . , n, and dj = dj(q(t)).

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 9 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let us associate

• with each firm Pi a production cost fi(t), t ∈ [0, T], i = 1, 2, . . . , m,

and fi = fi(p(t));

• with each demand market Qj a demand price dj(t), t ∈ [0, T],

j = 1, 2, . . . , n, and dj = dj(q(t)).

• Then

f : L2([0, T], Rm

+) → L2([0, T], Rm +),

d : L2([0, T], Rn

+) → L2([0, T], Rn +).

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 9 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let cij(t), i = 1, 2, . . . , m, j = 1, 2, . . . , n, denote the transaction cost at time t ∈ [0, T], between firm Pi and demand market Qj, and cij = cij(x(t)), so we have c : L2([0, T], Rmn

+ ) → L2([0, T], Rnm + ).

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 10 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem

Let cij(t), i = 1, 2, . . . , m, j = 1, 2, . . . , n, denote the transaction cost at time t ∈ [0, T], between firm Pi and demand market Qj, and cij = cij(x(t)), so we have c : L2([0, T], Rmn

+ ) → L2([0, T], Rnm + ).

The profit vi(t, x(t)), t ∈ [0, T], i = 1, 2, . . . , m, of firm Pi at the same time t ∈ [0, T] is vi(t, x(t)) =

n

• j=1

dj(p(t))xij(t) − fi(q(t)) −

n

• j=1

cij(x(t))xij(t).

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 10 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Dynamic oligopolistic market equilibrium problem Definition

A commodity shipment distribution x ∈ L2([0, T], Rmn) is a dynamic

• ligopolistic market equilibrium if and only if for each i = 1, 2, . . . , m and

j = 1, 2, . . . , n and a.e. in [0, T] there holds: vi(t, x∗(t)) ≥ vi(t, xi(t), x∗

i (t)),

∀x ∈ K, a.e. in [0, T], where we denote by

• x∗

i = (x∗ 1, . . . , x∗ i−1, x∗ i+1, . . . , x∗ m).

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 11 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Duality mapping and variational inequalities in Hilbert spaces

In the Hilbert space L2([0, T], Rk), let us recall that ≪ φ, y ≫:= T < φ(t), y(t) > dt, is its duality mapping, where φ ∈ (L2([0, T], Rk))∗ = L2([0, T], Rk) and y ∈ L2([0, T], Rk).

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 12 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Duality mapping and variational inequalities in Hilbert spaces

In the Hilbert space L2([0, T], Rk), let us recall that ≪ φ, y ≫:= T < φ(t), y(t) > dt, is its duality mapping, where φ ∈ (L2([0, T], Rk))∗ = L2([0, T], Rk) and y ∈ L2([0, T], Rk). Let K be a convex subset of L2([0, T], Rk) and let f : K → L2([0, T], Rk) be a mapping. Recall that an evolutionary variational inequality is the problem of finding a point x ∈ K such that ≪ f (x), y − x ≫≥ 0, ∀y ∈ K.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 12 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Variational formulation of the dynamic equilibrium oligopolistic market problem

Next, we use the variational inequality theory to obtain the existence of solutions for the oligopolistic problem formulated as a Cournot-Nash game.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 13 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Variational formulation of the dynamic equilibrium oligopolistic market problem

Next, we use the variational inequality theory to obtain the existence of solutions for the oligopolistic problem formulated as a Cournot-Nash game.

Definition

A function v is called pseudoconcave with respect to xi, i = 1, 2, . . . , m, if the following holds a.e. in [0, T]: ∂v ∂xi (t, x1, . . . , xi, . . . xn), xi − yi ≥ 0 = ⇒ v(t, x1, . . . , xi, . . . xn) ≥ v(t, x1, . . . , yi, . . . xn),

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 13 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Variational formulation of the dynamic equilibrium oligopolistic market problem Theorem

Assume that for each firm Pi the profit function vi(t, x(t)) is pseudoconcave with respect to the variables {xi1, xi2, . . . , xin}, i = 1, 2, . . . , m, and continuously differentiable for a.e. t ∈ [0, T]. Assume that ∇v is a Carath` eodory function such that ∃h ∈ L2([0, T], R) : ∇v(t, u(t)) ≤ h(t). (1) Then x∗ ∈ K is a dynamic Cournot-Nash equilibrium if and only if it satisfies the evolutionary variational inequality ≪ −∇v(t, x∗(t)), x − x∗ ≫≥ 0, ∀x ∈ K. (2)

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 14 / 36

Introduction Dynamic oligopolistic market equilibrium problem

Equivalent formulation Corollary

Assume that for each firm Pi the profit function vi(t, x(t)) is pseudoconcave with respect to the variables {xi1, xi2, . . . , xin}, and continuously differentiable, for a.e. t ∈ [0, T], and ∇v is a Carath` eodory function such that ∃h ∈ L2([0, T], R) : ∇v(t, u(t)) ≤ h(t). Then the evolutionary variational inequality (2) is equivalent to −∇v(t, x∗(t)), x(t) − x∗(t) ≥ 0, ∀x(t) ∈ K(t), a.e. in [0, T]. (3)

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 15 / 36

Existence results Existence theorem

Some definitions

Let E be a real topological vector space, K ⊆ E convex. Then φ : K → E ∗ is said to be:

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 16 / 36

Existence results Existence theorem

Some definitions

Let E be a real topological vector space, K ⊆ E convex. Then φ : K → E ∗ is said to be:

• pseudomonotone in the sense of Karamardian if and only if

∀u1, u2 ∈ K φ(u2), u1 − u2 ≥ 0 = ⇒ φ(u1), u1 − u2 ≥ 0;

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 16 / 36

Existence results Existence theorem

Some definitions

Let E be a real topological vector space, K ⊆ E convex. Then φ : K → E ∗ is said to be:

• pseudomonotone in the sense of Karamardian if and only if

∀u1, u2 ∈ K φ(u2), u1 − u2 ≥ 0 = ⇒ φ(u1), u1 − u2 ≥ 0;

• hemicontinuous in the sense of Fan if and only if

∀u ∈ K the function z → φ(z), z − u is upper semicontinuous on K;

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 16 / 36

Existence results Existence theorem

Some definitions

Let E be a real topological vector space, K ⊆ E convex. Then φ : K → E ∗ is said to be:

• pseudomonotone in the sense of Karamardian if and only if

∀u1, u2 ∈ K φ(u2), u1 − u2 ≥ 0 = ⇒ φ(u1), u1 − u2 ≥ 0;

• hemicontinuous in the sense of Fan if and only if

∀u ∈ K the function z → φ(z), z − u is upper semicontinuous on K;

• hemicontinuous along line segments if and only if

∀u1, u2 ∈ K the function z → φ(z), u2 − u1 is upper semicontinuous on the line segment [u1, u2].

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 16 / 36

Existence results Existence theorem

Existence theorem Theorem

Each of the following conditions is sufficient to ensure the existence of the solution to ≪ −∇v(x∗), x − x∗ ≫≥ 0, ∀x ∈ K,

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 17 / 36

Existence results Existence theorem

Existence theorem Theorem

Each of the following conditions is sufficient to ensure the existence of the solution to ≪ −∇v(x∗), x − x∗ ≫≥ 0, ∀x ∈ K,

• −∇v is hemicontinuous with respect to the strong topology and there

exist A ⊆ K compact and B ⊆ K compact, convex with respect to the strong topology such that ∀x1 ∈ K \ A ∃x2 ∈ B : ≪ −∇v(x1), x2 − x1 ≫< 0;

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 17 / 36

Existence results Existence theorem

Existence theorem Theorem

Each of the following conditions is sufficient to ensure the existence of the solution to ≪ −∇v(x∗), x − x∗ ≫≥ 0, ∀x ∈ K,

• −∇v is hemicontinuous with respect to the strong topology and there

exist A ⊆ K compact and B ⊆ K compact, convex with respect to the strong topology such that ∀x1 ∈ K \ A ∃x2 ∈ B : ≪ −∇v(x1), x2 − x1 ≫< 0;

• −∇v is K-pseudomonotone and hemicontinuous along line segments;
• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 17 / 36

Existence results Existence theorem

Existence theorem Theorem

Each of the following conditions is sufficient to ensure the existence of the solution to ≪ −∇v(x∗), x − x∗ ≫≥ 0, ∀x ∈ K,

• −∇v is hemicontinuous with respect to the strong topology and there

exist A ⊆ K compact and B ⊆ K compact, convex with respect to the strong topology such that ∀x1 ∈ K \ A ∃x2 ∈ B : ≪ −∇v(x1), x2 − x1 ≫< 0;

• −∇v is K-pseudomonotone and hemicontinuous along line segments;
• −∇v is F-hemicontinuous on K with respect to the weak topology.
• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 17 / 36

Regularity results Mosco’s convergence

Mosco’s convergence Definition

Let (X, · ) be an Hilbert space and K a closed, nonempty, convex subset

• f X. A sequence of nonempty, closed, convex sets Kn converges to K in

Mosco’s sense, as n → +∞, i.e. Kn → K, if and only if (M1) for any x ∈ K, there exists a sequence {xn}n∈N strongly converging to x in X such that xn lies in Kn for all n ∈ N, (M2) for any subsequence {xkn}n∈N weakly converging to x in X, such that xkn lies in Kkn for all n ∈ N, then the weak limit x belongs to K.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 18 / 36

Regularity results Mosco’s convergence

Set of feasible vectors Lemma

Let x, x ∈ C([0, T], Rmn

+ ), and let {tn}n∈N be a sequence such that

tn → t ∈ [0, T], as n → +∞. Then, the sequence of sets K(tn) =

• x(tn) ∈ Rmn : x(tn) ≤ x(tn) ≤ x(tn)
• ,

∀n ∈ N, converges to K(t) =

• x(t) ∈ Rmn : x(t) ≤ x(t) ≤ x(t)
• ,

as n → +∞, in Mosco’s sense.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 19 / 36

Regularity results Mosco’s convergence

Set of feasible vectors Lemma

Let x, x ∈ C([0, T], Rmn

+ ), and let {tn}n∈N be a sequence such that

tn → t ∈ [0, T], as n → +∞. Then, the sequence of sets K(tn) =

• x(tn) ∈ Rmn : x(tn) ≤ x(tn) ≤ x(tn)
• ,

∀n ∈ N, converges to K(t) =

• x(t) ∈ Rmn : x(t) ≤ x(t) ≤ x(t)
• ,

as n → +∞, in Mosco’s sense.

Lemma

Let x, x ∈ C([0, T], Rnm

+ ) be matrix-functions. Then, the set

K(t) = {x(t) ∈ Rnm : 0 ≤ x(t) ≤ x(t) ≤ x(t)}. (4) is uniformly bounded for t ∈ [0, T].

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 19 / 36

Regularity results Regularity results

Regularity result for the dynamic equilibrium oligopolistic market problem Theorem

Assume that for each firm Pi the profit function vi(t, x(t)) is strictly pseudoconcave with respect to the variables {xi1, xi2, . . . , xin} for a.e. t ∈ [0, T], and belongs to C 1([0, T] × Rmn

+ , R). Assume ∇v is a

Carath` eodory function such that ∃h ∈ L2([0, T], R) : ∇v(t, u(t)) ≤ h(t). (5) Then, the unique dynamic Cournot-Nash equilibrium x∗ ∈ K is continuous in [0, T].

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 20 / 36

Duality theorem and inverse problem Duality theory

Duality theory

Let us apply the Lagrange and the duality theory which plays an extraordinary role in economic theory in order to characterize the dynamic

• ligopolistic market equilibrium conditions in terms of the Lagrange

multipliers.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 21 / 36

Duality theorem and inverse problem Duality theory

Duality theory

Let us apply the Lagrange and the duality theory which plays an extraordinary role in economic theory in order to characterize the dynamic

• ligopolistic market equilibrium conditions in terms of the Lagrange

multipliers.

Definition

Let X denote a real normed space and C ⊆ X. The set TC(x) =

• h ∈ X : h = lim

n→∞ λn(xn − x), λn ∈ R and λn > 0, ∀n ∈ N,

xn ∈ C ∀n ∈ N and lim

n→∞ xn = x

• is called tangent cone to C at x.
• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 21 / 36

Duality theorem and inverse problem Duality theory

Some definitions Proposition

If TC(x) = ∅, then x ∈ Cl C. If C is convex we have: TC(x) = Cl Cone(C − {x}) where Cone(C) = {λx : x ∈ C, λ ∈ R, λ ≥ 0}.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 22 / 36

Duality theorem and inverse problem Duality theory

Some definitions Proposition

If TC(x) = ∅, then x ∈ Cl C. If C is convex we have: TC(x) = Cl Cone(C − {x}) where Cone(C) = {λx : x ∈ C, λ ∈ R, λ ≥ 0}.

Definition

Let S be a nonempty subset of a real linear space X and let Y be a real linear space partially ordered by the cone C. A mapping f : S → Y is called convex-like if the set f (S) + C is convex.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 22 / 36

Duality theorem and inverse problem Duality theory

Some definitions Proposition

If TC(x) = ∅, then x ∈ Cl C. If C is convex we have: TC(x) = Cl Cone(C − {x}) where Cone(C) = {λx : x ∈ C, λ ∈ R, λ ≥ 0}.

Definition

Let S be a nonempty subset of a real linear space X and let Y be a real linear space partially ordered by the cone C. A mapping f : S → Y is called convex-like if the set f (S) + C is convex.

Definition

Let C be a convex subset of X. The quasi-relative interior of C, denoted by qri C, is the set of those x ∈ C for which TC(x) is a linear subspace of X.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 22 / 36

Duality theorem and inverse problem Duality theory

Some definitions

Let X be a real linear topological space and S a nonempty subset of X; let (Y , · ) be a real normed space partially ordered by a convex cone C. Let f : S → R and g : S → Y be two functions such that the function (f , g) is convex-like with respect to the cone R+ × C of R × Y . Let us consider the problem min

x∈K f (x)

(6) where K = {x ∈ S : g(x) ∈ −C} and the dual problem: max

u∈C ∗ inf x∈S{f (x) + u, g(x)},

(7) where C ∗ = {u ∈ Y ∗ : u, y ≥ 0 ∀y ∈ C} is the dual cone of C.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 23 / 36

Duality theorem and inverse problem Duality theory

General results Assumptions S

We say that Assumption S is fulfilled at a point x0 ∈ K if it results: T e

M(f (x0), θY ) ∩ (] − ∞, 0[×{θY }) = ∅,

(8) where

• M = {(f (x) − f (x0) + α, g(x) + y) : x ∈ S \ K, α ≥ 0, y ∈ C}.
• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 24 / 36

Duality theorem and inverse problem Duality theory

General results Theorem

Let us assume that the function (f , g) : S → R × Y is convex-like. Then if problem (6) is solvable and Assumption S is fulfilled at the extremal solution x0 ∈ S, also problem (7) is solvable, the extreme values of both problems are equal and it results: λ, g(x0) = 0 where λ is the extremal point of problem (7).

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 25 / 36

Duality theorem and inverse problem Duality theory

General results Theorem

Assume that the assumptions of the previous Theorem are fulfilled. Then x0 ∈ K is a minimal solution to problem (6) if and only if there exist u ∈ C ∗ and v ∈ Z ∗ such that (x0, u, v) is a saddle point of the Lagrange functional, namely L(x0, u, v) ≤ L(x0, u, v) ≤ L(x, u, v), ∀x ∈ S, u ∈ C ∗, v ∈ Z ∗ and, moreover, it results that u, g(x0) = 0.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 26 / 36

Duality theorem and inverse problem Duality theory

Duality for dynamic oligopolistic market equilibrium problem

Let x∗ ∈ K a solution to variational inequality (2) and let us set ψ(x) =≪ −∇v(x∗), x − x∗ ≫, ∀x ∈ K and let us observe that ψ(x) ≥ 0 ∀x ∈ K and min

x∈K ψ(x) = ψ(x∗) = 0.

(9)

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 27 / 36

Duality theorem and inverse problem Duality theory

Duality for dynamic oligopolistic market equilibrium problem

Let x∗ ∈ K a solution to variational inequality (2) and let us set ψ(x) =≪ −∇v(x∗), x − x∗ ≫, ∀x ∈ K and let us observe that ψ(x) ≥ 0 ∀x ∈ K and min

x∈K ψ(x) = ψ(x∗) = 0.

(9) We associate with variational inequality (2) the following Lagrange functional, ∀x ∈ L2([0, T], Rnm), (α, β) ∈ C ∗, L(x, α, β) = ψ(x)+ ≪ α, x − x ≫ + ≪ β, x − x ≫, where C ∗ =

• α, β
• ∈ L2([0, T], Rmn) × L2([0, T], Rmn) : α(t) ≥

0, β(t) ≥ 0, a.e. in [0, T]

• is the dual cone of ordering cone C of

L2([0, T], Rmn) × L2([0, T], Rmn); we have that C = C ∗ and the function ψ(x), g1(t) = x(t) − x(t), g2(t) = x(t) − x(t) are linear.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 27 / 36

Duality theorem and inverse problem Duality theory

Duality for dynamic oligopolistic market equilibrium problem Lemma

Let x∗ ∈ K a solution to variational inequality (2) and let us set, for i = 1, 2, . . . , m and j = 1, 2, . . . , n, E ij

− = {t ∈ [0, T] : x∗ ij(t) = xij(t)},

E ij

0 = {t ∈ [0, T] : xij(t) < x∗ ij(t) < xij(t)},

E ij

+ = {t ∈ [0, T] : x∗ ij(t) = xij(t)}.

Then we have ∂v(t, x(t)) ∂xij ≤ 0 a.e. in E ij

−,

∂v(t, x∗(t)) ∂xij = 0 a.e. in E ij

0 ,

∂v(t, x(t)) ∂xij ≥ 0 a.e. in E ij

+.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 28 / 36

Duality theorem and inverse problem Duality theory

Duality for dynamic oligopolistic market equilibrium problem Theorem

Problem (9) verifies Assumption S at the minimal point x∗ ∈ K.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 29 / 36

Duality theorem and inverse problem Duality theory

Duality for dynamic oligopolistic market equilibrium problem Theorem

Problem (9) verifies Assumption S at the minimal point x∗ ∈ K.

Theorem

x∗ ∈ K is a solution to variational problem (2) if and only if there exist α∗, β∗ ∈ L2([0, T], Rmn) such that: (i) α∗(t), β∗(t) ≥ 0 a.e. in [0, T]; (ii) α∗(t)(x(t) − x∗(t)) = 0 a.e. in [0, T], β∗(t)(x∗(t) − x(t)) = 0 a.e. in [0, T]; (iii) −∇v(t, x∗(t)) + β∗(t) = α∗(t) a.e. in [0, T].

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 29 / 36

Duality theorem and inverse problem Inverse problem

Inverse problem

The previous result allows us to calculate the Lagrange variables α and β and to consider an inverse problem in which we determine the maximal and the minimal production in correspondence of the fixed marginal profit.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 30 / 36

Duality theorem and inverse problem Inverse problem

Inverse problem Theorem

Suppose that the assumptions of existence Theorem are fulfilled and let x∗ ∈ K be the solution to evolutionary variational inequality (3). Then, considering the Lagrange multipliers α∗, β∗ ∈ L2([0, T], Rmn), whose are guaranteed from Theorem 18. Setting

• Aij = {t ∈ [0, T] : α∗

ij(t) > 0},

• Bij = {t ∈ [0, T] : β∗

ij(t) > 0},

then, we have

• Aij = E −

ij

α∗

ij =

• −∂v(t,x(t)

∂xij

t ∈ Aij

• therwise

(10)

• Bij = E +

ij

β∗

ij =

• −∂v(t,x(t)

∂xij

t ∈ Bij

• therwise

. (11)

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 31 / 36

Duality theorem and inverse problem Inverse problem

Lagrange variables

The importance of the function α∗ and β∗ is in their ability to describe the behavior of the dynamic oligopolistic market equilibrium problem.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 32 / 36

Duality theorem and inverse problem Inverse problem

Lagrange variables

The importance of the function α∗ and β∗ is in their ability to describe the behavior of the dynamic oligopolistic market equilibrium problem. In fact, let us note that from ii) and α∗

ij(t) ≥ 0, β∗ ij(t) ≥ 0, xij(t) − x∗ ij(t) ≤ 0 and

x∗

ij(t) − xij(t) ≤ 0, for i = 1, 2, . . . , m, j = 1, 2, . . . , n, a.e. in [0, T], we get

α∗

ij(t)(xij(t) − x∗ ij(t))

= 0, ∀i = 1, . . . , m, ∀j = 1, . . . , n a.e. in [0, T], β∗

ij(t)(x∗ ij(t) − xij(t))

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

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 32 / 36

Duality theorem and inverse problem Inverse problem

Lagrange variables

The importance of the function α∗ and β∗ is in their ability to describe the behavior of the dynamic oligopolistic market equilibrium problem. In fact, let us note that from ii) and α∗

ij(t) ≥ 0, β∗ ij(t) ≥ 0, xij(t) − x∗ ij(t) ≤ 0 and

x∗

ij(t) − xij(t) ≤ 0, for i = 1, 2, . . . , m, j = 1, 2, . . . , n, a.e. in [0, T], we get

α∗

ij(t)(xij(t) − x∗ ij(t))

= 0, ∀i = 1, . . . , m, ∀j = 1, . . . , n a.e. in [0, T], β∗

ij(t)(x∗ ij(t) − xij(t))

= 0, ∀i = 1, . . . , m, ∀j = 1, . . . , n a.e. in [0, T]. Then the set ˜ Aij = {t ∈ [0, T] : α∗

ij(t) > 0}

indicates the time when the trade between the firm i and the demand market j is minimum and −∂v(t, x(t)) ∂xij = α∗

ij(t),

a.e. in ˜ Aij, namely, α∗

ij represents the marginal utility.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 32 / 36

Duality theorem and inverse problem Inverse problem

Lagrange variables

Analogously ˜ Bij = {t ∈ [0, T] : β∗

ij(t) > 0}

indicates the time when the trade between the firm i and the demand market j is maximum and ∂v(t, x(t)) ∂xij = β∗

ij(t),

a.e. in ˜ Bij, namely, −β∗

ij is the marginal utility.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 33 / 36

Duality theorem and inverse problem Inverse problem

Lagrange variables

Analogously ˜ Bij = {t ∈ [0, T] : β∗

ij(t) > 0}

indicates the time when the trade between the firm i and the demand market j is maximum and ∂v(t, x(t)) ∂xij = β∗

ij(t),

a.e. in ˜ Bij, namely, −β∗

ij is the marginal utility.

Now, let us consider the set ˜ Cij = {t ∈ [0, T] : xij < x∗

ij < xij}.

In C ij it results α∗

ij(t) = β∗ ij(t) = 0, a.e. in [0, T], then

∂v(t, x(t)) ∂xij = 0, a.e. in ˜ Cij.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 33 / 36

Duality theorem and inverse problem Inverse problem

References

• A. Barbagallo and M.-G. Cojocaru, Dynamic equilibrium formulation
• f oligopolistic market problem, Math. Comput. Model., 2008.
• A. Barbagallo and A. Maugeri, Duality theory for the dynamic
• ligopolistic market equilibrium problem, submitted.
• A. Barbagallo, Existence of continuous solutions to time-dependent

variational inequalities, J. Nonlinear Convex Anal., 7, 2006, pp. 343-354.

• P. Daniele, S. Giuffr`

e, G. Idone and A. Maugeri, Infinite Dimensional Duality and Applications, Mathematische Annalen, 339, 2007, pp. 221–239.

• A. Maugeri and F. Raciti, Remarks on Infinite Dimensional Duality, J.

Global Optim. 2009.

• A. Barbagallo (University of Catania)

VI,IDD, IP and applications to OMEP AIP 2009 34 / 36

Duality theorem and inverse problem Inverse problem

References

• A. Barbagallo, Regularity results for evolutionary nonlinear variational

and quasi-variational inequalities with applications to dynamic equilibrium problems, J. Global Optim. 40, 2008, pp. 29–39.

• A. Barbagallo and M.-G. Cojocaru, Continuity of solutions for

parametric variational inequalities in Banach space, J. Math. Anal.

• Appl. 351, 2009, pp. 707–720.
• A. Barbagallo (University of Catania)

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Duality theorem and inverse problem Inverse problem

Thank you for your attention!

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VI,IDD, IP and applications to OMEP AIP 2009 36 / 36