Variational Inequalities Learn about basic networks economics in - - PDF document

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 1

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Analysis Laboratory

Helsinki University of Technology

Mat-2.142 Seminar on optimization

Variational Inequalities in Network Economics

Pierre-Olivier Pineau

Session 1: Introduction

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 2

S ystems

Analysis Laboratory

Helsinki University of Technology

Goals of the seminar

• Learn about basic networks economics

and variational inequalities.

• Improve understanding of different

algorithmic approaches.

• Develop oral and written English

communication skills.

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 3

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Analysis Laboratory

Helsinki University of Technology

Plan of the session

• 1. Introduction to network economics
• 2. Equilibrium in networks economics
• 3. Variational inequalities
• 4. Related mathematical concepts
• 5. One application
• 6. Organization of the seminar

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 4

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Analysis Laboratory

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• 1. Introduction to network

economics (1)

Network economics deals with the range of economic problems where flows and paths cannot be neglected. Physical network Mathematical network The same economic thinking applies.

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 5

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Analysis Laboratory

Helsinki University of Technology

• 1. Introduction to network

economics (2)

G(V,A) ≡ Network with set of arcs A set of nodes V fa ≡ flow on arc a hp ≡ flow on path p ca(f) ≡ cost function on arc a TAA’ (.) ≡ Demand for transport between A and A’ Traffic assignment problem

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 6

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Analysis Laboratory

Helsinki University of Technology

• 1. Introduction to network

economics (3)

To find the steady-state volume of traffic in the network *It can be user driven (min. of travel cost) *It can by system driven (total cost min.) Traffic assignment problem

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 7

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Analysis Laboratory

Helsinki University of Technology

• 1. Introduction to network

economics (4)

πi ≡ price in node i Di (π) ≡ demand function at node i Si (π) ≡ supply function at node I With shipment costs, how will demand and supply behave? Spatial markets

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 8

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Analysis Laboratory

Helsinki University of Technology

• 1. Introduction to network

economics (5)

• Utility associated with locations
• Utility is dependant on the

population distribution pattern

• Cost and “psychic” cost of moving

How will the population migrate ? Migration problem

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 9

S ystems

Analysis Laboratory

Helsinki University of Technology

• 1. Introduction to network

economics (6)

Assets - Liabilities - Flow of funds

• m sectors of activities
• n financial “instruments” in each sector
• max. mean but min. variance

How to behave in investments? Portfolio management

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 10

S ystems

Analysis Laboratory

Helsinki University of Technology

• 2. Equilibrium in network

economics (1)

An equilibrium is reached when none

• f the agents / players have incentives

to deviate from their current plans (of actions).

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 11

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Analysis Laboratory

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• 2. Equilibrium in network

economics (2)

Equilibria Partial General

Perfect competition Imperfect competition

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 12

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Analysis Laboratory

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• 2. Equilibrium in network

economics (3)

⇒ How to characterize an equilibrium? ⇒ We’ll be looking for conditions of the equilibrium ⇒ But most of the time, these conditions are already given. ⇒ We “simply” use them to compute the equilibrium.

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 13

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Analysis Laboratory

Helsinki University of Technology

• 2. Equilibrium in network

economics (4)

• Simple network transportation problem in

linear programming ⇒ optimality conditions

• Addition of supply and demand functions ⇒

equilibrium conditions

• They correspond to Karush-Khun-Tucker of

a nonlinear program Example (p. 4-7 Harker)

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 14

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Analysis Laboratory

Helsinki University of Technology

• 2. Equilibrium in network

economics (5)

Key questions in equilibrium analysis:

• Existence
• Uniqueness
• Stability (sensitivity analysis /

comparative static)

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 15

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Analysis Laboratory

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• 3. Variational inequalities (1)

They are one methodology to study and solve network equilibrium problem. Other are:

• Optimization
• Fixed point approaches
• Nonlinear complementary formulations

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 16

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Analysis Laboratory

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• 3. Variational inequalities (2)

Variational Inequalities

Infinite-dimensional metric space Finite-dimensional Euclidean space

Computational efficiency for solving large-scale equilibrium problems

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 17

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• 3. Variational inequalities (3)

Let X be a nonempty subset of Rn and let F be a mapping from Rn into itself. The variational inequality problem, denoted by VI(X,F), is to find a vector x* in X such that F(x*)T(y - x*) ≥ ≥ 0 for all y in X

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 18

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Analysis Laboratory

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• 3. Variational inequalities (4)

The solution x* of the system of equations F(x) = 0 where F maps Rn into itself is the solution to VI(X,F) because only F(x*) will satisfy F(x*)T(y - x*) ≥ 0 for all y in X Simplest example

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 19

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Analysis Laboratory

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• 3. Variational inequalities (5)

Let f be a smooth real function on I=[a,b] We seek the points x* in I for which f(x*) = min f(x) (i) if a < x* < b, then f’(x*) = 0 (ii) if x* = a, then f’(x*) ≥ 0 (iii) if x* = b, then f’(x*) ≤ 0 Optimization problem

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 20

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Analysis Laboratory

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• 3. Variational inequalities (6)

Statements (i) to (iii) can be summarized by the VI f’(x*)(x - x*) ≥ 0 for all x in I A solution to VI(I,f’) is then also a solution to the original optimization problem Optimization problem

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 21

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• 3. Variational inequalities (7)

VI can also be used to create alternative formulations for

• Nonlinear complementary problems
• Fixed-point problems (projection

formulation) Other uses

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 22

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• 4. Related mathematical

concepts (1)

• Notions of mathematical programming
• Linear programming
• Kuhn-Tucker conditions
• First order optimality conditions
• Elementary topology
• Convexity
• Open, closed, compact sets

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 23

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Analysis Laboratory

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• 4. Related mathematical

concepts (2)

• Notions of analysis
• Continuity
• Taylor’s series
• Gradient, Jacobian matrix
• Linear algebra
• Fixed point theorem

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 24

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Analysis Laboratory

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• 5. One application (1)

The situation is:

• n firms i compete
• the cost function ci(xi) is stricly increasing and convex
• the market price p(Q) (inverse demand curve, with

Q = Σ xi) is stricly decreasing

• Qp(Q) is concave

Oligopolistic equilibrium (no network)

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 25

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Analysis Laboratory

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• 5. One application (2)

Oligopolistic equilibrium (no network)

The Nash-Cournot equilibrium x* is such that the profit function f i(xi*) = xi* p(xi* + Σ xj*) - ci(xi*) satisfies f i(xi*) > f i(xi) for all xi given xj*

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 26

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Analysis Laboratory

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• 5. One application (3)

Oligopolistic equilibrium (no network)

The equilibrium conditions are p(Q) + xi*p’(Q) - ci’(xi*) ≤ 0 xi*[p(Q) + xi*p’(Q) - ci’(xi*)] = 0 for all i.

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 27

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Analysis Laboratory

Helsinki University of Technology

• 5. One application (4)

Oligopolistic equilibrium (no network)

These equilibrium conditions correspond to a sequence of optimization programs that can be solved. But it’s also possible to write the VI(R+, F) where F(x) = - ∇f(x) F(x*)T•(x - x*) ≥ 0

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 28

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Analysis Laboratory

Helsinki University of Technology

• 6. Organization of the seminar

(1)

• Presentation of some theory

and applications.

• Presentation of assignments
• Active participation

In class: Home:

• Readings
• Assignments

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 29

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Analysis Laboratory

Helsinki University of Technology

• 6. Organization of the seminar

(2)

Algorithms (S4-S8)

• Locally convergent methods
• Globally convergent methods

Part 1: Applications (S9-S13) Part 2: Part 0: Intro (S1-S3)

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 30

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Analysis Laboratory

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Next week readings

• Lecture 1 of Harker 1993
• Haker, “Alternative models of

spatial competition”, Operations Research, vol. 34 (3), 1986.

Pierre-Olivier Pineau Seminar on V.I. in Network Economics - Spring 1999 / 31

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References

• Network Economics: A Variational Inequality approach,

Nagurney 1993.

• The Theory of Choice: A Critical Guide, Heap and al.,

1992.

• Lectures on Computational of Equilibria with Equation-

Based Methods, Harker, 1993.

• Finite-Dimensional VI and NLC problems: A Survey of

Theory, Algorithms and Applications, Harker and Pang, 1990.