Variational Inequalities Learn about basic networks economics in - PDF document

Mat-2.142 Seminar on optimization Goals of the seminar Variational Inequalities • Learn about basic networks economics in Network Economics and variational inequalities. • Improve understanding of different Session 1: Introduction algorithmic approaches. • Develop oral and written English Pierre-Olivier Pineau communication skills. S ystems S ystems Analysis Laboratory Analysis Laboratory Pierre-Olivier Pineau Pierre-Olivier Pineau Helsinki University of Technology Helsinki University of Technology Seminar on V.I. in Network Economics - Spring 1999 / 1 Seminar on V.I. in Network Economics - Spring 1999 / 2 1. Introduction to network Plan of the session economics (1) 1. Introduction to network economics Network economics deals with the 2. Equilibrium in networks economics range of economic problems where 3. Variational inequalities flows and paths cannot be neglected. 4. Related mathematical concepts Physical network 5. One application Mathematical network 6. Organization of the seminar The same economic thinking applies. S ystems S ystems Analysis Laboratory Analysis Laboratory Pierre-Olivier Pineau Pierre-Olivier Pineau Helsinki University of Technology Helsinki University of Technology Seminar on V.I. in Network Economics - Spring 1999 / 3 Seminar on V.I. in Network Economics - Spring 1999 / 4 1. Introduction to network 1. Introduction to network economics (2) economics (3) Traffic assignment problem Traffic assignment problem G(V,A) ≡ Network with set of arcs A To find the steady-state volume of traffic in set of nodes V the network f a ≡ flow on arc a h p ≡ flow on path p *It can be user driven (min. of travel cost) c a (f) ≡ cost function on arc a *It can by system driven (total cost min.) T AA’ (.) ≡ Demand for transport between A and A’ S ystems S ystems Analysis Laboratory Analysis Laboratory Pierre-Olivier Pineau Pierre-Olivier Pineau Helsinki University of Technology Helsinki University of Technology Seminar on V.I. in Network Economics - Spring 1999 / 5 Seminar on V.I. in Network Economics - Spring 1999 / 6

1. Introduction to network 1. Introduction to network economics (4) economics (5) Spatial markets Migration problem π i ≡ price in node i •Utility associated with locations D i ( π ) ≡ demand function at node i •Utility is dependant on the S i ( π ) ≡ supply function at node I population distribution pattern •Cost and “psychic” cost of moving How will the population migrate ? With shipment costs, how will demand and supply behave? S ystems S ystems Analysis Laboratory Analysis Laboratory Pierre-Olivier Pineau Pierre-Olivier Pineau Helsinki University of Technology Helsinki University of Technology Seminar on V.I. in Network Economics - Spring 1999 / 7 Seminar on V.I. in Network Economics - Spring 1999 / 8 1. Introduction to network 2. Equilibrium in network economics (6) economics (1) Portfolio management An equilibrium is reached when none Assets - Liabilities - Flow of funds of the agents / players have incentives • m sectors of activities to deviate from their current plans (of • n financial “instruments” in each sector actions). • max. mean but min. variance How to behave in investments? S ystems S ystems Analysis Laboratory Analysis Laboratory Pierre-Olivier Pineau Pierre-Olivier Pineau Helsinki University of Technology Helsinki University of Technology Seminar on V.I. in Network Economics - Spring 1999 / 9 Seminar on V.I. in Network Economics - Spring 1999 / 10 2. Equilibrium in network 2. Equilibrium in network economics (2) economics (3) ⇒ How to characterize an equilibrium? Equilibria ⇒ We’ll be looking for conditions of the equilibrium Partial General ⇒ But most of the time, these conditions are already given. Perfect Imperfect ⇒ We “simply” use them to compute the competition competition equilibrium. S ystems S ystems Analysis Laboratory Analysis Laboratory Pierre-Olivier Pineau Pierre-Olivier Pineau Helsinki University of Technology Helsinki University of Technology Seminar on V.I. in Network Economics - Spring 1999 / 11 Seminar on V.I. in Network Economics - Spring 1999 / 12

2. Equilibrium in network 2. Equilibrium in network economics (4) economics (5) Example (p. 4-7 Harker) Key questions in equilibrium analysis: •Simple network transportation problem in linear programming ⇒ optimality conditions •Existence •Addition of supply and demand functions ⇒ •Uniqueness •Stability (sensitivity analysis / equilibrium conditions comparative static) •They correspond to Karush-Khun-Tucker of a nonlinear program S ystems S ystems Analysis Laboratory Analysis Laboratory Pierre-Olivier Pineau Pierre-Olivier Pineau Helsinki University of Technology Helsinki University of Technology Seminar on V.I. in Network Economics - Spring 1999 / 13 Seminar on V.I. in Network Economics - Spring 1999 / 14 3. Variational inequalities (1) 3. Variational inequalities (2) Infinite-dimensional Variational Inequalities metric space They are one methodology to study and solve network equilibrium problem. Finite-dimensional Other are: Euclidean space - Optimization Computational efficiency for solving - Fixed point approaches large-scale equilibrium problems - Nonlinear complementary formulations S ystems S ystems Analysis Laboratory Analysis Laboratory Pierre-Olivier Pineau Pierre-Olivier Pineau Helsinki University of Technology Helsinki University of Technology Seminar on V.I. in Network Economics - Spring 1999 / 15 Seminar on V.I. in Network Economics - Spring 1999 / 16 3. Variational inequalities (3) 3. Variational inequalities (4) Simplest example Let X be a nonempty subset of R n and let The solution x* of the system of equations F be a mapping from R n into itself. The F(x) = 0 variational inequality problem , denoted where F maps R n into itself is the solution by VI (X , F ), is to find a vector x* in X to VI (X , F ) because only F(x*) will satisfy such that F(x*) T (y - x*) ≥ ≥ 0 for all y in X F(x*) T (y - x*) ≥ 0 for all y in X S ystems S ystems Analysis Laboratory Analysis Laboratory Pierre-Olivier Pineau Pierre-Olivier Pineau Helsinki University of Technology Helsinki University of Technology Seminar on V.I. in Network Economics - Spring 1999 / 17 Seminar on V.I. in Network Economics - Spring 1999 / 18

3. Variational inequalities (5) 3. Variational inequalities (6) Optimization problem Optimization problem Statements (i) to (iii) can be summarized by Let f be a smooth real function on I = [ a,b ] the VI We seek the points x* in I for which f’(x*)(x - x*) ≥ 0 for all x in I f(x*) = min f(x) (i) if a < x* < b , then f’(x*) = 0 (ii) if x* = a , then f’(x*) ≥ 0 A solution to VI(I, f’ ) is then also a solution to the original optimization problem (iii) if x* = b , then f’(x*) ≤ 0 S ystems S ystems Analysis Laboratory Analysis Laboratory Pierre-Olivier Pineau Pierre-Olivier Pineau Helsinki University of Technology Helsinki University of Technology Seminar on V.I. in Network Economics - Spring 1999 / 19 Seminar on V.I. in Network Economics - Spring 1999 / 20 4. Related mathematical 3. Variational inequalities (7) concepts (1) Other uses •Notions of mathematical programming VI can also be used to create alternative •Linear programming •Kuhn-Tucker conditions formulations for •First order optimality conditions • Nonlinear complementary problems •Elementary topology • Fixed-point problems (projection •Convexity formulation) •Open, closed, compact sets S ystems S ystems Analysis Laboratory Analysis Laboratory Pierre-Olivier Pineau Pierre-Olivier Pineau Helsinki University of Technology Helsinki University of Technology Seminar on V.I. in Network Economics - Spring 1999 / 21 Seminar on V.I. in Network Economics - Spring 1999 / 22 4. Related mathematical 5. One application (1) concepts (2) Oligopolistic equilibrium (no network) •Notions of analysis The situation is: •Continuity • n firms i compete •Taylor’s series • the cost function c i (x i ) is stricly increasing and convex •Gradient, Jacobian matrix •Linear algebra • the market price p(Q) (inverse demand curve, with Q = Σ x i ) is stricly decreasing •Fixed point theorem • Qp(Q) is concave S ystems S ystems Analysis Laboratory Analysis Laboratory Pierre-Olivier Pineau Pierre-Olivier Pineau Helsinki University of Technology Helsinki University of Technology Seminar on V.I. in Network Economics - Spring 1999 / 23 Seminar on V.I. in Network Economics - Spring 1999 / 24