Computer-Assisted Proof of Existence of Generalized Nash Equilibrium - - PowerPoint PPT Presentation

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case

Computer-Assisted Proof of Existence of Generalized Nash Equilibrium

Zhengyu Wang

Department of Mathematics, Nanjing University

The Asian Symposium on Computer Mathematics (ASCM) On The Latest Progress In Verified Computation October 27, 2012, Beijing

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case

Outline

1

Motivation of the Study Generalized Nash Equilibrium Problem Central Questions

2

Computer-Assisted Proof Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

3

Comments on Dynamic Case

2 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Outline

1

Motivation of the Study Generalized Nash Equilibrium Problem Central Questions

2

Computer-Assisted Proof Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

3

Comments on Dynamic Case

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

The Problem

Definition (Generalized Nash equilibrium problem) A multi-agent optimization problem in which the players

• ptimize their individual objective functions subject to resource

constraints, both the objective functions and the resource constraints depend on the other rivals’ strategies. Significance: natural extension of the Nash equilibrium problem, powerful and unifying setting for models with competition

economics, environmental pollution control, · · · ; transportation programming, · · · ;

• ptimal control problems with multi-criteria, like in

aero-structural aircraft wing shape design.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

The Problem

Definition (Generalized Nash equilibrium problem) A multi-agent optimization problem in which the players

• ptimize their individual objective functions subject to resource

constraints, both the objective functions and the resource constraints depend on the other rivals’ strategies. Significance: natural extension of the Nash equilibrium problem, powerful and unifying setting for models with competition

economics, environmental pollution control, · · · ; transportation programming, · · · ;

• ptimal control problems with multi-criteria, like in

aero-structural aircraft wing shape design.

4 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

The Problem

Definition (Generalized Nash equilibrium problem) A multi-agent optimization problem in which the players

• ptimize their individual objective functions subject to resource

constraints, both the objective functions and the resource constraints depend on the other rivals’ strategies. Significance: natural extension of the Nash equilibrium problem, powerful and unifying setting for models with competition

economics, environmental pollution control, · · · ; transportation programming, · · · ;

• ptimal control problems with multi-criteria, like in

aero-structural aircraft wing shape design.

4 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

The Problem

Definition (Generalized Nash equilibrium problem) A multi-agent optimization problem in which the players

• ptimize their individual objective functions subject to resource

constraints, both the objective functions and the resource constraints depend on the other rivals’ strategies. Significance: natural extension of the Nash equilibrium problem, powerful and unifying setting for models with competition

economics, environmental pollution control, · · · ; transportation programming, · · · ;

• ptimal control problems with multi-criteria, like in

aero-structural aircraft wing shape design.

4 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

The Problem

Definition (Generalized Nash equilibrium problem) A multi-agent optimization problem in which the players

• ptimize their individual objective functions subject to resource

constraints, both the objective functions and the resource constraints depend on the other rivals’ strategies. Significance: natural extension of the Nash equilibrium problem, powerful and unifying setting for models with competition

economics, environmental pollution control, · · · ; transportation programming, · · · ;

• ptimal control problems with multi-criteria, like in

aero-structural aircraft wing shape design.

4 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

The Problem

Definition (Generalized Nash equilibrium problem) A multi-agent optimization problem in which the players

• ptimize their individual objective functions subject to resource

constraints, both the objective functions and the resource constraints depend on the other rivals’ strategies. Significance: natural extension of the Nash equilibrium problem, powerful and unifying setting for models with competition

economics, environmental pollution control, · · · ; transportation programming, · · · ;

• ptimal control problems with multi-criteria, like in

aero-structural aircraft wing shape design.

4 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

The Problem

Definition (Generalized Nash equilibrium problem) A multi-agent optimization problem in which the players

• ptimize their individual objective functions subject to resource

constraints, both the objective functions and the resource constraints depend on the other rivals’ strategies. Significance: natural extension of the Nash equilibrium problem, powerful and unifying setting for models with competition

economics, environmental pollution control, · · · ; transportation programming, · · · ;

• ptimal control problems with multi-criteria, like in

aero-structural aircraft wing shape design.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Notations (for a N-agent non-cooperative game)

strategy profiles for the ν-th player:

strategy xν ∈ Rnν of the ν-th player profile of all the players’ strategies x = (xν)N

ν=1 ∈ Rn;

profile of the rivals’ strategies x−ν = (xν′)ν′=ν;

state profiles for the ν-th player:

state uν ∈ Rmν of the ν-th player profile of all the players’ states u = (uν)N

ν=1;

profile of the rivals’ state u−ν = (uν′)ν′=ν;

the state equation Gν(xν, uν) = 0 (differential equation); the objective functional: θν(·, x−ν, u); the strategy set Sν(x−ν) := {xν : gν(xν, x−ν) ≤ 0}.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Notations (for a N-agent non-cooperative game)

strategy profiles for the ν-th player:

strategy xν ∈ Rnν of the ν-th player profile of all the players’ strategies x = (xν)N

ν=1 ∈ Rn;

profile of the rivals’ strategies x−ν = (xν′)ν′=ν;

state profiles for the ν-th player:

state uν ∈ Rmν of the ν-th player profile of all the players’ states u = (uν)N

ν=1;

profile of the rivals’ state u−ν = (uν′)ν′=ν;

the state equation Gν(xν, uν) = 0 (differential equation); the objective functional: θν(·, x−ν, u); the strategy set Sν(x−ν) := {xν : gν(xν, x−ν) ≤ 0}.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Notations (for a N-agent non-cooperative game)

strategy profiles for the ν-th player:

strategy xν ∈ Rnν of the ν-th player profile of all the players’ strategies x = (xν)N

ν=1 ∈ Rn;

profile of the rivals’ strategies x−ν = (xν′)ν′=ν;

state profiles for the ν-th player:

state uν ∈ Rmν of the ν-th player profile of all the players’ states u = (uν)N

ν=1;

profile of the rivals’ state u−ν = (uν′)ν′=ν;

the state equation Gν(xν, uν) = 0 (differential equation); the objective functional: θν(·, x−ν, u); the strategy set Sν(x−ν) := {xν : gν(xν, x−ν) ≤ 0}.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Notations (for a N-agent non-cooperative game)

strategy profiles for the ν-th player:

strategy xν ∈ Rnν of the ν-th player profile of all the players’ strategies x = (xν)N

ν=1 ∈ Rn;

profile of the rivals’ strategies x−ν = (xν′)ν′=ν;

state profiles for the ν-th player:

state uν ∈ Rmν of the ν-th player profile of all the players’ states u = (uν)N

ν=1;

profile of the rivals’ state u−ν = (uν′)ν′=ν;

the state equation Gν(xν, uν) = 0 (differential equation); the objective functional: θν(·, x−ν, u); the strategy set Sν(x−ν) := {xν : gν(xν, x−ν) ≤ 0}.

5 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Notations (for a N-agent non-cooperative game)

strategy profiles for the ν-th player:

strategy xν ∈ Rnν of the ν-th player profile of all the players’ strategies x = (xν)N

ν=1 ∈ Rn;

profile of the rivals’ strategies x−ν = (xν′)ν′=ν;

state profiles for the ν-th player:

state uν ∈ Rmν of the ν-th player profile of all the players’ states u = (uν)N

ν=1;

profile of the rivals’ state u−ν = (uν′)ν′=ν;

the state equation Gν(xν, uν) = 0 (differential equation); the objective functional: θν(·, x−ν, u); the strategy set Sν(x−ν) := {xν : gν(xν, x−ν) ≤ 0}.

5 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Notations (for a N-agent non-cooperative game)

strategy profiles for the ν-th player:

strategy xν ∈ Rnν of the ν-th player profile of all the players’ strategies x = (xν)N

ν=1 ∈ Rn;

profile of the rivals’ strategies x−ν = (xν′)ν′=ν;

state profiles for the ν-th player:

state uν ∈ Rmν of the ν-th player profile of all the players’ states u = (uν)N

ν=1;

profile of the rivals’ state u−ν = (uν′)ν′=ν;

the state equation Gν(xν, uν) = 0 (differential equation); the objective functional: θν(·, x−ν, u); the strategy set Sν(x−ν) := {xν : gν(xν, x−ν) ≤ 0}.

5 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Notations (for a N-agent non-cooperative game)

strategy profiles for the ν-th player:

strategy xν ∈ Rnν of the ν-th player profile of all the players’ strategies x = (xν)N

ν=1 ∈ Rn;

profile of the rivals’ strategies x−ν = (xν′)ν′=ν;

state profiles for the ν-th player:

state uν ∈ Rmν of the ν-th player profile of all the players’ states u = (uν)N

ν=1;

profile of the rivals’ state u−ν = (uν′)ν′=ν;

the state equation Gν(xν, uν) = 0 (differential equation); the objective functional: θν(·, x−ν, u); the strategy set Sν(x−ν) := {xν : gν(xν, x−ν) ≤ 0}.

5 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Notations (for a N-agent non-cooperative game)

strategy profiles for the ν-th player:

strategy xν ∈ Rnν of the ν-th player profile of all the players’ strategies x = (xν)N

ν=1 ∈ Rn;

profile of the rivals’ strategies x−ν = (xν′)ν′=ν;

state profiles for the ν-th player:

state uν ∈ Rmν of the ν-th player profile of all the players’ states u = (uν)N

ν=1;

profile of the rivals’ state u−ν = (uν′)ν′=ν;

the state equation Gν(xν, uν) = 0 (differential equation); the objective functional: θν(·, x−ν, u); the strategy set Sν(x−ν) := {xν : gν(xν, x−ν) ≤ 0}.

5 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Notations (for a N-agent non-cooperative game)

strategy profiles for the ν-th player:

strategy xν ∈ Rnν of the ν-th player profile of all the players’ strategies x = (xν)N

ν=1 ∈ Rn;

profile of the rivals’ strategies x−ν = (xν′)ν′=ν;

state profiles for the ν-th player:

state uν ∈ Rmν of the ν-th player profile of all the players’ states u = (uν)N

ν=1;

profile of the rivals’ state u−ν = (uν′)ν′=ν;

the state equation Gν(xν, uν) = 0 (differential equation); the objective functional: θν(·, x−ν, u); the strategy set Sν(x−ν) := {xν : gν(xν, x−ν) ≤ 0}.

5 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Notations (for a N-agent non-cooperative game)

strategy profiles for the ν-th player:

strategy xν ∈ Rnν of the ν-th player profile of all the players’ strategies x = (xν)N

ν=1 ∈ Rn;

profile of the rivals’ strategies x−ν = (xν′)ν′=ν;

state profiles for the ν-th player:

state uν ∈ Rmν of the ν-th player profile of all the players’ states u = (uν)N

ν=1;

profile of the rivals’ state u−ν = (uν′)ν′=ν;

the state equation Gν(xν, uν) = 0 (differential equation); the objective functional: θν(·, x−ν, u); the strategy set Sν(x−ν) := {xν : gν(xν, x−ν) ≤ 0}.

5 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Notations (for a N-agent non-cooperative game)

strategy profiles for the ν-th player:

strategy xν ∈ Rnν of the ν-th player profile of all the players’ strategies x = (xν)N

ν=1 ∈ Rn;

profile of the rivals’ strategies x−ν = (xν′)ν′=ν;

state profiles for the ν-th player:

state uν ∈ Rmν of the ν-th player profile of all the players’ states u = (uν)N

ν=1;

profile of the rivals’ state u−ν = (uν′)ν′=ν;

the state equation Gν(xν, uν) = 0 (differential equation); the objective functional: θν(·, x−ν, u); the strategy set Sν(x−ν) := {xν : gν(xν, x−ν) ≤ 0}.

5 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Mathematical Formulation

Find a GNE (or called a solution of the GNEP): (x∗, u∗) = (x∗

ν, u∗ ν)N ν=1 such that for each ν, x∗ ν solves

minxν θν(xν, x∗

−ν, u∗)

s.t. Gν(x∗

ν, u∗ ν) = 0

gν(xν, x∗

−ν) ≤ 0

Different Settings for State Equation Gν(x∗

ν, u∗ ν) = 0:

static problem: no state variables involved; dynamic problem with Gν(x∗

ν, u∗ ν) = 0 being

• rdinary differential equation;

partial differential equations.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Mathematical Formulation

Find a GNE (or called a solution of the GNEP): (x∗, u∗) = (x∗

ν, u∗ ν)N ν=1 such that for each ν, x∗ ν solves

minxν θν(xν, x∗

−ν, u∗)

s.t. Gν(x∗

ν, u∗ ν) = 0

gν(xν, x∗

−ν) ≤ 0

Different Settings for State Equation Gν(x∗

ν, u∗ ν) = 0:

static problem: no state variables involved; dynamic problem with Gν(x∗

ν, u∗ ν) = 0 being

• rdinary differential equation;

partial differential equations.

6 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Mathematical Formulation

Find a GNE (or called a solution of the GNEP): (x∗, u∗) = (x∗

ν, u∗ ν)N ν=1 such that for each ν, x∗ ν solves

minxν θν(xν, x∗

−ν, u∗)

s.t. Gν(x∗

ν, u∗ ν) = 0

gν(xν, x∗

−ν) ≤ 0

Different Settings for State Equation Gν(x∗

ν, u∗ ν) = 0:

static problem: no state variables involved; dynamic problem with Gν(x∗

ν, u∗ ν) = 0 being

• rdinary differential equation;

partial differential equations.

6 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Mathematical Formulation

Find a GNE (or called a solution of the GNEP): (x∗, u∗) = (x∗

ν, u∗ ν)N ν=1 such that for each ν, x∗ ν solves

minxν θν(xν, x∗

−ν, u∗)

s.t. Gν(x∗

ν, u∗ ν) = 0

gν(xν, x∗

−ν) ≤ 0

Different Settings for State Equation Gν(x∗

ν, u∗ ν) = 0:

static problem: no state variables involved; dynamic problem with Gν(x∗

ν, u∗ ν) = 0 being

• rdinary differential equation;

partial differential equations.

6 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Mathematical Formulation

Find a GNE (or called a solution of the GNEP): (x∗, u∗) = (x∗

ν, u∗ ν)N ν=1 such that for each ν, x∗ ν solves

minxν θν(xν, x∗

−ν, u∗)

s.t. Gν(x∗

ν, u∗ ν) = 0

gν(xν, x∗

−ν) ≤ 0

Different Settings for State Equation Gν(x∗

ν, u∗ ν) = 0:

static problem: no state variables involved; dynamic problem with Gν(x∗

ν, u∗ ν) = 0 being

• rdinary differential equation;

partial differential equations.

6 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Mathematical Formulation

Find a GNE (or called a solution of the GNEP): (x∗, u∗) = (x∗

ν, u∗ ν)N ν=1 such that for each ν, x∗ ν solves

minxν θν(xν, x∗

−ν, u∗)

s.t. Gν(x∗

ν, u∗ ν) = 0

gν(xν, x∗

−ν) ≤ 0

Different Settings for State Equation Gν(x∗

ν, u∗ ν) = 0:

static problem: no state variables involved; dynamic problem with Gν(x∗

ν, u∗ ν) = 0 being

• rdinary differential equation;

partial differential equations.

6 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Dynamic Model

A dynamic game model governed by an ordinary differential equation, the minimization problems for the players: minx1 θ1(x1, x∗

2, u∗ 1, u∗ 2)

s.t. ˙ u1 = F1(x∗

1, x∗ 2, u1, u∗ 2)

u1(0) = x0

1

and minx2 θ2(x∗

1, x2, u∗ 1, u∗ 2)

s.t. ˙ u2 = F2(x∗

1, x∗ 2, u∗ 1, u2)

u2(0) = u0

2

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Dynamic Model (continued)

A dynamic game governed by a partial differential equation, the minimization problems for the players: minx1 θ1(x1, x∗

2, u)

s.t. −∆u = F1(x∗

1, x∗ 2, u) in Ω

u = 0 on ∂Ω x1 ∈ X1 and minx2 θ2(x∗

1, x2, u)

s.t. −∆u = F2(x∗

1, x∗ 2, u) in Ω

u = 0 on ∂Ω x2 ∈ X2

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Outline

1

Motivation of the Study Generalized Nash Equilibrium Problem Central Questions

2

Computer-Assisted Proof Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

3

Comments on Dynamic Case

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Central Questions

Existence of GNE in a given domain is of importance, but how to rigorously validate it via practical manner? How to compute an approximate solution with a guaranteed error bound accompanied; How to get an initial point close enough to an exact GNE for starting an algorithm? A realistic GNEP normally contains a data vector, which is unknown or known inexactly when the solution has to be determined, how to cope with various realization of the data?

10 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Central Questions

Existence of GNE in a given domain is of importance, but how to rigorously validate it via practical manner? How to compute an approximate solution with a guaranteed error bound accompanied; How to get an initial point close enough to an exact GNE for starting an algorithm? A realistic GNEP normally contains a data vector, which is unknown or known inexactly when the solution has to be determined, how to cope with various realization of the data?

10 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Central Questions

Existence of GNE in a given domain is of importance, but how to rigorously validate it via practical manner? How to compute an approximate solution with a guaranteed error bound accompanied; How to get an initial point close enough to an exact GNE for starting an algorithm? A realistic GNEP normally contains a data vector, which is unknown or known inexactly when the solution has to be determined, how to cope with various realization of the data?

10 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Generalized Nash Equilibrium Problem Central Questions

Central Questions

Existence of GNE in a given domain is of importance, but how to rigorously validate it via practical manner? How to compute an approximate solution with a guaranteed error bound accompanied; How to get an initial point close enough to an exact GNE for starting an algorithm? A realistic GNEP normally contains a data vector, which is unknown or known inexactly when the solution has to be determined, how to cope with various realization of the data?

10 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Outline

1

Motivation of the Study Generalized Nash Equilibrium Problem Central Questions

2

Computer-Assisted Proof Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

3

Comments on Dynamic Case

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Idea of Computer-Assisted Proof

Definition A computer-assisted proof is a mathematical proof that has been at least partially generated by computer. The idea is to use a computer program to perform lengthy computations, and to provide a proof that the result of these computations implies the given theorem. A computer-assisted proof of existence is to rigorously validate the conditions of some existence theorems by numeric calculations for controlling the round-off and propagation errors through the interval arithmetic technique, and symbolic calculations.

12 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Idea of Computer-Assisted Proof

Definition A computer-assisted proof is a mathematical proof that has been at least partially generated by computer. The idea is to use a computer program to perform lengthy computations, and to provide a proof that the result of these computations implies the given theorem. A computer-assisted proof of existence is to rigorously validate the conditions of some existence theorems by numeric calculations for controlling the round-off and propagation errors through the interval arithmetic technique, and symbolic calculations.

12 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Idea of Computer-Assisted Proof

Definition A computer-assisted proof is a mathematical proof that has been at least partially generated by computer. The idea is to use a computer program to perform lengthy computations, and to provide a proof that the result of these computations implies the given theorem. A computer-assisted proof of existence is to rigorously validate the conditions of some existence theorems by numeric calculations for controlling the round-off and propagation errors through the interval arithmetic technique, and symbolic calculations.

12 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Idea of Computer-Assisted Proof

Definition A computer-assisted proof is a mathematical proof that has been at least partially generated by computer. The idea is to use a computer program to perform lengthy computations, and to provide a proof that the result of these computations implies the given theorem. A computer-assisted proof of existence is to rigorously validate the conditions of some existence theorems by numeric calculations for controlling the round-off and propagation errors through the interval arithmetic technique, and symbolic calculations.

12 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Interval Arithmetic Technique

Interval Arithmetic for Error Tracking Computation is reduced to a sequence of elementary

• perations, say (+,-,*,/); the result of an elementary
• peration is rounded off by the computer precision.

Construct an interval provided by upper and lower bounds

• n the result of an elementary operation, and replace

numbers with intervals and perform elementary operations between intervals of representable numbers. Existing Software, i.e.: IntLab, MatLab toolbox, by Rump’s team, and C-XSC, by Karlsruhe, Wuppertal University.

13 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Interval Arithmetic Technique

Interval Arithmetic for Error Tracking Computation is reduced to a sequence of elementary

• perations, say (+,-,*,/); the result of an elementary
• peration is rounded off by the computer precision.

Construct an interval provided by upper and lower bounds

• n the result of an elementary operation, and replace

numbers with intervals and perform elementary operations between intervals of representable numbers. Existing Software, i.e.: IntLab, MatLab toolbox, by Rump’s team, and C-XSC, by Karlsruhe, Wuppertal University.

13 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Interval Arithmetic Technique

Interval Arithmetic for Error Tracking Computation is reduced to a sequence of elementary

• perations, say (+,-,*,/); the result of an elementary
• peration is rounded off by the computer precision.

Construct an interval provided by upper and lower bounds

• n the result of an elementary operation, and replace

numbers with intervals and perform elementary operations between intervals of representable numbers. Existing Software, i.e.: IntLab, MatLab toolbox, by Rump’s team, and C-XSC, by Karlsruhe, Wuppertal University.

13 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Interval Arithmetic Technique

Interval Arithmetic for Error Tracking Computation is reduced to a sequence of elementary

• perations, say (+,-,*,/); the result of an elementary
• peration is rounded off by the computer precision.

Construct an interval provided by upper and lower bounds

• n the result of an elementary operation, and replace

numbers with intervals and perform elementary operations between intervals of representable numbers. Existing Software, i.e.: IntLab, MatLab toolbox, by Rump’s team, and C-XSC, by Karlsruhe, Wuppertal University.

13 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Interval Arithmetic Technique

Interval Arithmetic for Error Tracking Computation is reduced to a sequence of elementary

• perations, say (+,-,*,/); the result of an elementary
• peration is rounded off by the computer precision.

Construct an interval provided by upper and lower bounds

• n the result of an elementary operation, and replace

numbers with intervals and perform elementary operations between intervals of representable numbers. Existing Software, i.e.: IntLab, MatLab toolbox, by Rump’s team, and C-XSC, by Karlsruhe, Wuppertal University.

13 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Outline

1

Motivation of the Study Generalized Nash Equilibrium Problem Central Questions

2

Computer-Assisted Proof Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

3

Comments on Dynamic Case

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Computer-Assisted Proof and Validated Solution

Validated Solution An approximation ˆ x of an exact solution x∗ accompanied with an interval vector [e] with the property e(ˆ x) = x∗ − ˆ x ∈ [e]. Here [e] covers all the possible errors aroused in performing the algorithms on floating point system, and therefore offers a guaranteed error bound. The validated solution answers the existence of a GNE in a given domain; yields safe practical solutions;

• ffers an initial vector close enough to an unknown exact

solution, for starting locally convergent algorithms.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Computer-Assisted Proof and Validated Solution

Validated Solution An approximation ˆ x of an exact solution x∗ accompanied with an interval vector [e] with the property e(ˆ x) = x∗ − ˆ x ∈ [e]. Here [e] covers all the possible errors aroused in performing the algorithms on floating point system, and therefore offers a guaranteed error bound. The validated solution answers the existence of a GNE in a given domain; yields safe practical solutions;

• ffers an initial vector close enough to an unknown exact

solution, for starting locally convergent algorithms.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Computer-Assisted Proof and Validated Solution

Validated Solution An approximation ˆ x of an exact solution x∗ accompanied with an interval vector [e] with the property e(ˆ x) = x∗ − ˆ x ∈ [e]. Here [e] covers all the possible errors aroused in performing the algorithms on floating point system, and therefore offers a guaranteed error bound. The validated solution answers the existence of a GNE in a given domain; yields safe practical solutions;

• ffers an initial vector close enough to an unknown exact

solution, for starting locally convergent algorithms.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Computer-Assisted Proof and Validated Solution

Validated Solution An approximation ˆ x of an exact solution x∗ accompanied with an interval vector [e] with the property e(ˆ x) = x∗ − ˆ x ∈ [e]. Here [e] covers all the possible errors aroused in performing the algorithms on floating point system, and therefore offers a guaranteed error bound. The validated solution answers the existence of a GNE in a given domain; yields safe practical solutions;

• ffers an initial vector close enough to an unknown exact

solution, for starting locally convergent algorithms.

15 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Computer-Assisted Proof and Validated Solution

Validated Solution An approximation ˆ x of an exact solution x∗ accompanied with an interval vector [e] with the property e(ˆ x) = x∗ − ˆ x ∈ [e]. Here [e] covers all the possible errors aroused in performing the algorithms on floating point system, and therefore offers a guaranteed error bound. The validated solution answers the existence of a GNE in a given domain; yields safe practical solutions;

• ffers an initial vector close enough to an unknown exact

solution, for starting locally convergent algorithms.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Computer-Assisted Proof and Validated Solution (continued)

A validated solution can be obtained by: starting from solution validation, and then by two procedures for sharpening, based

• n the computer-assisted proof:

Solution Validation: proves that x∗ ∈ ˆ x + [e] for an interval vector [e], where the proof should be rigorously performed by computation; Bounding Procedure: computes an interval vector, in which the validation procedure gives a positive answer; Interval Iteration: computes a sequence of interval vectors when the initial one can be provided by the bounding procedure.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Computer-Assisted Proof and Validated Solution (continued)

A validated solution can be obtained by: starting from solution validation, and then by two procedures for sharpening, based

• n the computer-assisted proof:

Solution Validation: proves that x∗ ∈ ˆ x + [e] for an interval vector [e], where the proof should be rigorously performed by computation; Bounding Procedure: computes an interval vector, in which the validation procedure gives a positive answer; Interval Iteration: computes a sequence of interval vectors when the initial one can be provided by the bounding procedure.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Computer-Assisted Proof and Validated Solution (continued)

A validated solution can be obtained by: starting from solution validation, and then by two procedures for sharpening, based

• n the computer-assisted proof:

Solution Validation: proves that x∗ ∈ ˆ x + [e] for an interval vector [e], where the proof should be rigorously performed by computation; Bounding Procedure: computes an interval vector, in which the validation procedure gives a positive answer; Interval Iteration: computes a sequence of interval vectors when the initial one can be provided by the bounding procedure.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Computer-Assisted Proof and Validated Solution (continued)

A validated solution can be obtained by: starting from solution validation, and then by two procedures for sharpening, based

• n the computer-assisted proof:

Solution Validation: proves that x∗ ∈ ˆ x + [e] for an interval vector [e], where the proof should be rigorously performed by computation; Bounding Procedure: computes an interval vector, in which the validation procedure gives a positive answer; Interval Iteration: computes a sequence of interval vectors when the initial one can be provided by the bounding procedure.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Computer-Assisted Proof and Validated Solution (continued)

A validated solution can be obtained by: starting from solution validation, and then by two procedures for sharpening, based

• n the computer-assisted proof:

Solution Validation: proves that x∗ ∈ ˆ x + [e] for an interval vector [e], where the proof should be rigorously performed by computation; Bounding Procedure: computes an interval vector, in which the validation procedure gives a positive answer; Interval Iteration: computes a sequence of interval vectors when the initial one can be provided by the bounding procedure.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Computer-Assisted Proof and Validated Solution (continued)

A validated solution can be obtained by: starting from solution validation, and then by two procedures for sharpening, based

• n the computer-assisted proof:

Solution Validation: proves that x∗ ∈ ˆ x + [e] for an interval vector [e], where the proof should be rigorously performed by computation; Bounding Procedure: computes an interval vector, in which the validation procedure gives a positive answer; Interval Iteration: computes a sequence of interval vectors when the initial one can be provided by the bounding procedure.

16 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Computer-Assisted Proof and Validated Solution (continued)

A validated solution can be obtained by: starting from solution validation, and then by two procedures for sharpening, based

• n the computer-assisted proof:

Solution Validation: proves that x∗ ∈ ˆ x + [e] for an interval vector [e], where the proof should be rigorously performed by computation; Bounding Procedure: computes an interval vector, in which the validation procedure gives a positive answer; Interval Iteration: computes a sequence of interval vectors when the initial one can be provided by the bounding procedure.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Outline

1

Motivation of the Study Generalized Nash Equilibrium Problem Central Questions

2

Computer-Assisted Proof Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

3

Comments on Dynamic Case

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Equation Reformulation of GNEP

GNEP: Find x∗ = (x∗

ν)N ν=1 such that for each ν, x∗ ν solves

minxν θν(xν, x∗

−ν)

s.t. gν(xν, x∗

−ν) ≤ 0

Write its KKT system: ∇xνθν(xν, x−ν) = 0 0 ≤ yν⊥ − gν(xν, x−ν) ≥ 0 and concentrate all the KKT systems, it gives the mixed complementarity system: F(x) = 0 0 ≤ y⊥G(x) ≥ 0

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Equation Reformulation of GNEP

GNEP: Find x∗ = (x∗

ν)N ν=1 such that for each ν, x∗ ν solves

minxν θν(xν, x∗

−ν)

s.t. gν(xν, x∗

−ν) ≤ 0

Write its KKT system: ∇xνθν(xν, x−ν) = 0 0 ≤ yν⊥ − gν(xν, x−ν) ≥ 0 and concentrate all the KKT systems, it gives the mixed complementarity system: F(x) = 0 0 ≤ y⊥G(x) ≥ 0

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Equation Reformulation of GNEP

GNEP: Find x∗ = (x∗

ν)N ν=1 such that for each ν, x∗ ν solves

minxν θν(xν, x∗

−ν)

s.t. gν(xν, x∗

−ν) ≤ 0

Write its KKT system: ∇xνθν(xν, x−ν) = 0 0 ≤ yν⊥ − gν(xν, x−ν) ≥ 0 and concentrate all the KKT systems, it gives the mixed complementarity system: F(x) = 0 0 ≤ y⊥G(x) ≥ 0

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Equation Reformulation of GNEP (continued)

Complementarity problem can be equivalently reformulated into 0 = H(z) :=

• F(x)

min{y, G(x)}

• ,

where z = (x, y), F : Rn → Rn and G : Rn → Rm. Interval Slope Interval slope δF(ˆ x, [x]): interval matrix such that F(x) − F(ˆ x) ∈ δF(ˆ x, [x])([x] − ˆ x), ∀x ∈ [x]. Differentiable case: can be computed by existing software; Non-differentiable case: ???

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Equation Reformulation of GNEP (continued)

Complementarity problem can be equivalently reformulated into 0 = H(z) :=

• F(x)

min{y, G(x)}

• ,

where z = (x, y), F : Rn → Rn and G : Rn → Rm. Interval Slope Interval slope δF(ˆ x, [x]): interval matrix such that F(x) − F(ˆ x) ∈ δF(ˆ x, [x])([x] − ˆ x), ∀x ∈ [x]. Differentiable case: can be computed by existing software; Non-differentiable case: ???

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Equation Reformulation of GNEP (continued)

Complementarity problem can be equivalently reformulated into 0 = H(z) :=

• F(x)

min{y, G(x)}

• ,

where z = (x, y), F : Rn → Rn and G : Rn → Rm. Interval Slope Interval slope δF(ˆ x, [x]): interval matrix such that F(x) − F(ˆ x) ∈ δF(ˆ x, [x])([x] − ˆ x), ∀x ∈ [x]. Differentiable case: can be computed by existing software; Non-differentiable case: ???

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Equation Reformulation of GNEP (continued)

Complementarity problem can be equivalently reformulated into 0 = H(z) :=

• F(x)

min{y, G(x)}

• ,

where z = (x, y), F : Rn → Rn and G : Rn → Rm. Interval Slope Interval slope δF(ˆ x, [x]): interval matrix such that F(x) − F(ˆ x) ∈ δF(ˆ x, [x])([x] − ˆ x), ∀x ∈ [x]. Differentiable case: can be computed by existing software; Non-differentiable case: ???

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Computer-Assisted Proof by Moore Test

Theorem (Moore Test) For a [z] = ([x], [y]) and a fixed ˆ z ∈ [z], if ˆ z − AH(ˆ z) + (I − AδH(ˆ z, [z])([z] − ˆ z) =: K(ˆ z, [z], A) ⊆ [z], H(z) = 0 has a solution in [z], and then [x] contains a GNE . Alefeld, Chen and Potra, Numer. Math., 1999 and 2002. Interval slope obtained by solving optimization problems. Other computer-assisted proof, more readily giving positive answer?

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Computer-Assisted Proof by Moore Test

Theorem (Moore Test) For a [z] = ([x], [y]) and a fixed ˆ z ∈ [z], if ˆ z − AH(ˆ z) + (I − AδH(ˆ z, [z])([z] − ˆ z) =: K(ˆ z, [z], A) ⊆ [z], H(z) = 0 has a solution in [z], and then [x] contains a GNE . Alefeld, Chen and Potra, Numer. Math., 1999 and 2002. Interval slope obtained by solving optimization problems. Other computer-assisted proof, more readily giving positive answer?

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Computer-Assisted Proof by Moore Test

Theorem (Moore Test) For a [z] = ([x], [y]) and a fixed ˆ z ∈ [z], if ˆ z − AH(ˆ z) + (I − AδH(ˆ z, [z])([z] − ˆ z) =: K(ˆ z, [z], A) ⊆ [z], H(z) = 0 has a solution in [z], and then [x] contains a GNE . Alefeld, Chen and Potra, Numer. Math., 1999 and 2002. Interval slope obtained by solving optimization problems. Other computer-assisted proof, more readily giving positive answer?

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Computer-Assisted Proof by Moore Test

Theorem (Moore Test) For a [z] = ([x], [y]) and a fixed ˆ z ∈ [z], if ˆ z − AH(ˆ z) + (I − AδH(ˆ z, [z])([z] − ˆ z) =: K(ˆ z, [z], A) ⊆ [z], H(z) = 0 has a solution in [z], and then [x] contains a GNE . Alefeld, Chen and Potra, Numer. Math., 1999 and 2002. Interval slope obtained by solving optimization problems. Other computer-assisted proof, more readily giving positive answer?

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Miranda Theorem

Other computer-assisted proof, more readily giving positive answer? Theorem (Miranda) Let [z] = ˆ z + [−r, r], define its n pairs of parallel and opposite faces [z]+

i

= {z ∈ [z] : zi = ˆ zi + ri}, [z]−

i

= {z ∈ [z] : zi = ˆ zi − ri}, where i = 1, . . . , n. H(z) = 0 has a solution in [z] if for any i (AH(z))i ≤ 0 if z ∈ [z]+

i ,

≥ 0 if z ∈ [z]−

i .

Hard to be directly validated, if not completely impossibly!

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Miranda Theorem

Other computer-assisted proof, more readily giving positive answer? Theorem (Miranda) Let [z] = ˆ z + [−r, r], define its n pairs of parallel and opposite faces [z]+

i

= {z ∈ [z] : zi = ˆ zi + ri}, [z]−

i

= {z ∈ [z] : zi = ˆ zi − ri}, where i = 1, . . . , n. H(z) = 0 has a solution in [z] if for any i (AH(z))i ≤ 0 if z ∈ [z]+

i ,

≥ 0 if z ∈ [z]−

i .

Hard to be directly validated, if not completely impossibly!

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Miranda Theorem

Other computer-assisted proof, more readily giving positive answer? Theorem (Miranda) Let [z] = ˆ z + [−r, r], define its n pairs of parallel and opposite faces [z]+

i

= {z ∈ [z] : zi = ˆ zi + ri}, [z]−

i

= {z ∈ [z] : zi = ˆ zi − ri}, where i = 1, . . . , n. H(z) = 0 has a solution in [z] if for any i (AH(z))i ≤ 0 if z ∈ [z]+

i ,

≥ 0 if z ∈ [z]−

i .

Hard to be directly validated, if not completely impossibly!

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Computation of Interval Slope

Theorem Let ˆ z = (ˆ x, ˆ y), ˆ ui = ˆ yi − Gi(ˆ x), i = 1, . . . , m, and ui and ui be the floating point numbers such that {yi − Gi(x) : z = (x, y) ∈ [z]} ⊆ [ui, ui]. We have an interval slope δH(ˆ z, [z]) of H, where δHi(ˆ x, [x]) =    δFi(ˆ x, [x]) 1 ≤ i ≤ n ei − [αi](ei − δGi(ˆ x, [x])) n + 1 ≤ i ≤ n + m,

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Computation of Interval Slope (continued)

Theorem (continued) [αi] =                            [0, 0] if ui ≤ ˆ ui ≤ ui ≤ 0

• 0,

ui ui − ˆ ui

• if ui ≤ ˆ

ui ≤ 0 < ui

• ˆ

ui ˆ ui − ui , 1

• if ui ≤ 0 < ˆ

ui ≤ ui [1, 1] if 0 < ui ≤ ˆ ui ≤ ui Moreover, we have [αi] ⊆ [0, 1].

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Comments for Computation of Interval Slope

Trivial to compute the range [ui, ui] by existing software, e.g. IntLab. Extended in a naive manner to the interval-valued function by evaluating {yi − [G]i(x) : z = (x, y) ∈ [z]} ⊆ [ui, ui]. namely, computing an interval matrix δ[H](ˆ z, [z]) such that [H](z) − [H](ˆ z) ⊂ δ[H](ˆ z, [z])([z] − ˆ z). Basis for the existence proof for the uncertain GNEP .

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Comments for Computation of Interval Slope

Trivial to compute the range [ui, ui] by existing software, e.g. IntLab. Extended in a naive manner to the interval-valued function by evaluating {yi − [G]i(x) : z = (x, y) ∈ [z]} ⊆ [ui, ui]. namely, computing an interval matrix δ[H](ˆ z, [z]) such that [H](z) − [H](ˆ z) ⊂ δ[H](ˆ z, [z])([z] − ˆ z). Basis for the existence proof for the uncertain GNEP .

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Computer-Assisted Proof Procedure

Theorem Compute K +

i

= [AH(ˆ z + riei)]i + [δAH(ˆ z + riei, [z]+

i )]i([z]+ i − ˆ

z − riei), K −

i

= [AH(ˆ z − riei)]i + [δAH(ˆ z − riei, [z]−

i )]i([z]− i − ˆ

z + riei). If sup

• K −

i

• ≤ 0 ≤ inf
• K +

i

• ,

then (AH(z))i ≤ 0 if z ∈ [z]+

i ,

≥ 0 if z ∈ [z]−

i ,

and then H(z) = 0 has a solution z∗ ∈ [z], and then [x] (the x-component of [z]) contains a GNE.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Comments on the Proof

If ˆ z is close an exact z∗, then sup

• K −

i

• ≤ 0 ≤ inf
• K +

i

• .

holds for a sharp [z] = ˆ z + [−r, r]. If ˆ z − AH(ˆ z) + (I − AδH(ˆ z, [z])([z] − ˆ z) =: K(ˆ z, [z], A) ⊆ [z], then sup

• K −

i

• ≤ 0 ≤ inf
• K +

i

• .

but not vice versa.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case

Comments on the Proof

If ˆ z is close an exact z∗, then sup

• K −

i

• ≤ 0 ≤ inf
• K +

i

• .

holds for a sharp [z] = ˆ z + [−r, r]. If ˆ z − AH(ˆ z) + (I − AδH(ˆ z, [z])([z] − ˆ z) =: K(ˆ z, [z], A) ⊆ [z], then sup

• K −

i

• ≤ 0 ≤ inf
• K +

i

• .

but not vice versa.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case

Idea of Computer-Assisted Proof for Dynamic Model

Validate (e.g. Schauder) fixed point theorem by extending Taylor model, a sum of an approximation of Taylor polynomial and interval vector, e.g., for Optimal Condition ˙ u(t) = F(t, x(t), u(t)) ∈ G(x(t), u(t)), Interval vector covering the remainder, Taylor coefficients obtained by symbolic and numerical computation, requiring smooth solution (x(t), u(t), therefore, we need:

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case

Idea of Computer-Assisted Proof for Dynamic Model

Validate (e.g. Schauder) fixed point theorem by extending Taylor model, a sum of an approximation of Taylor polynomial and interval vector, e.g., for Optimal Condition ˙ u(t) = F(t, x(t), u(t)) ∈ G(x(t), u(t)), Interval vector covering the remainder, Taylor coefficients obtained by symbolic and numerical computation, requiring smooth solution (x(t), u(t), therefore, we need:

27 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case

Idea of Computer-Assisted Proof for Dynamic Model

Validate (e.g. Schauder) fixed point theorem by extending Taylor model, a sum of an approximation of Taylor polynomial and interval vector, e.g., for Optimal Condition ˙ u(t) = F(t, x(t), u(t)) ∈ G(x(t), u(t)), Interval vector covering the remainder, Taylor coefficients obtained by symbolic and numerical computation, requiring smooth solution (x(t), u(t), therefore, we need:

27 / 30

Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case

Idea of Computer-Assisted Proof for Dynamic Model

Validate (e.g. Schauder) fixed point theorem by extending Taylor model, a sum of an approximation of Taylor polynomial and interval vector, e.g., for Optimal Condition ˙ u(t) = F(t, x(t), u(t)) ∈ G(x(t), u(t)), Interval vector covering the remainder, Taylor coefficients obtained by symbolic and numerical computation, requiring smooth solution (x(t), u(t), therefore, we need:

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case

Idea of Computer-Assisted Proof (continued)

regularization for guaranteeing the unique solvability, giving a regularized system ˙ u(t) = F(t, x(t), u(t)) = Gλ(x(t), u(t)), and smoothing for guaranteeing the smooth solution, giving a regularized-smoothing system: ˙ u(t) = F(t, x(t), u(t)) = Gλ,µ(x(t), u(t)), identification of smooth pieces.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case

Idea of Computer-Assisted Proof (continued)

regularization for guaranteeing the unique solvability, giving a regularized system ˙ u(t) = F(t, x(t), u(t)) = Gλ(x(t), u(t)), and smoothing for guaranteeing the smooth solution, giving a regularized-smoothing system: ˙ u(t) = F(t, x(t), u(t)) = Gλ,µ(x(t), u(t)), identification of smooth pieces.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case

Idea of Computer-Assisted Proof (continued)

regularization for guaranteeing the unique solvability, giving a regularized system ˙ u(t) = F(t, x(t), u(t)) = Gλ(x(t), u(t)), and smoothing for guaranteeing the smooth solution, giving a regularized-smoothing system: ˙ u(t) = F(t, x(t), u(t)) = Gλ,µ(x(t), u(t)), identification of smooth pieces.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case

References

F . Facchinei and C. Kanzow, Generalized Nash equilibrium

• problems. Ann. Oper. Res. 175, 177-211(2010).
• X. Chen and Z. Wang, Computational error bounds for

differential linear variational inequality, IMA J. Numer. Anal. 32, 957-982(2012).

• X. Chen and Z. Wang, Regularization and smoothing

approximation of dynamic games, preprint, Hong Kong Polytechinic University, 2012.

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Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case

Acknowledgements

Thank You!

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