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 1 / 30
Motivation of the Study Computer-Assisted Proof Comments on Dynamic Case Outline 1 Motivation of the Study Generalized Nash Equilibrium Problem Central Questions Computer-Assisted Proof 2 Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case Comments on Dynamic Case 3 2 / 30
Motivation of the Study Generalized Nash Equilibrium Problem Computer-Assisted Proof Central Questions Comments on Dynamic Case Outline 1 Motivation of the Study Generalized Nash Equilibrium Problem Central Questions Computer-Assisted Proof 2 Philosophical Description Significance for Numerical Solution Computer-Assisted Proof for Static Case Comments on Dynamic Case 3 3 / 30
Motivation of the Study Generalized Nash Equilibrium Problem Computer-Assisted Proof Central Questions Comments on Dynamic Case The Problem Definition (Generalized Nash equilibrium problem) A multi-agent optimization problem in which the players optimize 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, · · · ; optimal control problems with multi-criteria, like in aero-structural aircraft wing shape design. 4 / 30
Motivation of the Study Generalized Nash Equilibrium Problem Computer-Assisted Proof Central Questions Comments on Dynamic Case The Problem Definition (Generalized Nash equilibrium problem) A multi-agent optimization problem in which the players optimize 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, · · · ; optimal control problems with multi-criteria, like in aero-structural aircraft wing shape design. 4 / 30
Motivation of the Study Generalized Nash Equilibrium Problem Computer-Assisted Proof Central Questions Comments on Dynamic Case The Problem Definition (Generalized Nash equilibrium problem) A multi-agent optimization problem in which the players optimize 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, · · · ; optimal control problems with multi-criteria, like in aero-structural aircraft wing shape design. 4 / 30
Motivation of the Study Generalized Nash Equilibrium Problem Computer-Assisted Proof Central Questions Comments on Dynamic Case The Problem Definition (Generalized Nash equilibrium problem) A multi-agent optimization problem in which the players optimize 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, · · · ; optimal control problems with multi-criteria, like in aero-structural aircraft wing shape design. 4 / 30
Motivation of the Study Generalized Nash Equilibrium Problem Computer-Assisted Proof Central Questions Comments on Dynamic Case The Problem Definition (Generalized Nash equilibrium problem) A multi-agent optimization problem in which the players optimize 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, · · · ; optimal control problems with multi-criteria, like in aero-structural aircraft wing shape design. 4 / 30
Motivation of the Study Generalized Nash Equilibrium Problem Computer-Assisted Proof Central Questions Comments on Dynamic Case The Problem Definition (Generalized Nash equilibrium problem) A multi-agent optimization problem in which the players optimize 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, · · · ; optimal control problems with multi-criteria, like in aero-structural aircraft wing shape design. 4 / 30
Motivation of the Study Generalized Nash Equilibrium Problem Computer-Assisted Proof Central Questions Comments on Dynamic Case The Problem Definition (Generalized Nash equilibrium problem) A multi-agent optimization problem in which the players optimize 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, · · · ; optimal control problems with multi-criteria, like in aero-structural aircraft wing shape design. 4 / 30
Motivation of the Study Generalized Nash Equilibrium Problem Computer-Assisted Proof Central Questions Comments on Dynamic Case Notations (for a N -agent non-cooperative game) strategy profiles for the ν -th player: strategy x ν ∈ R n ν of the ν -th player profile of all the players’ strategies x = ( x ν ) N ν = 1 ∈ R n ; profile of the rivals’ strategies x − ν = ( x ν ′ ) ν ′ � = ν ; state profiles for the ν -th player: state u ν ∈ R m ν of the ν -th player profile of all the players’ states u = ( u ν ) N ν = 1 ; profile of the rivals’ state u − ν = ( u ν ′ ) ν ′ � = ν ; the state equation G ν ( x ν , u ν ) = 0 (differential equation); the objective functional: θ ν ( · , x − ν , u ) ; the strategy set S ν ( x − ν ) := { x ν : g ν ( x ν , x − ν ) ≤ 0 } . 5 / 30
Motivation of the Study Generalized Nash Equilibrium Problem Computer-Assisted Proof Central Questions Comments on Dynamic Case Notations (for a N -agent non-cooperative game) strategy profiles for the ν -th player: strategy x ν ∈ R n ν of the ν -th player profile of all the players’ strategies x = ( x ν ) N ν = 1 ∈ R n ; profile of the rivals’ strategies x − ν = ( x ν ′ ) ν ′ � = ν ; state profiles for the ν -th player: state u ν ∈ R m ν of the ν -th player profile of all the players’ states u = ( u ν ) N ν = 1 ; profile of the rivals’ state u − ν = ( u ν ′ ) ν ′ � = ν ; the state equation G ν ( x ν , u ν ) = 0 (differential equation); the objective functional: θ ν ( · , x − ν , u ) ; the strategy set S ν ( x − ν ) := { x ν : g ν ( x ν , x − ν ) ≤ 0 } . 5 / 30
Motivation of the Study Generalized Nash Equilibrium Problem Computer-Assisted Proof Central Questions Comments on Dynamic Case Notations (for a N -agent non-cooperative game) strategy profiles for the ν -th player: strategy x ν ∈ R n ν of the ν -th player profile of all the players’ strategies x = ( x ν ) N ν = 1 ∈ R n ; profile of the rivals’ strategies x − ν = ( x ν ′ ) ν ′ � = ν ; state profiles for the ν -th player: state u ν ∈ R m ν of the ν -th player profile of all the players’ states u = ( u ν ) N ν = 1 ; profile of the rivals’ state u − ν = ( u ν ′ ) ν ′ � = ν ; the state equation G ν ( x ν , u ν ) = 0 (differential equation); the objective functional: θ ν ( · , x − ν , u ) ; the strategy set S ν ( x − ν ) := { x ν : g ν ( x ν , x − ν ) ≤ 0 } . 5 / 30
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