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| Preface to the Dover Edition | |
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| Preface (1982) | |
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| Summary of Results: A Guideline for the Reader | |
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| Contents of Other Possible Courses | |
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| Notations | |
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| Optimization and Convex Analysis | |
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| Minimization Problems and Convexity | |
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| Strategy sets and loss functions | |
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| Optimization problem | |
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| Allocation of available commodities | |
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| Resource and service operators | |
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| Extension of loss functions | |
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| Sections and epigraphs | |
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| Decomposition principle | |
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| Product of a loss function by a linear operator | |
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| Example: Inf-convolution of functions | |
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| Decomposition principle | |
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| Another decomposition principle | |
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| Mixed strategies and convexity | |
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| Motivation: extension of strategy sets and loss functions | |
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| Mixed strategies and linearized loss functions | |
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| Interpretation of mixed strategies | |
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| Case of finite strategy sets | |
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| Representation by infinite sequences of pure strategies | |
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| Linearized extension of maps and the barycentric operator | |
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| Interpretation of convex functions in terms of risk aversion | |
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| Elementary properties of convex subsets and functions | |
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| Indicators, support functions and gauges | |
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| Indicators and support functions | |
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| Reformulation of the Hahn-Banach theorem | |
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| The bipolar theorem | |
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| Recession cones and barrier cones | |
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| Interpretation: production sets and profit functions | |
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| Gauges | |
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| Existence, Uniqueness and Stability of Optimal Solutions | |
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| Existence and uniqueness of an optimal solution | |
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| Structure of the optimal set | |
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| Existence of an optimal solution | |
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| Continuity versus compactness | |
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| Lower semi-continuity of convex functions in infinite dimensional spaces | |
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| Fundamental property of lower semi-continuous and compact functions | |
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| Uniqueness of an optimal solution | |
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| Non-satiation property | |
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| Minimization of quadratic functionals on convex sets | |
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| Hilbert spaces | |
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| Existence and uniqueness of the minimal solution | |
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| Characterization of the minimal solution | |
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| Projectors of best approximation | |
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| The duality map from an Hilbert space onto its dual | |
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| Minimization of quadratic functionals on subspaces | |
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| The fundamental formula | |
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| Orthogonal right inverse | |
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| Orthogonal left inverse | |
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| Another decomposition property | |
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| Interpretation | |
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| Perturbation by linear forms: conjugate functions | |
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| Conjugate functions | |
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| Characterization of lower semi-continuous convex functions | |
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| Examples of conjugate functions | |
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| Elementary properties of conjugate functions | |
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| Interpretation: cost and profit functions | |
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| Stability properties: an introduction to correspondences | |
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| Upper semi-continuous correspondences | |
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| Lower semi-continuous correspondences | |
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| Closed correspondences | |
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| Construction of upper semi-continuous correspondences | |
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| Compactness and Continuity Properties | |
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| Lower semi-compact functions | |
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| Coercive and semi-coercive functions | |
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| Functions such that f* is continuous at 0 | |
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| Lower semi-compactness of linear forms | |
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| Constraint qualification hypothesis | |
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| Case of infinite dimensional spaces | |
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| Extension to compact subsets of mixed strategies | |
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| Proper maps and preimages of compact subsets | |
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| Proper maps | |
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| Compactness of some strategy sets | |
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| Examples where the map L* + 1 is proper | |
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| Continuous convex functions | |
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| A characterization of lower semi-continuous convex functions | |
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| A characterization of continuous convex functions | |
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| Examples of continuous convex functions | |
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| Continuity of gL and Lf | |
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| Continuous convex functions (continuation) | |
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| Strong continuity of lower semi-continuous convex functions | |
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| Estimates of lower semi-continuous convex functions | |
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| Characterization of continuous convex functions | |
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| Continuity of support functions | |
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| Maximum of a convex function: extremal points | |
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| Differentiability and Subdifferentiability: Characterization of Optimal Solutions | |
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| Subdifferentiability | |
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| Definitions | |
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| Examples of subdifferentials | |
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| Subdifferentiability of continuous convex functions | |
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| Upper semi-continuity of the subdifferential | |
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| Characterization of subdifferentiable convex functions | |
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| Differentiability and variational inequalities | |
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| Definitions | |
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| Differentiability and subdifferentiability | |
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| Legendre transform | |
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| Interpretation: marginal profit | |
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| Variational inequalities | |
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| Differentiability from the right | |
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| Definition and main inequalities | |
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| Derivatives from the right and the support function of the subdifferential | |
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| Derivative of a pointwise supremum | |
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| Local [epsilon]-subdifferentiability and perturbed minimization problems | |
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| Approximate optimal solutions in Banach spaces | |
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| The approximate variational principle | |
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| Local [epsilon]-subdifferentiability | |
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| Perturbation of minimization problems | |
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| Proof of Ekeland-Lebourg's theorem | |
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| Introduction to Duality Theory | |
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| Dual problem and Lagrange multipliers | |
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| Lagrangian | |
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| Lagrange multipliers and dual problem | |
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| Marginal interpretation of Lagrange multipliers | |
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| Example | |
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| Case of linear constraints: extremality relations | |
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| Generalized minimization problem | |
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| Extremality relations | |
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| The fundamental formula | |
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| Minimization problem under linear constraints | |
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| Minimization of a quadratic functional under linear constraints | |
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| Minimization problem under linear equality constraints | |
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| Duality and the decomposition principle | |
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| The decentralization principle | |
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| Conjugate function of gL | |
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| Conjugate function of f[subscript 1]+f[subscript 2] | |
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| Minimization of the projection of a function | |
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| Minimization on the diagonal of a product | |
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| Existence of Lagrange multipliers in the case of a finite number of constraints | |
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| The Fenchel existence theorem | |
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| Stability properties | |
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| Applications to subdifferentiability | |
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| Case of nonlinear constraints: The Uzawa existence theorem | |
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| Game Theory and the Walras Model of Allocation of Resources | |
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| Two-Person Games: An Introduction | |
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| Some solution concepts | |
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| Description of the game | |
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| Shadow minimum | |
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| Conservative solutions and values | |
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| Non-cooperative equilibrium | |
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| Pareto minimum | |
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| Core of a two-person game | |
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| Selection of strategy of the core | |
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| Examples: some finite games | |
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| Example | |
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| Coordination game | |
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| Prisoner's dilemma | |
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| Game of chicken | |
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| The battle of the sexes | |
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| Example: Analysis of duopoly | |
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| The model of a duopoly | |
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| The set of Pareto minima | |
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| Conservative solutions | |
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| Non-cooperative equilibria | |
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| Stackelberg equilibria | |
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| Stackelberg disequilibrium | |
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| Example: Edgeworth economic game | |
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| The set of feasible allocations | |
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| The biloss operator | |
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| The Edgeworth box | |
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| Pareto minima | |
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| Core | |
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| Walras equilibria | |
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| Two-person zero-sum games | |
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| Duality gap and value | |
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| Saddle point | |
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| Perturbation by linear functions | |
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| Case of finite strategy sets: Matrix games | |
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| Two-Person Zero-Sum Games: Existence Theorems | |
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| The fundamental existence theorems | |
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| Existence of conservative solutions | |
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| Decision rules | |
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| Finite topology on convex subsets | |
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| Existence of an optimal decision rule | |
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| The Ky-Fan inequality | |
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| The Lasry theorem | |
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| The minisup theorem | |
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| The Nikaido theorem | |
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| Existence of saddle points | |
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| Another existence theorem for saddle points | |
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| Extension of games without and with exchange of informations | |
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| Definition of extensions of games | |
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| Mixed extensions | |
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| Extensions without exchange of information | |
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| Sequential extensions | |
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| Extensions with exchange of information | |
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| Iterated games | |
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| Iterated extensions | |
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| The Moulin theorem | |
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| Proof of playability of iterated extensions | |
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| A system of functional equations | |
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| A lemma on successive approximations | |
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| Proof of existence of saddle decision rules | |
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| The Fundamental Economic Model: Walras Equilibria | |
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| Description of the model | |
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| The subset of available commodities | |
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| Appropriation of the economy | |
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| Demand correspondences | |
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| Walras equilibrium | |
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| Examples of subsets of available commodities and of appropriations | |
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| Example: Quadratic demand functions | |
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| Existence of a Walras equilibrium | |
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| Existence of a Walras pre-equilibrium | |
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| Surjectivity of correspondences: the Debreu-Gale-Nikaido theorem | |
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| Demand correspondences defined by loss functions | |
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| Statement of the existence theorem | |
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| Upper semi-continuity of the demand correspondence | |
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| Compactification of an economy | |
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| Proof of the existence of a Walras equilibrium | |
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| Economies with producers | |
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| Description of the model | |
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| Statement of the existence theorem | |
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| Compactification | |
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| Proof of the existence of a Walras equilibrium | |
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| Non-Cooperative n-Person Games | |
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| Existence of a non-cooperative equilibrium | |
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| Games described in strategic form | |
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| Conservative values and multistrategies | |
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| Non-cooperative equilibria | |
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| The Nash theorem | |
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| Stability | |
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| Associated variational inequalities | |
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| Case of quadratic loss functions; application to Walras-Cournot equilibria | |
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| Non-cooperative games with quadratic loss functions | |
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| Existence of solutions of variational inequalities | |
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| Examples | |
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| Multistrategy sets defined by linear constraints | |
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| Walras-Cournot equilibria | |
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| Constrained non-cooperative games and fixed point theorems | |
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| Selection of a fixed point | |
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| Equilibria of constrained non-cooperative games | |
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| Fixed-point theorems | |
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| Non-cooperative Walras equilibria | |
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| Description of the model | |
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| Existence of a non-cooperative Walras equilibrium: the Arrow-Debreu theorem | |
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| Non-cooperative Walras equilibria of economies with producers | |
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| Main Solution Concepts of Cooperative Games | |
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| Behavior of the whole set of players: Pareto strategies | |
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| Pareto strategies | |
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| Rates of transfer | |
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| Pareto multipliers | |
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| Pareto allocations | |
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| Selection of Pareto strategies and imputations | |
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| Normalized games | |
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| Pareto strategies obtained by using selection functions | |
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| Closest strategy to the shadow minimum | |
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| The best compromise | |
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| Existence of Pareto strategies | |
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| Interpretation: threat functionals | |
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| Imputations: the Nash bargaining solution | |
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| Behavior of coalitions of players: the core | |
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| Coalitions | |
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| Cooperative game described in strategic form and its core | |
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| The multiloss operator F[superscript A]# of the coalition A | |
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| Examples of multistrategy sets X(A) | |
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| Economic games and core of an economy | |
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| Cooperative game described in characteristic form and its core | |
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| Behavior of fuzzy coalitions: the fuzzy core | |
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| Fuzzy coalitions | |
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| Extension of a family of coalitions | |
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| Debreu-Scarf coalitions | |
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| Fuzzy coalitions on a continuum of players | |
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| Fuzzy games described in characteristic form | |
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| Characterization of the core of a (fuzzy) game | |
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| Fuzzy economic games and fuzzy core of an economy | |
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| Fuzzy games described in strategic form and fuzzy core | |
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| Selection of elements of the core: cooperative equilibrium and nucleolus | |
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| Canonical cooperative equilibrium | |
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| Least-core | |
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| Nucleolus | |
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| Games With Side-Payments | |
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| Core of a fuzzy game with side-payments | |
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| Core of a game with side-payments | |
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| Linear games | |
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| Non-emptiness of the core of fuzzy games with side-payments | |
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| Core of fuzzy market games | |
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| Core of a game with side-payments | |
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| Convex cover of a game | |
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| Non-emptiness of the core of a balanced game | |
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| Balanced family of multistrategy sets | |
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| Balanced characteristic functions and convex loss functions | |
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| Further properties of convex functions and balances | |
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| Values of fuzzy games | |
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| The diagonal property | |
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| Sequence of fuzzy values | |
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| Existence and uniqueness of a sequence of fuzzy values | |
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| Relations between core and fuzzy value | |
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| Best approximation property of fuzzy values | |
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| Generalized solution to locally Lipschitz games | |
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| Shapley value and nucleolus of games with side-payments | |
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| The Shapley value | |
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| Existence and uniqueness of a Shapley value | |
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| Simple games | |
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| Nucleolus of games with side-payments | |
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| Games Without Side-Payments | |
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| Equivalence between the fuzzy core and the set of equilibria | |
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| Representation of a game | |
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| Equilibrium of a representation | |
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| Cover associated with a representation | |
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| Fuzzy core of a representation | |
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| The equivalence theorem | |
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| Non-emptiness of the fuzzy core of a balanced game | |
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| Statement of theorems of non-emptiness of the fuzzy core | |
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| Upper semi-continuity of the associated side-payment games | |
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| Existence of approximate cooperative equilibria | |
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| Proof of the non-emptiness of the core | |
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| Equivalence between the fuzzy core of an economy and the set of Walras allocations | |
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| Representation of economic games | |
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| Fuzzy core and Walras allocations | |
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| The equivalence theorem | |
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| Non-Linear Analysis and Optimal Control Theory | |
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| Minimax Type Inequalities, Monotone Correspondences and [gamma]-Convex Functions | |
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| Relaxation of compactness assumptions | |
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| Existence of a conservative solution | |
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| Proof of existence of a conservative solution | |
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| Existence of optimal decision rules and minisup under weaker compactness assumptions | |
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| Relaxation of continuity assumptions: variational inequalities for monotone correspondences | |
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| Variational inequalities | |
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| Existence of a solution to variational inequalities for completely upper semi-continuous correspondences | |
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| Pseudo-monotone functions: the Brezis-Nirenberg-Stampacchia theorem | |
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| Existence of a solution to variational inequalities for pseudo-monotone maps | |
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| Pseudo-monotonicity of monotone maps | |
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| Monotone and cyclically monotone correspondences | |
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| Maximal monotone correspondences | |
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| Relaxation of convexity assumptions | |
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| Definition of [gamma]-convex functions | |
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| The fundamental characteristic property of families of [gamma]-convex functions | |
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| The minisup theorem for [gamma subscript x]-convex-[gamma subscript y]-concave functions | |
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| Existence of optimal decision rules for functions [gamma subscript y]-concave with respect to y | |
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| Example: Image of a cone of convex functions by [pi]* | |
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| Relations between convexity and [gamma]-convexity | |
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| Example: [beta]-convex set functions | |
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| Example: Convex functions of atomless vector measures | |
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| Introduction to Calculus of Variations and Optimal Control | |
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| Duality in infinite dimensional spaces | |
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| Lagrangian of a minimization problem under linear constraints | |
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| Extremality relations | |
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| Existence of a Lagrange multiplier under the Slater condition | |
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| Relaxation of the Slater condition | |
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| Generalized Lagrangian of a minimization problem | |
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| Characterization of a Lagrangian by perturbations of the minimization problem | |
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| Duality in the case of non-convex integral criterion and contraints | |
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| Modulus of non-convexity of a function | |
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| Estimate of the duality gap | |
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| The Shapley-Folkman theorem | |
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| Sharp estimate of the duality gap | |
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| Applications | |
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| Extremality relations | |
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| The Aumann-Perles duality theorem | |
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| The approximation procedure | |
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| Duality in calculus of variations | |
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| The Green formula | |
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| Abstract problem of calculus of variations | |
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| The Hamiltonian system | |
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| Lagrangian of a problem of calculus of variations | |
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| Existence of a Lagrange multiplier | |
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| Example: the Dirichlet variational problem | |
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| The maximum principle for optimal control problems | |
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| Optimal control and impulsive control problems | |
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| The Hamilton-Jacobi-Bellman equation of a control problem | |
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| Construction of the closed loop control | |
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| The principle of optimality | |
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| The quadratic case: Riccati equations | |
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| The Bensoussan-Lions variational inequalities of a stopping time problem | |
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| Construction of the optimal stopping time | |
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| The Bensoussan-Lions quasi-variational inequalities of an impulsive control problem | |
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| Construction of the optimal impulsive control | |
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| Fixed Point Theorems, Quasi-Variational Inequalities and Correspondences | |
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| Fixed point and surjectivity theorems for correspondences | |
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| The Browder-Ky-Fan existence theorem for critical points | |
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| Properties of inward and outward correspondences | |
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| Critical points of homotopic correspondences | |
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| Other existence theorems for critical points | |
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| Quasi-variational inequalities | |
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| Selection of fixed point by pseudo-monotone functions | |
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| Fixed point theorem for increasing maps | |
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| Quasi-variational inequalities for increasing correspondences | |
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| Other properties and examples of upper and lower semi-continuous correspondences | |
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| Lower semi-continuity of preimages of linear operators | |
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| Lower semi-continuity of correspondences defined by constraints | |
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| Continuous selection theorem | |
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| Weak Hausdorff topology on the family of closed subsets of topological vector spaces | |
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| Relations between hemi-continuity and semi-continuity | |
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| Summary of Linear Functional Analysis | |
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| Hahn-Banach theorems | |
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| Paired spaces | |
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| Topologies of uniform convergence | |
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| Topologies associated with a duality pairing | |
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| The Banach-Steinhauss theorem | |
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| The Knaster-Kuratowski-Mazurkiewicz Lemma | |
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| Barycentric subdivision of simplexes | |
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| Sequence of barycentric subdivisions | |
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| The Sperner lemma | |
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| The Knaster-Kuratowski-Mazurkiewicz lemma | |
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| The Brouwer theorem | |
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| Lyapunov's Theorem on the Range of A Vector Valued Measure | |
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| Comments | |
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| References | |
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| Subject Index | |