The purpose of this book is to study those tools of nonlinear (convex) analysis required for solving optimization problems, and for finding various kinds of equilibria.
In a first part the existence of solutions to minimization problems is treated, first in a general context, then in the case of a convex function. The projection theorem, the various separation theorems and the duality relations between convex functions and their conjugate functions are also studied.
In a second part Fermat’s rule is proved, viz. that the gradient of a function vanishes in a minimum. This rule is generalized to nondifferentiable functions using subdifferentials and generalized gradients.
Next, two fundamental theorems of two-person game theory are proved: the von Neumann minimax theorem, and the inequality of Ky Fan. This last inequality is used to derive a series of existence theorems of solutions of nonlinear equations.
These results are applied to prove classical existence theorems of competitive equilibria. The von Neumann growth model is also studied, as well as the Perron-Frobenius theorems on positive matrices.
The last two chapters deal with n-person game theory (cooperative and non cooperative).