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Bauer's maximum principle is the following theorem in mathematical optimization: Any function that is convex and continuous, and defined on a set that is convex and compact, attains its maximum at some extreme point of that set. It is attributed to the German mathematician Heinz Bauer. [1]
There is no single or most general maximum principle which applies to all situations at once. In the field of convex optimization, there is an analogous statement which asserts that the maximum of a convex function on a compact convex set is attained on the boundary. [2]
A function (in black) is convex if and only if the region above its graph (in green) is a convex set. A graph of the bivariate convex function x 2 + xy + y 2. Convex vs. Not convex
A convex optimization problem is defined by two ingredients: [5] [6] The objective function, which is a real-valued convex function of n variables, :;; The feasible set, which is a convex subset.
If is strictly quasiconcave in for each and is convex-valued, then is single-valued, and thus is a continuous function rather than a correspondence. [ 15 ] If f {\displaystyle f} is concave in X × Θ {\displaystyle X\times \Theta } and C {\displaystyle C} has a convex graph, then f ∗ {\displaystyle f^{*}} is concave and C ∗ {\displaystyle ...
Convex programming studies the case when the objective function is convex (minimization) or concave (maximization) and the constraint set is convex. This can be viewed as a particular case of nonlinear programming or as generalization of linear or convex quadratic programming.
The 1971 Ph.D. Thesis by Dimitri P. Bertsekas (Proposition A.22) [3] proves a more general result, which does not require that (,) is differentiable. Instead it assumes that (,) is an extended real-valued closed proper convex function for each in the compact set , that ( ()), the interior of the effective domain of , is nonempty, and that is continuous on the set ( ()).
The extreme value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930. Bolzano's proof consisted of showing that a continuous function on a closed interval was bounded, and then showing that the function attained a maximum and a minimum value.