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  2. Bauer maximum principle - Wikipedia

    en.wikipedia.org/wiki/Bauer_maximum_principle

    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]

  3. Maximum principle - Wikipedia

    en.wikipedia.org/wiki/Maximum_principle

    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]

  4. Convex function - Wikipedia

    en.wikipedia.org/wiki/Convex_function

    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

  5. Convex optimization - Wikipedia

    en.wikipedia.org/wiki/Convex_optimization

    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.

  6. Maximum theorem - Wikipedia

    en.wikipedia.org/wiki/Maximum_theorem

    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 ...

  7. Mathematical optimization - Wikipedia

    en.wikipedia.org/wiki/Mathematical_optimization

    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.

  8. Danskin's theorem - Wikipedia

    en.wikipedia.org/wiki/Danskin's_theorem

    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 ⁡ (⁡ ()).

  9. Extreme value theorem - Wikipedia

    en.wikipedia.org/wiki/Extreme_value_theorem

    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.