Optimization is the branch of mathematics concerned with selecting the best element, according to some measurable criterion, from a set of available alternatives, and with the methods for finding it efficiently. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Central QuestionGiven a criterion to maximize or minimize and a set of allowed choices, does a best choice exist, and can it be found or verified without checking every alternative in turn. 1 Key DebateHow much structure a problem needs before its best solution can be certified rather than merely searched for. The Simplex algorithm, published in 1947, gave a practical method for linear problems, and John von Neumann's work on the theory of duality around the same time showed that a linear program and a related dual program can certify each other's optimal value, a guarantee that most harder, nonlinear optimization problems still lack. 1 Classification
Pure or Applied Connections
Associated With
Includes
Source Bauer maximum principle (Wikipedia)
Source Bellman Equation (Wikipedia)
Source Danskin's Theorem (Wikipedia)
Source Envelope theorem - Wikipedia
Source Maximum theorem (Wikipedia)
Additional Source Wikipedia: Minimax TheoremLead section
Source No free lunch theorem, Wikipedia
Source Wikipedia: Richard Bellman
Source Rosenbrock Function (Wikipedia)
Source Wolfram MathWorld: Softmax Function
Source Wikipedia: Theodore Motzkin
Source Value Function (Wikipedia)
Source Who Solved the Hirsch Conjecture, article by Günter M. Ziegler (Documenta Mathematica)
Sources
1. Wikipedia: Mathematical Optimization
Wikimedia FoundationLead section
the selection of a best element, with regard to some criteria, from some set of available alternatives
Sufficient conditions for optimality section
If a candidate solution satisfies the first-order conditions, then the satisfaction of the second-order conditions as well is sufficient to establish at least local optimality.
History section
Dantzig published the Simplex algorithm in 1947, and also John von Neumann and other researchers worked on the theoretical aspects of linear programming (like the theory of duality) around the same time.
View the Source Wikipedia: Minimax Theorem
Wikimedia FoundationIncludes: Minimax Theorem, Lead sectionQuote, Includes: Minimax Theorem, Lead section
In the mathematical area of game theory and of convex optimization
Danskin's Theorem (Wikipedia)
Includes: Danskin's Theorem, LeadQuote, Includes: Danskin's Theorem, Lead
In convex analysis, Danskin's theorem is a theorem which provides information about the derivatives of a function
Who Solved the Hirsch Conjecture, article by Günter M. Ziegler (Documenta Mathematica)
Includes: Warren M. Hirsch, PDF page 2Quote, Includes: Warren M. Hirsch, PDF page 2
a key problem in the modern theory of polyhedra, motivated by linear programming
Envelope theorem - Wikipedia
Includes: Envelope Theorem, Lead sentenceQuote, Includes: Envelope Theorem, Lead sentence
operties of the value function of a parameterized optimization problem.
Bauer maximum principle (Wikipedia)
Includes: Bauer Maximum Principle, Lead sentenceQuote, Includes: Bauer Maximum Principle, Lead sentence
rinciple is the following theorem in mathematical optimization: Any function that is convex and continuous, and defined on a set t
Value Function (Wikipedia)
Includes: Value Function, Lead sentenceQuote, Includes: Value Function, Lead sentence
The value function of an optimization problem gives the value attained by the objective function at a solution, while only dependi
Rosenbrock Function (Wikipedia)
Includes: Rosenbrock Function, Lead sentenceQuote, Includes: Rosenbrock Function, Lead sentence
In mathematical optimization, the Rosenbrock function is a non-convex function, introduced by Howard H.
Wikipedia: Theodore Motzkin
Includes: Theodore Motzkin, LeadQuote, Includes: Theodore Motzkin, Lead
Motzkin's dissertation contained an important contribution to the nascent theory of linear programming
Wikipedia: Richard Bellman
Includes: Richard Bellman, LeadQuote, Includes: Richard Bellman, Lead
Richard Ernest Bellman (August 26, 1920, March 19, 1984) was an American applied mathematician, who introduced dynamic programming in 1953, and made important contributions in other fields of mathematics, such as biomathematics.
No free lunch theorem, Wikipedia
Includes: No Free Lunch Theorem, Lead paragraphQuote, Includes: No Free Lunch Theorem, Lead paragraph
In this case the intended meaning is that on the average there are no easy shortcuts to designing optimization algorithms.
Bellman Equation (Wikipedia)
Includes: Bellman Equation, Lead paragraphQuote, Includes: Bellman Equation, Lead paragraph
Bellman, is a technique in dynamic programming which breaks an optimization problem into a sequence of simpler subproblems, as Bellman's "principle of optimality" prescribes.
Wolfram MathWorld: Softmax Function
Wolfram MathWorldIncludes: Softmax Function, Subject classificationsQuote, Includes: Softmax Function, Subject classifications
Applied Mathematics Optimization
Maximum theorem (Wikipedia)
Includes: Maximum Theorem, Lead sectionQuote, Includes: Maximum Theorem, Lead section
The maximum theorem provides conditions for the continuity of an optimized function and the set of its maximizers with respect to
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