Convex Analysis - Convex Set, Convex Hull, Legendre Transformation, Convex Function, Quasiconvex Function, Convex Optimization (Paperback)

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 31. Chapters: Convex set, Convex hull, Legendre transformation, Convex function, Quasiconvex function, Convex optimization, R. Tyrrell Rockafellar, Convex conjugate, Convex cone, Farkas' lemma, Concave function, Pseudolinear function, Pseudoconvex function, Subderivative, Minkowski functional, Logarithmically concave function, Gauss-Lucas theorem, Modulus and characteristic of convexity, Dual cone and polar cone, Popoviciu's inequality, Hilbert projection theorem, Ekeland's variational principle, Danskin's theorem, Shephard's problem, Strictly convex space, Fenchel's duality theorem, Uniformly convex space, Moreau's theorem, Tonelli's theorem, Characteristic function, Karamata's inequality, Invex function, Kachurovskii's theorem, Epigraph, Proper convex function, Linear separability, Polyconvex function, Hypograph, Schur-convex function, Closed convex function. Excerpt: In mathematics, the Legendre transformation or Legendre transform, named after Adrien-Marie Legendre, is an operation that transforms one real-valued function of a real variable into another. Specifically, the Legendre transform of a convex function is the function defined by If is differentiable, then (p) can be interpreted as the negative of the y-intercept of the tangent line to the graph of that has slope p. In particular, the value of x that attains the maximum has the property that That is, the derivative of the function becomes the argument to the function . In particular, if is convex (or concave up), then satisfies the functional equation The Legendre transform is its own inverse. Like the familiar Fourier transform, the Legendre transform takes a function (x) and produces a function of a different variable p. However, while the Fourier transform consists of an integration with a kernel, the Legendre transform uses maximization as the tra...