[Boyd & Vandenberghe] Chapter 3 Convex Functions

lecture 3 video
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@ Stanford Engineering Everywhere
Linear Systems and Optimization | Convex Optimization I
Instructor: Boyd, Stephen 

EE364a: Convex Optimization I
Professor Stephen Boyd, Stanford University, Winter Quarter 2007–08

3.1 Basic properties and examples

3.1.1 Definition

http://en.wikipedia.org/wiki/Convex_function
a function is convex if and only if its epigraph (the set of points lying on or above the graph) is a convex set

Pictorially, a function is called 'convex' if the function lies below the straight line segment connecting two points, for any two points in the interval


3.1.2 Extended-value extensions

http://en.wikipedia.org/wiki/Indicator_function



3.1.3 First-order conditions

http://en.wikipedia.org/wiki/Taylor_approximation



3.1.4 Second-order conditions

3.1.5 Examples

3.1.6 Sublevel sets

3.1.7 Epigraph

3.1.8 Jensen's inequality and extensions

3.1.9 Inequalities

3.2 Operations that preserve convexity

3.2.1 Nonnegative weighted sums

3.2.2 Composition with an affine mapping

3.2.3 Pointwise maximum and supremum

3,2.4 Composition

3.2.5 Minimization

3.2.6 Perspective of a function

3.3 The conjugate function

3.3.1 Definition and examples

http://en.wikipedia.org/wiki/Convex_conjugate



3.3.2 Basic properties

3.4 Quasiconvex functions

3.4.1 Definition and examples

http://en.wikipedia.org/wiki/Quasiconvex_function
A quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form  is a convex set.


3.4.2 Basic properties

3.4.3 Differentiable quasiconvex functions

3.4.4 Operations that preserve quasiconvexity

3.4.5 Representation via family of convex functions

3.5 Log-concave and log-convex functions

3.5.1 Definition

http://en.wikipedia.org/wiki/Logarithmically_concave_function

http://en.wikipedia.org/wiki/Logarithmically_convex_function

an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X.

http://en.wikipedia.org/wiki/Geometric_mean


http://en.wikipedia.org/wiki/Bohr%E2%80%93Mollerup_theorem

ref.
http://mathworld.wolfram.com/GammaFunction.html

Convexity in the theory of the gamma function
International Journal of Applied Mathematics & Statistics ,  Nov, 2007   by Milan Merkle

THU NGỌC DƯƠNG, THE GAMMA FUNCTION

Mohamed A. Khamsi, The Gamma Function
 



http://en.wikipedia.org/wiki/Determinant
http://mathworld.wolfram.com/Determinant.html


http://en.wikipedia.org/wiki/Lebesgue_measure


http://en.wikipedia.org/wiki/Laplace_transform
http://mathworld.wolfram.com/LaplaceTransform.html


ref.
Ernesto Schirmacher <Log-Concavity and the Exponential Formula>

András Prékopa, Logarithmic Concave Measures with Applications to Stochastic Programming, 1971
András Prékopa, On logarithmic concave measures and functions, 1973
András Prékopa, Logarithmic Concave Measures and Related Topics, 1980

Ole E. Barndorff-Nielsen,

Bruce E. Sagan, Log Concave Sequences of Symmetric Functions and Analogs of the Jacobi-Trudi Determinants. 1992
Thomas M. Cover and A. Thomas, Determinant Inequalities via Information Theory




3.5.2 Properties

http://en.wikipedia.org/wiki/Cumulative_distribution_function

http://en.wikipedia.org/wiki/Polyhedron

3.6 Convexity with respect to generalized inequalities

3.6.1 Monotonocity with respect to a generalized inequality

3.6.2 Convexity with respect to a generalized inequality