1
Convex Sets
▶
1.1
Generalities
▶
1.1.1
Definitions and first examples
1.1.2
Convexity preserving operations on sets
1.1.3
Convex combinations and convex hulls
1.1.4
Closed convex sets and hulls
1.2
Convex Sets attached to a Convex Set
▶
1.2.1
Relative interior
1.2.2
The asymptotic cone
1.2.3
Extreme points
1.2.4
Exposed faces
1.3
Projection Onto Closed Convex Set
▶
1.3.1
The projection operator
1.3.2
Projection onto a closed convex cone
1.4
Separation and Applications
▶
1.4.1
Separation between convex sets
1.4.2
First consequences of the separation properties
1.4.3
The lemma of Minkowski and Farkas
1.5
Conical approximations of convex sets
▶
1.5.1
Convenient definitions of tangent cones
1.5.2
The tangent cones and normal cones to a convex set
1.5.3
Some properties of tangent and normal cones
2
Convex functions
▶
2.1
Basic Definitions and Examples
▶
2.1.1
The definitions of a Convex Function
2.1.2
Special convex functions: Affinity and Closedness
2.1.3
First examples
2.2
Functional Operations Preserving Convexity
▶
2.2.1
Operations preserving closedness
2.2.2
Dilations and perspectives of a functions
2.2.3
Infimal convolution
2.2.4
Image of a function under a linear mapping
2.2.5
Convex hull and closed convex hull of a function
2.3
Local and Global Behavior of a Convex Function
▶
2.3.1
Continuity properties
2.3.2
Behavior at infinity
2.4
First- and Second-order Differentiation
▶
2.4.1
Differentiable convex functions
2.4.2
Nondifferentiable convex functions
2.4.3
Second-order differentiation
3
Sublinearity and Support Functions
▶
3.1
Sublinear Functions
▶
3.1.1
Definitions and first properties
3.1.2
Some examples
3.1.3
The convex cone of all closed sublinear functions
3.2
The Support Function of a Nonempty Set
▶
3.2.1
Definitions, Interpretations
3.2.2
Basic properties
3.2.3
Examples
3.3
The Isomorphism Between Closed Convex Sets and Closed Sublinear Functions
▶
3.3.1
The fundamental correspondence
3.3.2
Example: Norms and Their Duals, Polarity
3.3.3
Calculus with support functions
3.3.4
Example: Support functions of closed convex polyhedra
4
Subdifferentials of Finite Convex Functions
▶
4.1
The Subdifferential: Definitions and Interpretations
▶
4.1.1
First definition: Directional derivative
4.1.2
Second definition: Minorization by affine functions
4.1.3
Geometric constructions and interpretations
4.2
Local Properties of the Subdifferential
▶
4.2.1
First-order developments
4.2.2
Minimality conditions
4.2.3
Mean-value theorems
4.3
First Examples
4.4
Calculus Rules with Subdifferentials
▶
4.4.1
Positive combinations of functions
4.4.2
Pre-composition with an affine mapping
4.4.3
Post-composition with increasing convex function of several variables
4.4.4
Supremum of convex functions
4.4.5
Image of a function under a linear mapping
4.5
Further Examples
▶
4.5.1
Largest eigenvalue of a symmetric matrix
4.5.2
Nested optimization
4.5.3
Best approximation of a convex function on a compact interval
4.6
Subdifferential as a multifunction
▶
4.6.1
Monotonicity property of the subdifferential
4.6.2
Continuity properties of the subdifferential
4.6.3
Subdifferentials and limits of subgradients
5
Conjugacy in Convex Analysis
▶
5.1
The Convex Conjugate of a Function
▶
5.1.1
Definition and first examples
5.1.2
Interpretations
5.1.3
First properties
5.1.4
Subdifferentials and extended-valued functions
5.2
Conjugacy Rules on the Conjugacy Operation
▶
5.2.1
Image of a function under a linear mapping
5.2.2
Pre-composition with an affine mapping
5.2.3
Sum of two functions
5.2.4
Infima and suprema
5.2.5
Post-composition with an increasing convex function
5.3
Various Examples
▶
5.3.1
The Cramer transform
5.3.2
The conjugate of a convex partially quadratic function
5.3.3
Polyhedral functions
5.4
Differentiability of a Conjugate Function
▶
5.4.1
First-order differentiability
5.4.2
Lipschitz continuity of the gradient mapping
6
Algorithms
▶
6.1
Convex optimization in finite dimension
▶
6.1.1
Center of gravity
6.1.2
Ellipsoid method
6.2
Dimension-free convex optimization
▶
6.2.1
Subgradient descent
6.2.2
Gradient descent
6.2.3
Nesterov
6.2.4
Frank-Wolfe
6.3
Almost dimension-free convex optimization in non-Euclidean spaces
▶
6.3.1
Mirror descent
6.3.2
Dual averaging
6.3.3
Mirror prox
6.4
Beyond the black-box model
▶
6.4.1
Newton’s method
6.4.2
Interior point methods
Dependency graph
Convex Eval
Audrey Xie
1
Convex Sets
1.1
Generalities
1.1.1
Definitions and first examples
1.1.2
Convexity preserving operations on sets
1.1.3
Convex combinations and convex hulls
1.1.4
Closed convex sets and hulls
1.2
Convex Sets attached to a Convex Set
1.2.1
Relative interior
1.2.2
The asymptotic cone
1.2.3
Extreme points
1.2.4
Exposed faces
1.3
Projection Onto Closed Convex Set
1.3.1
The projection operator
1.3.2
Projection onto a closed convex cone
1.4
Separation and Applications
1.4.1
Separation between convex sets
1.4.2
First consequences of the separation properties
1.4.3
The lemma of Minkowski and Farkas
1.5
Conical approximations of convex sets
1.5.1
Convenient definitions of tangent cones
1.5.2
The tangent cones and normal cones to a convex set
1.5.3
Some properties of tangent and normal cones
2
Convex functions
2.1
Basic Definitions and Examples
2.1.1
The definitions of a Convex Function
2.1.2
Special convex functions: Affinity and Closedness
2.1.3
First examples
2.2
Functional Operations Preserving Convexity
2.2.1
Operations preserving closedness
2.2.2
Dilations and perspectives of a functions
2.2.3
Infimal convolution
2.2.4
Image of a function under a linear mapping
2.2.5
Convex hull and closed convex hull of a function
2.3
Local and Global Behavior of a Convex Function
2.3.1
Continuity properties
2.3.2
Behavior at infinity
2.4
First- and Second-order Differentiation
2.4.1
Differentiable convex functions
2.4.2
Nondifferentiable convex functions
2.4.3
Second-order differentiation
3
Sublinearity and Support Functions
3.1
Sublinear Functions
3.1.1
Definitions and first properties
3.1.2
Some examples
3.1.3
The convex cone of all closed sublinear functions
3.2
The Support Function of a Nonempty Set
3.2.1
Definitions, Interpretations
3.2.2
Basic properties
3.2.3
Examples
3.3
The Isomorphism Between Closed Convex Sets and Closed Sublinear Functions
3.3.1
The fundamental correspondence
3.3.2
Example: Norms and Their Duals, Polarity
3.3.3
Calculus with support functions
3.3.4
Example: Support functions of closed convex polyhedra
4
Subdifferentials of Finite Convex Functions
4.1
The Subdifferential: Definitions and Interpretations
4.1.1
First definition: Directional derivative
4.1.2
Second definition: Minorization by affine functions
4.1.3
Geometric constructions and interpretations
4.2
Local Properties of the Subdifferential
4.2.1
First-order developments
4.2.2
Minimality conditions
4.2.3
Mean-value theorems
4.3
First Examples
4.4
Calculus Rules with Subdifferentials
4.4.1
Positive combinations of functions
4.4.2
Pre-composition with an affine mapping
4.4.3
Post-composition with increasing convex function of several variables
4.4.4
Supremum of convex functions
4.4.5
Image of a function under a linear mapping
4.5
Further Examples
4.5.1
Largest eigenvalue of a symmetric matrix
4.5.2
Nested optimization
4.5.3
Best approximation of a convex function on a compact interval
4.6
Subdifferential as a multifunction
4.6.1
Monotonicity property of the subdifferential
4.6.2
Continuity properties of the subdifferential
4.6.3
Subdifferentials and limits of subgradients
5
Conjugacy in Convex Analysis
5.1
The Convex Conjugate of a Function
5.1.1
Definition and first examples
5.1.2
Interpretations
5.1.3
First properties
5.1.4
Subdifferentials and extended-valued functions
5.2
Conjugacy Rules on the Conjugacy Operation
5.2.1
Image of a function under a linear mapping
5.2.2
Pre-composition with an affine mapping
5.2.3
Sum of two functions
5.2.4
Infima and suprema
5.2.5
Post-composition with an increasing convex function
5.3
Various Examples
5.3.1
The Cramer transform
5.3.2
The conjugate of a convex partially quadratic function
5.3.3
Polyhedral functions
5.4
Differentiability of a Conjugate Function
5.4.1
First-order differentiability
5.4.2
Lipschitz continuity of the gradient mapping
6
Algorithms
6.1
Convex optimization in finite dimension
6.1.1
Center of gravity
6.1.2
Ellipsoid method
6.2
Dimension-free convex optimization
6.2.1
Subgradient descent
6.2.2
Gradient descent
6.2.3
Nesterov
6.2.4
Frank-Wolfe
6.3
Almost dimension-free convex optimization in non-Euclidean spaces
6.3.1
Mirror descent
6.3.2
Dual averaging
6.3.3
Mirror prox
6.4
Beyond the black-box model
6.4.1
Newton’s method
6.4.2
Interior point methods