Useful convex analysis stuff: support functions & normal and tangent cones
This is a collection of some simple, yet useful results from convex analysis. We will give examples of support functions of convex sets, and normal cones. Of course special focus will be given to the two most popular sets of convex analysis: balls (Euclidean balls and ellipsoids) and polyhedra.
Continue reading →Generalized Directional Derivatives: Some Examples
Earlier we defined the directional derivative of a function 
provided that the limit exists. It turns out that all convex functions are directionally differentiable on the interior (actually, the core) of their domains and 
Pointwise maximum function differentiability
These are some notes on some differentiability properties of the maximum of a finite number of functions based on some results taken mainly from the book of Borwein and Lewis and Rockafellar and Wets’s “Variational Analysis”.
Continue reading →Notes on the Rayleigh Quotient
So here I am after a short of period of absence. This will be a short blog post on the Rayleigh quotient of a symmetric matrix, 
Strict and strong convexity of maximum of two functions
This is a brief note on the strict and strong convexity properties of the maximum of two functions. The question is whether the maximum of two strictly/strongly convex functions is strintly/strongly convex. The proofs are very simple.
Interchangeability of infimum in risk measures
In this post we discuss the interchangeability of the infimum with (monotone) risk measures in finite probability spaces. In particular, we show that under the common monotonicity assumption (which is satisfied by all well-behaving risk measures), for a risk measure 
and 
Cone programs and self-dual embeddings
This post aims at providing some intuition into cone programs from different perspectives; in particular:
- Equivalence of different formulations of cone programs
- Fenchel duality
- Primal-dual optimality conditions (OC)
- OCs as variational inequalities
- Homogeneous self-dual embeddings (HSDEs)
- OCs for HSDEs
Continuity of argmin
Where here we ask what happens to the infima and sets of minimisers of sequences of functions 
Projection on epigraph via a proximal operator
A while ago I posted this article on how to project on the epigraph of a convex function where I derived the optimality conditions and the KKT conditions. This post comes as an addendum proving a third way to project on an epigraph. Do read the previous article first because I use the same notation here. Continue reading →
Lagrange vs Fenchel Duality
In this post we discuss the correspondence between the Lagrangian and the Fenchelian duality frameworks and we trace their common origin to the concept of convex conjugate functions and perturbed optimization problems. Continue reading →
Mathematix 