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 $\rho:\mathbb{R}^n\to\mathbb{R}$ and a mapping $f:\mathbb{R}^m\to\mathbb{R}^n$, we have

\begin{aligned} \rho\left(\inf_x f(x)\right) = \inf_x \rho(f(x)) \end{aligned}

and $\mathbf{argmin}_x f(x) \subseteq \mathbf{argmin}_x \rho(f(x))$, while, under additional conditions (which are typically met in finite-dimensional spaces), we haveĀ $\mathbf{argmin}_x f(x) = \mathbf{argmin}_x \rho(f(x))$ Continue reading →

Cone programs and self-dual embeddings

This post aims at providing some intuition into cone programs from different perspectives; in particular:

1. Equivalence of different formulations of cone programs
2. Fenchel duality
3. Primal-dual optimality conditions (OC)
4. OCs as variational inequalities
5. Homogeneous self-dual embeddings (HSDEs)
6. OCs for HSDEs

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 →

Projection on the epigraph of the squared Euclidean norm

As a follow-up on the previous post titled Projection on an epigraph, we here discuss how we can project on the epigraph of the squared norm function. Continue reading →

Metric subregularity for monotone inclusions

Metric sub-regularity is a local property of set-valued operators which turns out to be a key enabler for linear convergence in several operator-based algorithms. However, conditions are often stated for the fixed-point residual operator and become rather difficult to verify in practice. In this post we state sufficient metric sub-regularity conditions for a monotone inclusion which are easier to verify and we focus on the preconditioned proximal point method (P3M). Continue reading →

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