# 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 have $\mathbf{argmin}_x f(x) = \mathbf{argmin}_x \rho(f(x))$ 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

# 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 →

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