# Stochastic constraints

Where in this post we discuss various types of constraints on stochastic optimal control problems starting from the classic uniform constraints to chance/probabilistic constraints and more elaborate risk constraints. We point out some caveats and propose a few remedies. Continue reading →

# Conditional risk mappings: robust representations

Coherent risk measures can be written as the support function of a set of random variables or as a worst-case expectation over a set of probability measures. This is the so-called dual or robust representation of risk measures. These representations extend to the conditional variants of risk measures. In this post we revisit the construction and properties of conditional risk mappings and derive the dual representation of compositions of conditional risk mappings. Continue reading →

# Probability Theory Cookbook

I have been working on a probability theory cookbook which is intended to serve as a collection of important results in general probability theory. It can be used for a quick brush up, as a quick reference or cheat sheet, but not as primary tutorial material. This is a first draft version, but I will keep adding material

Feedback is more than welcome!

# Mean Square Stable MJLS, but higher moments diverge

It is well known that MJLS can be mean-stable (the expected value of the state’s norm converges to 0), but not mean-square stable. It makes sense to assume that an MJLS can be mean-square stable, but some higher-order norms do not converge to 0. Despite the fact that mean-square stability conditions for MJLS are well studied and are easy to check, this is not always the case for moments of other order. In this post we present a counterexample and state conditions for p-order mean stability.

# 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

# Continuity of argmin

Where here we ask what happens to the infima and sets of minimisers of sequences of functions $\{f_n\}_n$? under what conditions do these converge? what is an appropriate notion of convergence for functions which transfers the convergence to the corresponding sequence of its minima and minimizers? This poses a question of continuity for the infimum (as an operator) as well as the set of minimisers (as a multi-valued operator). We aim at characterising the continuity of these operators. Continue reading →

# 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 →

# Quadratic constraints to second-order conic ones

In the previous post we discussed how we can project onto the epigraph of the squared norm. However, when in an optimisation problem we encounter constraints of the form

\begin{aligned} (x,t)\in \mathrm{epi}_{\|{}\cdot{}\|^2} \end{aligned}

That is, quadratic constraints (of the form $\|x\|^2 \leq t$), these can be converted to second-order conic constraints. Continue reading →

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