Tag Archives: Analysis

Generalized Directional Derivatives: Some Examples

By psopasakis on October 13, 2021 | 1 Comment

Earlier we defined the directional derivative of a function $f: \mathbb{R}^n \to \overline{\mathbb{R}}$ as

$\begin{aligned}f'(x; h) = \lim_{t \downarrow 0} \frac{f(x+th)-f(x)}{t},\end{aligned}$

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 $f'(x; \cdot)$ is sublinear. However, the sublinearity property may fail when working with nonconvex functions. This motivates the definition of generalised directional derivatives which will hopefully be accompanied by some good calculus rules.

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Posted in: Optimisation | Tagged: Analysis, convergence, nonsmooth analysis, optimization

Pointwise maximum function differentiability

By psopasakis on October 10, 2021 | Leave a comment

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”.

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Posted in: Optimisation | Tagged: Analysis, nonsmooth analysis, optimization

Interchangeability of infimum in risk measures

By psopasakis on December 19, 2017 | Leave a comment

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 →

Posted in: Optimisation, Probability theory, Risk Measures, Variational analysis | Tagged: Analysis, duality, Optimisation, optimization, risk, risk-measures

Third and higher order Taylor expansions in several variables

By psopasakis on November 5, 2016 | Leave a comment

In this post we show that it is possible to derive third and higer-order Taylor expansions for functions of several variables. Given that the gradient of a function $f:\mathbb{R}^n \to\mathbb{R}$ is vector-valued and its Hessian is matrix-valued, it is natural to guess that its third-order gradient will be tensor-valued. However, not only is the use of tensors not very convenient, but in this context it is also unnecessary. Continue reading →

Posted in: Optimisation | Tagged: Analysis, Calculus, directional derivative, directional Hessian, Hessian, Taylor expansion, third-order gradient

Error bounds for second order approximation

By psopasakis on August 3, 2016 | Leave a comment

Where here we prove an approximation bound for twice continuously differentiable functions $f:\mathbb{R}^n \to \mathbb{R}$ with M-Lipschitzian Hessian, that is $\|\nabla^2 f(x) - \nabla^2 f(y)\| \leq M \|x-y\|$ for all $x,y$ . In particular, we show that for all $x,y$

$\begin{aligned} |f(y) - f(x) - \langle \nabla f(x), y-x\rangle - \frac{1}{2}\langle \nabla f^2(x)(y-x), y-x\rangle| \leq \frac{M}{6}\|x-y\|^3. \end{aligned}$

This is stated as Lemma 1.2.4 in: Y. Nesterov, Introductory Lectures on Convex Optimization – A basic course, Kluwer Ac. Publishers, 2004. Continue reading →

Posted in: Variational analysis | Tagged: Analysis, approximation, Hessian, Lipschitz, Lipschitzian, quadratic approximation

Weak closure points not attainable as limits of sequences

By psopasakis on July 24, 2016 | Leave a comment

Where in this post we discover an uncanny property of the weak topology: the points of the weak closure of a set cannot always be attained as limits of elements of the set. Naturally, the w-closure of a set is weakly closed. The so-called weak sequential closure of a set, on the other hand, is the set of cluster points of sequences made with elements from that set. The new set is not, however, weakly sequentially closed, which means that there may arise new cluster points; we may, in fact, have to take the weak sequential closure trans-finitely many time to obtain a set which is weakly sequentially closed – still, this may not be weakly closed. Continue reading →

Posted in: Analysis | Tagged: Analysis, convergence, topology, weak closure, weak sequential closure, weak topology

On Set Convergence I

By psopasakis on June 26, 2016 | 3 Comments

We give the definitions of inner and outer limits for sequences of sets in topological and normed spaces and we provide some important facts on set convergence on topological and normed spaces. We juxtapose the notions of the limit superior and limit inferior for sequences of sets and we outline some facts regarding the Painlevé-Kuratowski convergence of set sequences. Continue reading →

Posted in: Variational analysis | Tagged: Analysis, convergence, convergence of sets, inner limit, outer limit, Painleve-Kuratowski convergence, topology

Mathematix