Tag Archives: duality

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

Cone programs and self-dual embeddings

By psopasakis on June 5, 2017 | Leave a comment

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

Continue reading →

Posted in: Optimisation, Variational analysis | Tagged: cone programs, cones, convexity, duality, fenchel duality, optimality conditions, Optimisation, optimization

Lagrange vs Fenchel Duality

By psopasakis on May 7, 2017 | 2 Comments

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 →

Posted in: Optimisation, Variational analysis | Tagged: duality, fenchel duality, lagrange duality, Optimisation, optimization, perturbation functions

Mathematix

Tag Archives: duality

Interchangeability of infimum in risk measures

Cone programs and self-dual embeddings

Lagrange vs Fenchel Duality

Recent Posts

Archives

Categories

Mathematix

Blogs I Follow