**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.

Concept# Convex analysis

Summary

Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.
Convex set
A subset of some vector space is if it satisfies any of the following equivalent conditions:
If is real and then
If is real and with then
Convex function
Throughout, will be a map valued in the extended real numbers with a domain that is a convex subset of some vector space.
The map is a if
holds for any real and any with If this remains true of when the defining inequality () is replaced by the strict inequality
then is called .
Convex functions are related to convex sets. Specifically, the function is convex if and only if its
is a convex set. The epigraphs of extended real-valued functions play a role in convex analysis that is analogous to the role played by graphs of real-valued function in real analysis. Specifically, the epigraph of an extended real-valued function provides geometric intuition that can be used to help formula or prove conjectures.
The domain of a function is denoted by while its is the set
The function is called if and for Alternatively, this means that there exists some in the domain of at which and is also equal to In words, a function is if its domain is not empty, it never takes on the value and it also is not identically equal to If is a proper convex function then there exist some vector and some such that
for every
where denotes the dot product of these vectors.
Convex conjugate
The of an extended real-valued function (not necessarily convex) is the function from the (continuous) dual space of and
where the brackets denote the canonical duality The of is the map defined by for every
If denotes the set of -valued functions on then the map defined by is called the .
If and then the is
For example, in the important special case where is a norm on , it can be shown
that if then this definition reduces down to:
and
For any and which is called the . This inequality is an equality (i.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (151)

Related people (33)

Related concepts (9)

Related courses (11)

EE-556: Mathematics of data: from theory to computation

This course provides an overview of key advances in continuous optimization and statistical analysis for machine learning. We review recent learning formulations and models as well as their guarantees

ME-425: Model predictive control

Provide an introduction to the theory and practice of Model Predictive Control (MPC). Main benefits of MPC: flexible specification of time-domain objectives, performance optimization of highly complex

MICRO-512: Image processing II

Study of advanced image processing; mathematical imaging. Development of image-processing software and prototyping in Jupyter Notebooks; application to real-world examples in industrial vision and bio

Related lectures (105)

Related units (7)

Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.

In mathematics, the epigraph or supergraph of a function valued in the extended real numbers is the set, denoted by of all points in the Cartesian product lying on or above its graph. The strict epigraph is the set of points in lying strictly above its graph. Importantly, although both the graph and epigraph of consists of points in the epigraph consists of points in the subset which is not necessarily true of the graph of If the function takes as a value then will be a subset of its epigraph For example, if then the point will belong to but not to These two sets are nevertheless closely related because the graph can always be reconstructed from the epigraph, and vice versa.

In mathematics, a dual system, dual pair, or duality over a field is a triple consisting of two vector spaces and over and a non-degenerate bilinear map . Duality theory, the study of dual systems, is part of functional analysis. It is separate and distinct to Dual-system Theory in psychology. Pairings A or pair over a field is a triple which may also be denoted by consisting of two vector spaces and over (which this article assumes is the field either of real numbers or the complex numbers ).

Optimal Transport: Rockafellar Theorem

Explores the Rockafellar Theorem in optimal transport, focusing on c-cyclical monotonicity and convex functions.

Legendre Transform

Explores the Legendre transform, duality in convex analysis, and optimization problems.

Minkowski-Weyl: Convexity and Separation Theorem

Explores convex sets, Minkowski-Weyl theorem, and Separation theorem in convex analysis.

This paper develops a fast algorithm for computing the equilibrium assignment with the perturbed utility route choice (PURC) model. Without compromise, this allows the significant advantages of the PURC model to be used in large-scale applications. We form ...

Gian Florin Gentinetta, Stefan Woerner

Quantum support vector machines employ quantum circuits to define the kernel function. It has been shown that this approach offers a provable exponential speedup compared to any known classical algorithm for certain data sets. The training of such models c ...

Volkan Cevher, Efstratios Panteleimon Skoulakis, Leello Tadesse Dadi

Given a sequence of functions $f_1,\ldots,f_n$ with $f_i:\mathcal{D}\mapsto \mathbb{R}$, finite-sum minimization seeks a point ${x}^\star \in \mathcal{D}$ minimizing $\sum_{j=1}^nf_j(x)/n$. In this work, we propose a key twist into the finite-sum minimizat ...

2024