Lecture

Convex Optimization: Elementary Results

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In course

MGT-418: Convex optimization

This course introduces the theory and application of modern convex optimization from an engineering perspective. ...

Description

This lecture covers the elementary results in convex optimization, focusing on affine, convex, and conic hulls. It explains the definitions of these sets, their combinations, and transformations that preserve convexity. The lecture also delves into proper cones, conic inequalities, convex functions, and their conditions, providing examples and insights into convexity-preserving operations.

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Instructor

Daniel Kuhn

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

Convex optimization

Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard.

Affine transformation

In Euclidean geometry, an affine transformation or affinity (from the Latin, affinis, "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, an affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space onto itself while preserving both the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes, and so on) and the ratios of the lengths of parallel line segments.

Mathematical optimization

Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.

Affine space

In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point.

Absolutely convex set

In mathematics, a subset C of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk. The disked hull or the absolute convex hull of a set is the intersection of all disks containing that set. A subset of a real or complex vector space is called a and is said to be , , and if any of the following equivalent conditions is satisfied: is a convex and balanced set.

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