# Welcome

## Contents

# Welcome¶

## What is New¶

Linear Algebra and Analysis

Convex Sets and Functions

## Mathematical Optimization Problems¶

A mathematical optimization problem consists of maximizing or minimizing a real valued function under a set of constraints. We shall assume \(\VV\) to denote a finite dimensional real vector space. Typical examples of \(\VV\) are \(\RR^n\) and \(\SS^n\).

Formally, we express a mathematical optimization problem as:

\(\bx \in \VV\) is the

*optimization variable*of the problem.\(f_0 : \VV \to \RR\) is the

*objective function*.The functions \(f_i : \VV \to \RR, \; i=1,\dots, m\) are the (inequality)

*constraint functions*.The (real scalar) constants \(b_1, \dots, b_m\) are the limits for the inequality constraints.

A vector \(\bx \in \VV\) is called

*feasible*if it belongs to the domains of \(f_0, f_1, \dots, f_m\) and satisfies all the constraints.A vector \(\bx^*\) is called

*optimal*if is feasible and has the smallest objective value; i.e. for any feasible \(\bz\), we have \(f_0(\bz)\geq f_0(\bx^*)\).An optimal vector is also called a

*solution*to the optimization problem.An optimization problem is called

*infeasible*if there is no feasible vector. i.e. there is no vector \(\bx \in \VV\) which satisfies the inequality constraints.An infeasible problem doesn’t have a solution.

A feasible problem may not have a solution if the objective function is

*unbounded below*. i.e. for every feasible \(\bx\), there exists another feasible \(\bz\) such that \(f_0(\bz) < f_0(\bx)\).If a feasible problem is not unbounded below, then it may have one or more solutions.

Convex Optimization focuses on a special class of mathematical optimization problems where:

The real valued function being optimized (maximized or minimized) is convex.

The feasible set of values for the function is a closed convex set.

## Convex Optimization Problems¶

Convex optimization problems are usually further classified into

Least squares

Linear programming

Quadratic minimization with linear constraints

Quadratic minimization with convex quadratic constraints

Conic optimization

Geometric programming

Second order cone programming

Semidefinite programming

There are specialized algorithms available for each of these classes.

## Applications¶

Some of the applications of convex optimization include:

Portfolio optimization

Worst case risk analysis

Compressive sensing

Statistical regression

Model fitting

Combinatorial Optimization