# 1 Overview¶

An optimization problem such as

is an example of a linear optimization problem. In general such problems may be very large and are often compactly written in one of the standard forms such as:

A natural generalization of linear constraints \(Ax+b=0\) are constraints of the form

where \(\K\) is a more general nonlinear convex set.

Here enters the **MOSEK** Optimization Suite, which is a software package for solving large optimization problems with many constraints and variables, in particular:

linear problems as in (1.1),

nonlinear conic problems with constraints as in (1.2), where \(\K\) can be the second-order cone, power cone, exponential cone, semidefinite cone and some cones derived from them,

as well as, independently, problems with quadratic objective and constraints,

mixed-integer variants of those problems and problems with disjunctive constraints.

In addition to optimization algorithms the **MOSEK** Optimization Suite provides interfaces to mainstream programming languages such as C, Java, MATLAB, .NET, Python, R, Julia or Rust.

Below are the typical constraints and expressions which can be modeled using conic form:

Conic Quadratic Optimization: \(t\geq x^2\), sums of squares, 2-norm \(t\geq \|x\|_2\), variance, \(\|Ax-b\|_2\).

Power Cone Optimization: powers \(x^p\), geometric mean, products \(x^\alpha y^\beta\), \(p\)-norm.

Exponential Cone: \(e^x\), \(\ln{x}\), log-sum-exp, entropy \(x\ln{x}\), relative entropy, geometric programming.

Semidefinite Optimization: \(X\succeq 0\) is positive semidefinite.

**MOSEK** provides three solvers:

A simplex solver for linear problems.

An interior-point conic solver for linear and nonlinear continuous problems, conic problems and quadratic problems.

A mixed-integer solver when the problem contains integer variables or disjunctive constraints.

The **MOSEK** Optimization Suite includes

the low-level optimizer API for C, Java, .NET, Python, Julia and Rust.

the object-oriented

*Fusion*API for C++, Java, .NET and Python.an optimization toolbox for MATLAB.

an Rmosek package for R.

a command line tool.

optimizer server for remote optimization.

In addition **MOSEK** Optimization Suite provides

sensitivity analysis for linear problems.

infeasibility diagnostic tools.

problem analyses for diagnosing numerical issues in input data.

reading and writing optimization problems and solutions from/to files.

Most large optimization problems are very sparse i.e. most of the input data consists of zeros. Therefore, the APIs and optimization algorithms in **MOSEK** Optimization Suite is designed to exploit sparsity to reduce both storage and time.