1 Introduction

The MOSEK Optimization Suite 8.1.0.33 is a powerful software package capable of solving large-scale optimization problems of the following kind:

  • linear,
  • conic quadratic (also known as second-order cone),
  • convex quadratic,
  • semidefinite,
  • and general convex.

Integer constrained variables are supported for all problem classes except for semidefinite and general convex problems. In order to obtain an overview of features in the MOSEK Optimization Suite consult the product introduction guide.

The most widespread class of optimization problems is linear optimization problems, where all relations are linear. The tremendous success of both applications and theory of linear optimization can be ascribed to the following factors:

  • The required data are simple, i.e. just matrices and vectors.
  • Convexity is guaranteed since the problem is convex by construction.
  • Linear functions are trivially differentiable.
  • There exist very efficient algorithms and software for solving linear problems.
  • Duality properties for linear optimization are nice and simple.

Even if the linear optimization model is only an approximation to the true problem at hand, the advantages of linear optimization may outweigh the disadvantages. In some cases, however, the problem formulation is inherently nonlinear and a linear approximation is either intractable or inadequate. Conic optimization has proved to be a very expressive and powerful way to introduce nonlinearities, while preserving all the nice properties of linear optimization listed above.

The fundamental expression in linear optimization is a linear expression of the form

\[A x - b \in \K\]

where \(\K = \{y: y \geq 0\}\), i.e.,

\[\begin{split}\begin{array}{l} A x - b =y,\\ y \in \K. \end{array}\end{split}\]

In conic optimization a wider class of convex sets \(\K\) is allowed, for example in 3 dimensions \(\K\) may correspond to an ice cream cone. The conic optimizer in MOSEK supports three structurally different types of cones \(\K\), which allows a surprisingly large number of nonlinear relations to be modelled (as described in the MOSEK modeling cookbook), while preserving the nice algorithmic and theoretical properties of linear optimization.

1.1 Why the Optimization Server?

The MOSEK OptServer is a simple solver service. It can receive tasks over HTTP or HTTPS and return solutions, log and other information. It can be used either in

  • completely open mode, where no authentication is required,
  • closed mode, where authentication is required, or
  • semi-open mode, where authentication is required for administrative tasks, but optimizer tasks can be submitted anonymously.

The OptServer provides an API for submitting tasks and retrieving information. It makes it easy to offload heavy computations to a remote machine. This is useful for running MOSEK on a wider range of devices.