# 12 Problem Formulation and Solutions¶

In this chapter we will discuss the following issues:

• The formal, mathematical formulations of the problem types that MOSEK can solve and their duals.

• The solution information produced by MOSEK.

• The infeasibility certificate produced by MOSEK if the problem is infeasible.

For the underlying mathematical concepts, derivations and proofs see the Modeling Cookbook or any book on convex optimization. This chapter explains how the related data is organized specifically within the MOSEK API. Below is an outline of the various formats. For details see the corresponding subesctions.

## 12.1 Continuous problem formulations¶

• Sec. 12.2.1 (Linear Optimization)

A linear problem has the form

$\begin{split}\begin{array}{lccccl} \mbox{minimize} & & & c^T x+c^f & & \\ \mbox{subject to} & l^c & \leq & A x & \leq & u^c, \\ & l^x & \leq & x & \leq & u^x. \end{array}\end{split}$
• Sec. 12.2.2 (Conic Optimization)

Conic optimization extends linear optimization with affine conic constraints (ACC), so a conic problem has the form

$\begin{split}\begin{array}{lccccl} \mbox{minimize} & & & c^T x+c^f & & \\ \mbox{subject to} & l^c & \leq & A x & \leq & u^c, \\ & l^x & \leq & x & \leq & u^x, \\ & & & Fx+g & \in & \D, \end{array}\end{split}$

where $$\D$$ is a product of domains from Sec. 15.8 (Supported domains).

• Sec. 12.2.3 (Semidefinite Optimization)

A conic optimization problem can be further extended with semidefinite variables, leading to a semidefinite optimization problem of the form

$\begin{split}\begin{array}{lccccl} \mbox{minimize} & & & c^T x+ \langle \barC,\barX\rangle + c^f & & \\ \mbox{subject to} & l^c & \leq & A x + \langle \barA,\barX\rangle & \leq & u^c, \\ & l^x & \leq & x & \leq & u^x, \\ & & & Fx+\langle \barF,\barX\rangle +g & \in & \D, \\ & & & \barX & \in & \PSD, \end{array}\end{split}$

where $$\D$$ is a product of domains from Sec. 15.8 (Supported domains) and $$\PSD$$ is a product of PSD cones meaning that $$\barX$$ is a sequence of PSD matrix variables.

$\begin{split}\begin{array}{lccccl} \mbox{minimize} & & & \frac12 x^TQ^ox + c^T x+c^f & & \\ \mbox{subject to} & l^c & \leq & \frac12 x^TQ^cx+ A x & \leq & u^c, \\ & l^x & \leq & x & \leq & u^x. \end{array}\end{split}$
$x_I \in \integral$
for some index set $$I$$. A problem with at least one integer variable is solved by the mixed-integer optimizer.