# 8 Problem Formulation and Solutions¶

In this chapter we will discuss the following topics:

• 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 different problem types for quick reference.

Continuous problem formulations

• Linear optimization (LO)

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

Conic optimization extends linear optimization with affine conic constraints (ACC):

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

where $$\D$$ is a product of domains from Sec. 11.5 (Supported domains).

• Semidefinite optimization (SDO)

A conic optimization problem can be further extended with semidefinite variables:

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

where $$\D$$ is a product of domains from Sec. 11.5 (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 $$I$$. Available for problems of type LO, CO, QO and QCQO.