# 12.2 Conic Quadratic Optimization¶

*Conic quadratic optimization* is an extension of linear optimization (see Sec. 12.1 (Linear Optimization)) allowing conic domains to be specified for subsets of the problem variables. A conic quadratic optimization problem can be written as

where set \(\K\) is a Cartesian product of convex cones, namely \(\K = \K_1 \times \cdots \times \K_p\). Having the domain restriction, \(x \in \K\), is thus equivalent to

where \(x = (x^1, \ldots , x^p)\) is a partition of the problem variables. Please note that the \(n\)-dimensional Euclidean space \(\real^n\) is a cone itself, so simple linear variables are still allowed.

**MOSEK** supports only a limited number of cones, specifically:

- The \(\real^n\) set.
- The quadratic cone:

- The rotated quadratic cone:

Although these cones may seem to provide only limited expressive power they can be used to model a wide range of problems as demonstrated in [MOSEKApS12].

## 12.2.1 Duality for Conic Quadratic Optimization¶

The dual problem corresponding to the conic quadratic optimization problem (1) is given by

where the dual cone \(\K^*\) is a Cartesian product of the cones

where each \(\K^*_t\) is the dual cone of \(\K_t\). For the cone types **MOSEK** can handle, the relation between the primal and dual cone is given as follows:

- The \(\real^n\) set:

- The quadratic cone:

- The rotated quadratic cone:

Please note that the dual problem of the dual problem is identical to the original primal problem.

## 12.2.2 Infeasibility for Conic Quadratic Optimization¶

In case **MOSEK** finds a problem to be infeasible it reports a certificate of infeasibility. This works exactly as for linear problems (see Sec. 12.1.2 (Infeasibility for Linear Optimization)).

### 12.2.2.1 Primal Infeasible Problems¶

If the problem (1) is infeasible, **MOSEK** will report a certificate of primal infeasibility: The dual solution reported is the certificate of infeasibility, and the primal solution is undefined.

A certificate of primal infeasibility is a feasible solution to the problem

such that the objective value is strictly positive.

### 12.2.2.2 Dual infeasible problems¶

If the problem (2) is infeasible, **MOSEK** will report a certificate of dual infeasibility: The primal solution reported is the certificate of infeasibility, and the dual solution is undefined.

A certificate of dual infeasibility is a feasible solution to the problem

where

and

such that the objective value is strictly negative.