Conic optimization

Conic optimization is a subfield of convex optimization that studies problems consisting of minimizing a convex function over the intersection of an affine subspace and a convex cone.

The class of conic optimization problems includes some of the most well known classes of convex optimization problems, namely linear and semidefinite programming.

Definition
Given a real vector space X, a convex, real-valued function


 * $$f:C \to \mathbb R$$

defined on a convex cone $$C \subset X$$, and an affine subspace $$\mathcal{H}$$ defined by a set of affine constraints $$h_i(x) = 0 \ $$, a conic optimization problem is to find the point $$x$$ in $$C \cap \mathcal{H} $$ for which the number $$f(x)$$ is smallest.

Examples of $$ C $$ include the positive orthant $$\mathbb{R}_+^n = \left\{ x \in \mathbb{R}^n : \, x \geq \mathbf{0}\right\} $$, positive semidefinite matrices $$\mathbb{S}^n_{+}$$, and the second-order cone $$\left \{ (x,t) \in \mathbb{R}^{n}\times \mathbb{R} : \lVert x \rVert \leq t \right \} $$. Often $$f \ $$ is a linear function, in which case the conic optimization problem reduces to a linear program, a semidefinite program, and a second order cone program, respectively.

Duality
Certain special cases of conic optimization problems have notable closed-form expressions of their dual problems.

Conic LP
The dual of the conic linear program


 * minimize $$c^T x \ $$
 * subject to $$Ax = b, x \in C \ $$

is


 * maximize $$b^T y \ $$
 * subject to $$A^T y + s= c, s \in C^* \ $$

where $$C^*$$ denotes the dual cone of $$C \ $$.

Whilst weak duality holds in conic linear programming, strong duality does not necessarily hold.

Semidefinite Program
The dual of a semidefinite program in inequality form


 * minimize $$c^T x \ $$
 * subject to $$x_1 F_1 + \cdots + x_n F_n + G \leq 0$$

is given by


 * maximize $$\mathrm{tr}\ (GZ)\ $$
 * subject to $$\mathrm{tr}\ (F_i Z) +c_i =0,\quad i=1,\dots,n$$
 * $$Z \geq0$$