Conic optimization

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Conic optimization is a subfield of convex optimization. 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 \$ , the 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 semidefinite matrices $\mathbb{S}^n_{++}$ , the positive orthant $x \geq \mathbf{0}$ for $x \in \mathbb{R}^n$ , and the second-order cone $\left \{ (x,t) \in \mathbb{R}^{n+1} : \lVert x \rVert \leq t \right \}$ . Often $f \$ is a linear function, in which case the conic optimization problem reduces to a semidefinite program, a linear program, and a second order cone program, respectively.