Conic optimization

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

${\displaystyle f:C\to \mathbb {R} }$

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

Examples of ${\displaystyle C}$ include the positive orthant ${\displaystyle \mathbb {R} _{+}^{n}=\left\{x\in \mathbb {R} ^{n}:\,x\geq \mathbf {0} \right\}}$, positive semidefinite matrices ${\displaystyle \mathbb {S} _{+}^{n}}$, and the second-order cone ${\displaystyle \left\{(x,t)\in \mathbb {R} ^{n}\times \mathbb {R}$ :\lVert x\rVert \leq t\right\}} . Often ${\displaystyle f\ }$ 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.

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

• Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. ISBN 978-0-521-83378-3. Retrieved October 15, 2011.
• MOSEK Software capable of solving conic optimization problems.