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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.

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

$f:C\to ℝ$

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

Examples of $C$ include the positive orthant ${ℝ}_{+}^n=\{x\in {ℝ}^n:x\ge \mathrm{𝟎}\}$, positive semidefinite matrices ${𝕊}_{+}^n$, and the second-order cone $\left\{\right(x,t)\in {ℝ}^n\times ℝ:\Vert x\Vert \le t\}$. 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.

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.
• Splitting Conic Solver (SCS) Package for solving large-scale convex cone problems.
• MOSEK Software capable of solving conic optimization problems.