Conic optimization

Conic optimization

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.

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 y^T A + s= c, s in C^*

where C^* denotes the dual cone of C .

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

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