next up previous
Next: Computing Errors Up: Introduction Previous: Introduction

The Problems Solved

The primal and dual pair of conic optimization problems over a self-dual cone are defined as

\begin{displaymath}
\begin{array}{lrccrcr}
& \min & \langle c, x \rangle & \hsp...
...= b & \hspace{.75cm} & & \mathcal{A}^* y + z = c &
\end{array}\end{displaymath}

where In the case of a semidefinite-quadratic-linear program these are defined as follows: Thus (P) and (D) become

\begin{displaymath}\begin{array}{rlllllllllllll} \min & \sum_{j=1}^{n_s} c^s_j \...
...^q_k \geq_q 0 \; \forall k & \quad x^\ell \geq 0 \\ \end{array}\end{displaymath} ($ SQLP-P$)

and

\begin{displaymath}\begin{array}{rllllllllll} \max & \sum_{i=1}^{m} b_i y_i \\ {...
...ceq 0 & \quad z^q_k \geq_q 0&\quad z^\ell \geq 0 \\ \end{array}\end{displaymath} ($ SQLP-D$)


Thus the feasible set is a product of semidefinite, quadratic and nonnegative orthant cones, intersected with an affine subspace. It is possible that one or more of the three parts of the problem is absent, i.e., any of $ n_s$, $ n_q$, or the length of $ x^\ell$ may be zero.

The rotated quadratic cone is defined as

$\displaystyle \{ \, v \in \mathbb{R}^{k} \, \vert \, v_1 v_2 \geq \vert\!\vert v_{3:k} \vert\!\vert \, \}.
$

It is simply a rotation of the usual quadratic cone, but for the purpose of modeling quadratic inequalities, it is more convenient to use, thus several participating codes support this cone.


next up previous
Next: Computing Errors Up: Introduction Previous: Introduction
Hans D. Mittelmann 2002-08-17