LOQO

Authors: H. Y. Benson, Vanderbei
Version: 5.04, 8/2000;
Available: yes, from http://orfe.princeton.edu/ loqo/
Key paper: [27]
Features: NLP approach
Language, Input format: C; SDPA, Matlab, AMPL
Error computations: no
Solves: SOCP (SDP)

LOQO is a software package for solving general (smooth) nonlinear optimization problems of the form

$$\begin{array}{lrcl} \mbox{minimize } & f(x) & & \\ \mbox{... ... \\ & h_i (x) & \ge & 0, \quad i \in \mathcal{I}, \end{array}$$

where $x \in \mathbb{R}^n$, $f: \mathbb{R}^n \rightarrow \mathbb{R}$, $\mathcal{E}$ is the set of equalities, $\mathcal{I}$ is the set of inequalities, $g: \mathbb{R}^n \rightarrow \mathbb{R}^{\vert\mathcal{E}\vert}$, and $h: \mathbb{R}^n \rightarrow \mathbb{R}^{\vert\mathcal{I}\vert}$. It implements an infeasible-primal-dual path-following method and requires that the problem be smooth, that is $f$ and $h$ be twice differentiable, and $g$ be an affine function. Even though LOQO can handle nonconvex problems in general, it performs better with convex problems, where $f$ is convex, $g$ are affine, and $h$ are concave functions.

Stopping criteria

$$\Vert\mathcal{A}x - b\Vert _2 \leq 10^{-7}$$

$$\Vert\mathcal{A}^* y + z -c\Vert _2 \leq 10^{-8}$$

$$\log_{10} \left( \frac{ \vert \langle c, x \rangle - b^T y \vert }{\vert\langle c, x \rangle \vert + 1} \right) \leq -8$$

• LOQO can handle other types of nonlinear constraints in the problem.
• A weakness results from the use of the Cholesky factorization. LOQO works best with a sparse problem, and it does not exploit the special structure of an SOCP.
• Its SDP approach leads to dense problems.