Crate totsu

Totsu (凸 in Japanese) means convex.

This crate for Rust provides a first-order conic linear program solver.

Target problem

A common target problem is continuous scalar convex optimization such as LP, QP, QCQP, SOCP and SDP. Each of those problems can be represented as a conic linear program: \[ \begin{array}{ll} {\rm minimize} & c^T x \\ {\rm subject \ to} & A x + s = b \\ & s \in \mathcal{K}, \end{array} \] where

• variables \( x \in \mathbb{R}^n,\ s \in \mathbb{R}^m \)
• \( c \in \mathbb{R}^n \) as an objective linear operator
• \( A \in \mathbb{R}^{m \times n} \) and \( b \in \mathbb{R}^m \) as constraint linear operators
• a nonempty, closed, convex cone \( \mathcal{K} \).

Algorithm and design concepts

The author combines the two papers [1] [2] so that the homogeneous self-dual embedding matrix in [2] is formed as a linear operator in [1].

A core method solver::Solver::solve takes the following arguments:

• objective and constraint linear operators that implement operator::Operator trait and
• a projection onto a cone that implements cone::Cone trait.

Therefore solving a specific problem requires an implementation of those traits. You can use pre-defined implementations (see problem), as well as construct a user-defined tailored version for the reason of functionality and efficiency. Modules operator and cone include several basic structs that implement operator::Operator and cone::Cone trait.

Core linear algebra operations that solver::Solver requires are abstracted by linalg::LinAlg trait, while subtrait linalg::LinAlgEx is used for operator, cone and problem modules. This crate includes two linalg::LinAlgEx implementors:

• linalg::FloatGeneric - num::Float-generic implementation (pure Rust but slow)
• linalg::F64LAPACK - f64-specific implementation using cblas-sys and lapacke-sys.

Features

Using linalg::F64LAPACK

[dependencies.totsu]
version = "0.8.0"
features = ["f64lapack"]

In addition you need a BLAS/LAPACK source to link.

Without std

This crate can be used without the standard library (#![no_std]). Use this in Cargo.toml:

[dependencies.totsu]
version = "0.8.0"
default-features = false
features = ["libm"]

Some module and structs are not availale in this case:

• problem
• operator::MatBuild

Changelog

Changelog is available in CHANGELOG.md.

Examples

QP

use float_eq::assert_float_eq;
use totsu::prelude::*;
use totsu::operator::MatBuild;
use totsu::problem::ProbQP;

//env_logger::init();

type LA = FloatGeneric<f64>;
type AMatBuild = MatBuild<LA, f64>;
type AProbQP = ProbQP<LA, f64>;
type ASolver = Solver<LA, f64>;

let n = 2; // x0, x1
let m = 1;
let p = 0;

// (1/2)(x - a)^2 + const
let mut sym_p = AMatBuild::new(MatType::SymPack(n));
sym_p[(0, 0)] = 1.;
sym_p[(1, 1)] = 1.;

let mut vec_q = AMatBuild::new(MatType::General(n, 1));
vec_q[(0, 0)] = -(-1.); // -a0
vec_q[(1, 0)] = -(-2.); // -a1

// 1 - x0/b0 - x1/b1 <= 0
let mut mat_g = AMatBuild::new(MatType::General(m, n));
mat_g[(0, 0)] = -1. / 2.; // -1/b0
mat_g[(0, 1)] = -1. / 3.; // -1/b1

let mut vec_h = AMatBuild::new(MatType::General(m, 1));
vec_h[(0, 0)] = -1.;

let mat_a = AMatBuild::new(MatType::General(p, n));

let vec_b = AMatBuild::new(MatType::General(p, 1));

let s = ASolver::new().par(|p| {
   p.max_iter = Some(100_000);
});
let mut qp = AProbQP::new(sym_p, vec_q, mat_g, vec_h, mat_a, vec_b, s.par.eps_zero);
let rslt = s.solve(qp.problem()).unwrap();

assert_float_eq!(rslt.0[0..2], [2., 0.].as_ref(), abs_all <= 1e-3);

Other examples

You can find other tests of pre-defined problems. More practical examples are also available.

References

1. T. Pock and A. Chambolle. “Diagonal preconditioning for first order primal-dual algorithms in convex optimization.” 2011 International Conference on Computer Vision. IEEE, 2011.
2. B. O’donoghue, et al. “Conic optimization via operator splitting and homogeneous self-dual embedding.” Journal of Optimization Theory and Applications 169.3 (2016): 1042-1068.
3. N. Parikh and S. Boyd. “Proximal algorithms.” Foundations and Trends in optimization 1.3 (2014): 127-239.
4. Mosek ApS. “MOSEK modeling cookbook.” (2020).
5. S. Boyd and L. Vandenberghe. “Convex Optimization.” (2004).

Modules

cone

Convex cone

linalg

Linear algebra

operator

Linear operator

prelude

Prelude

problem

Pre-defined optimization problems

solver

First-order conic linear program solver