MosekCOModel is still an experimental project project. Use with caution, expect rough corners and untested cases, and the API is subject to change. Any comments and suggestions are welcome!

MosekCOModel

The MosekCOModel crate is a modeling package for building linear and conic optimization models. The crate does not directly include a solver - these are implemented in separate projects, currently:

• MOSEK
• HIGHS (linear and integer optimization)
• OptServer MOSEK via optserver (linear, conic and integer)

Published crates: https://crates.io/crates/mosekcomodel

Documentation for latest crates: https://docs.rs/mosekcomodel/latest/mosekcomodel/

Design principle

MosekModel allows building a model of the form

min/max   c^t x + c_fix
such that Ax_b ∊ K_c
          x ∊ K_x  

That is affine expressions and conic domains of constraints and variables.

The MosekModel package provides functionality to build the linear expressions.

The Model

The model object encapsulates all information describing the optimization problem, solver parameters, solutions etc. Variables and constraints are created through the model object.

Variables

A variable object is an N-dimensional object, either dense or sparse - basically a dense or sparse N-dimensional array of scalar variables. When a variable is created in a model, it is created with a domain, i.e. either linear bounds or a convex conic domain as well as integrality. The domain of a variable is immutable once created. Variables can be sliced and stacked to create new variables.

Variable objects can be used to create linear expressions and to access solution values once the problem is solved.

Constraints

Constraints are added to the model, created from a linear expression and a domain. Constraints cannot be integer constrained.

Constraint objects can be used to access solution values.

Simple conic example

Implementing the models

minimize y₁ + y₂ + y₃
such that
         x₁ + x₂2 + 2.0 x₃ = 1.0
                  x₁,x₂,x₃ ≥ 0.0
and
         (y₁,x₁,x₂) in C₃,
         (y₂,y₃,x₃) in K₃

where C₃ and K₃ are respectively the quadratic and rotated quadratic cone of size 3 defined as

    C₃ = { z₁,z₂,z₃ :      z₁ ≥ √(z₂² + z₃²) }
    K₃ = { z₁,z₂,z₃ : 2 z1 z₂ ≥ z₃²          }

This is the included model cqo1.rs included under mosekcomodel-mosek/examples:

extern crate mosekcomodel;
use mosekcomodel::*;
use mosekcomodel::expr::*;
use mosekcomodel_mosek::Model;

fn main() {
    let mut m = Model::new(Some("cqo1"));
    let x = m.variable(Some("x"), greater_than(vec![0.0;3]));
    let y = m.variable(Some("y"), 3);

    // Create the aliases
    //      z1 = [ y[0],x[0],x[1] ]
    //  and z2 = [ y[1],y[2],x[2] ]

    let z1 = Variable::vstack(&[&y.index(0..1), &x.index(0..2)]);
    let z2 = Variable::vstack(&[&y.index(1..3), &x.index(2..3)]);

    // Create the constraint
    //      x[0] + x[1] + 2.0 x[2] = 1.0
    let aval = &[1.0, 1.0, 2.0];
    let _ = m.constraint(Some("lc"), aval.dot(x.clone()), equal_to(1.0));

    // Create the constraints
    //      z1 belongs to C_3
    //      z2 belongs to K_3
    // where C_3 and K_3 are respectively the quadratic and
    // rotated quadratic cone of size 3, i.e.
    //                 z1[0] >= sqrt(z1[1]^2 + z1[2]^2)
    //  and  2.0 z2[0] z2[1] >= z2[2]^2
    let qc1 = m.constraint(Some("qc1"), &z1, in_quadratic_cone(3));
    let _qc2 = m.constraint(Some("qc2"), &z2, in_rotated_quadratic_cone(3));

    // Set the objective function to (y[0] + y[1] + y[2])
    m.objective(Some("obj"), Sense::Minimize, y.sum());

    // Solve the problem
    m.solve();

    // Get the linear solution values
    let solx = m.primal_solution(SolutionType::Default,&x);
    let soly = m.primal_solution(SolutionType::Default,&y);
    println!("x = {:?}", solx);
    println!("y = {:?}", soly);

    // Get primal and dual solution of qc1
    let qc1lvl = m.primal_solution(SolutionType::Default,&qc1);
    let qc1sn  = m.dual_solution(SolutionType::Default,&qc1);

    println!("qc1 levels = {:?}", qc1lvl);
    println!("qc1 dual conic var levels = {:?}", qc1sn);
}

Compiling, testing, running

First, to build simply type

cargo build

Optionally, also pass the --release flag as the Debug build is significantly slower that the release build.

Running examples requires the MOSEK library to be available and a valid MOSEK license file. The simplest solution is to download and unpack the MOSEK distro from MOSEK Downloads, unpack the distro and set relevant environment variable:

• On MS Windows: set PATH=C:\full\path\to\mosek\binaries;%PATH%
• On Mac OSX: export DYLD_LIBRARY_PATH=/full/path/to/mosek/binaries:$DYLD_LIBRARY_PATH
• On Linux: export LD_LIBRARY_PATH=/full/path/to/mosek/binaries:$LD_LIBRARY_PATH

To run tests, do

cargo test --all

A trial license can be obtained from MOSEK Trial license.

Demos

The project also contanis a set of graphical demos. These are in separate sub-projects in examples/demos since they depend on a lot of external libraries.

To run these, go to the sub-folder with demos

cd examples/demos

Then to run, say lowner-john-2d, do

cargo run --release -p lowner-john-2d

lowner-john-2d

For a set of moving, rotating polygons, computes the minimal bounding ellipsoid containing all polygons (or, in fact all corner points), and the maximum ellipsoid contained in the intersection (when the intersection is non-empty).

lowner-john-outer-3d

For a set of rotating and moving polyhedrons, compute the minimal bounding ellipsoid containing all polyhedrons (or their corner points).

ellipsoid-approximation

For a set of moving and rotating ellipses, compute the outer approximation (minimal ellipse containing all moving ellipses), and inner approximation (maximum ellipe contained in the intersection of all ellipses).

ellipsoid-approximation-3d

For a set of moving and rotating ellipsoids, compute the minimal bounding ellipsoid.


trigpoly

Simple visualization of trigonometric polynomial optimization.

truss

Simple 2D truss design model assigning material to bars in a truss construction. This requires a data file, e.g.

cargo run --release -p truss -- truss/data/bridge.trs

Total Variation

Implements a simple image enhancement method as a optimization problem. Originally, the method was designed for use with a large scale first-order method, so this is mostly just a visual demonstration of an optimization result.

Run with --help to see further command line options.

cargo run --release -p total-variation

Traveling Salesman

Very simple implementation of the traveling salesman problem. Run with --help to see further command line options.

cargo run --release -p tsp

Dependencies

The amount of external dependencies is minimal.

• The crate directly depend in itertools and mosek.

• Some examples depend no rand_distr

• Benchmarking tests require criterion and rand.