# Pajarito.jl

Pajarito is a **mixed-integer convex programming** (MICP) solver package written in Julia.

MICP problems are convex except for restrictions that some variables take binary or integer values.

Pajarito solves MICP problems in conic form, by constructing sequential polyhedral outer approximations of the conic feasible set.

The underlying algorithm has theoretical finite-time convergence under reasonable assumptions.

Pajarito accesses state-of-the-art MILP solvers and continuous conic solvers through MathOptInterface.

## Pavito

For algorithms that use a derivative-based nonlinear programming (NLP) solver (for example, Ipopt) instead of a conic solver, use Pavito.

Pavito is a convex mixed-integer nonlinear programming (convex MINLP) solver. Because Pavito relies on gradient cuts, it can fail near points of non-differentiability. Pajarito may be more robust than Pavito on non-smooth problems.

## License

`Pajarito.jl`

is licensed under the MPL 2.0 license.

## Installation

Install Pajarito using `Pkg.add`

:

```
import Pkg
Pkg.add("Pajarito")
```

## MIP and continuous solvers

The algorithm implemented by Pajarito itself is relatively simple, and most of the hard work is performed by the MIP outer approximation (OA) solver and the continuous conic solver.

Therefore, in addition to installing Pajarito, you must also install a mixed-integer linear programming solver and a continuous conic solver.

**The performance of Pajarito depends on these two types of solvers.**

The OA solver (typically a mixed-integer linear solver) is specified by the `oa_solver`

option. You must first load the Julia package that provides this solver, for example, `using Gurobi`

. The continuous conic solver is specified by the `conic_solver`

option.

See JuMP's list of supported solvers.

## Use with JuMP

To use Pajarito with JuMP, use:

```
using JuMP, Pajarito, HiGHS, Hypatia
model = Model(
optimizer_with_attributes(
Pajarito.Optimizer,
"oa_solver" => optimizer_with_attributes(
HiGHS.Optimizer,
MOI.Silent() => true,
"mip_feasibility_tolerance" => 1e-8,
"mip_rel_gap" => 1e-6,
),
"conic_solver" =>
optimizer_with_attributes(Hypatia.Optimizer, MOI.Silent() => true),
)
)
set_attribute(model, "time_limit", 60)
```

## Options

We list Pajarito's options below.

`verbose::Bool`

toggles printing`tol_feas::Float64`

is the feasibility tolerance for conic constraints`tol_rel_gap::Float64`

is the relative optimality gap tolerance`tol_abs_gap::Float64`

is the absolute optimality gap tolerance`time_limit::Float64`

sets the time limit (in seconds)`iteration_limit::Int`

sets the iteration limit (for the iterative method)`use_iterative_method::Union{Nothing,Bool}`

toggles the iterative algorithm; if`nothing`

is specified, Pajarito defaults to the OA-solver-driven (single tree) algorithm if lazy callbacks are supported by the OA solver`use_extended_form::Bool`

toggles the use of extended formulations for the second-order cone`solve_relaxation::Bool`

toggles solution of the continuous conic relaxation`solve_subproblems::Bool`

toggles solution of the continuous conic subproblems`use_init_fixed_oa::Bool`

toggles initial fixed OA cuts`oa_solver::Union{Nothing,MOI.OptimizerWithAttributes}`

is the OA solver`conic_solver::Union{Nothing,MOI.OptimizerWithAttributes}`

is the conic solver

**Pajarito may require tuning of parameters to improve convergence.**

For example, it often helps to tighten the OA solver's integrality tolerance. OA solver and conic solver options must be specified directly to those solvers.

Note: if `solve_subproblems`

is true, Pajarito usually returns a solution constructed from one of the conic solver's feasible solutions; since the conic solver is not subject to the same feasibility tolerances as the OA solver, Pajarito's solution will not necessarily satisfy `tol_feas`

.

## Cone interface

Pajarito has a generic cone interface (see the cones folder that allows the user to add support for new convex cones.

To illustrate, in the experimental package PajaritoExtras we have extended Pajarito by adding support for several cones recognized by Hypatia.jl (a continuous conic solver with its own generic cone interface).

The examples folder of PajaritoExtras also contains many applied mixed-integer convex problems that are solved using Pajarito.

## Bug reports and support

Please report any issues via the GitHub issue tracker.

All types of issues are welcome and encouraged; this includes bug reports, documentation typos, and feature requests. The Optimization (Mathematical) category on Discourse is appropriate for general discussion.

## References

If you find Pajarito useful in your work, we kindly request that you cite the following paper (arXiv preprint), which is recommended reading for advanced users:

```
@article{CoeyLubinVielma2020,
title={Outer approximation with conic certificates for mixed-integer convex problems},
author={Coey, Chris and Lubin, Miles and Vielma, Juan Pablo},
journal={Mathematical Programming Computation},
volume={12},
number={2},
pages={249--293},
year={2020},
publisher={Springer}
}
```

Note this paper describes a legacy MathProgBase version of Pajarito, which is available on the `mathprogbase`

branch of this repository. Starting with version v0.8.0, Pajarito supports MathOptInterface instead of MathProgBase.