Author joaquimg
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## Introduction

Non-convex quadratic programs are extremely hard to solve. This problem class can be solved by Global Solvers such as Couenne. Another possibility is to rely on binary expasion of products terms that appear in the problem, in this case the problem is approximated and can be solved by off-the-shelf MIP solvers such as Cbc, CPLEX, GLPK, Gurobi, HiGHS, Xpress.

## Example

If one wants to solve the optimization problem with this package:

```# Max 2x + y
# s.t. x * y <= 4 (c)
#      x, y >= 1```

One should model as a quadratic program and simply wrap the solver with a `QuadraticToBinary.Optimizer`, with one extra requirement: all variables appearing in quadratic terms must be bounded above and below.

Therefore, the new model can be:

```# Max 2x + y
# s.t. x * y <= 4 (c)
#      x, y >= 1
#      x, y <= 10```

### JuMP with HiGHS solver

```using JuMP
using MathOptInterface
using HiGHS

model = Model(
MOI.instantiate(HiGHS.Optimizer, with_bridge_type = Float64)))

@variable(model, 1 <= x <= 10)
@variable(model, 1 <= y <= 10)

@constraint(model, c, x * y <= 4)

@objective(model, Max, 2x + y)

optimize!(model)

termination_status(model)

primal_status(model)

objective_value(model) # ≈ 9.0

@assert value(x) ≈ 4.0
@assert value(y) ≈ 1.0

@assert value(c) ≈ 4.0```

### MathOptInterface with HiGHS solver

```using MathOptInterface
const MOI = MathOptInterface
using HiGHS

optimizer = MOI.instantiate(HiGHS.Optimizer, with_bridge_type = Float64)

c = MOI.add_constraint(model, 1.0 * x * y, MOI.LessThan(4.0))

MOI.set(model, MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
2.0 * x + y)
MOI.set(model, MOI.ObjectiveSense(), MOI.MAX_SENSE)

MOI.optimize!(model)

@assert MOI.get(model, MOI.TerminationStatus()) == MOI.OPTIMAL

@assert MOI.get(model, MOI.PrimalStatus()) == MOI.FEASIBLE_POINT

@assert MOI.get(model, MOI.ObjectiveValue()) ≈ 9.0

@assert MOI.get(model, MOI.VariablePrimal(), x) ≈ 4.0
@assert MOI.get(model, MOI.VariablePrimal(), y) ≈ 1.0

@assert MOI.get(model, MOI.ConstraintPrimal(), c) ≈ 4.0```

Note: that duals are not available because the problem was approximated as a MIP.

### Precision

It is possible to change the precision of the approximations to the number `val`, for all variables:

`MOI.set(model, QuadraticToBinary.GlobalVariablePrecision(), val)`

Or, for each variable `vi`:

`MOI.set(model, QuadraticToBinary.VariablePrecision(), vi, val)`

The precision for each varible will be `val * (UB - LB)`. Where `UB` and `LB` are, respectively, the upper and lower bound of the variable.

Note: binary expansion problem can be numerically challenging for high precision. You might need to modify solver options accordingly. In the case of HiGHS:

```tol = 1e-9
MOI.set(model, MOI.RawOptimizerAttribute("mip_feasibility_tolerance"), tol)
MOI.set(model, MOI.RawOptimizerAttribute("primal_feasibility_tolerance"), tol)```

### Bounds

For the sake of simplicity, the following two attributes are made available: `QuadraticToBinary.FallbackUpperBound` and `QuadraticToBinary.FallbackLowerBound`. As usual, these can be get and set with the `MOI.get` and `MOI.set` methods. These allow setting bounds used in variables that have no explicit upper bounds and need to be expanded.

## Reference

For more details on the formulation applied here see this paper:

``````@article{andrade2019enhancing,
title={Enhancing the normalized multiparametric disaggregation technique for mixed-integer quadratic programming},
author={Andrade, Tiago and Oliveira, Fabricio and Hamacher, Silvio and Eberhard, Andrew},
journal={Journal of Global Optimization},
volume={73},
number={4},
pages={701--722},
year={2019},
publisher={Springer}
}
``````

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