# LinearInequalityToPenalty¶

class LinearInequalityToPenalty(penalty=None)[source]

Bases: qiskit_optimization.converters.quadratic_program_converter.QuadraticProgramConverter

Convert linear inequality constraints to penalty terms of the objective function.

There are some linear constraints which do not require slack variables to construct penalty terms [1]. This class supports the following inequality constraints.

$\begin{split}\begin{array}{} \text { Inequality constraint } & & \text { Penalty term } \\ x \leq y & \rightarrow & P(x-x y) \\ x \geq y & \rightarrow & P(y-x y) \\ \sum_{i=1}^n x_i \leq 1, n \geq 2 & \rightarrow & P \sum_{i, j : i < j} x_i x_j\\ \sum_{i=1}^n x_i \geq n-1, n \geq 2 & \rightarrow & P \sum_{i, j : i < j} (1 - x_i) (1 - x_j) \end{array}\end{split}$

Note that x, y, z and $$x_i$$ are binary variables, and P is a penalty factor, where the value of P is automatically determined or supplied by users.

If constraints match with any of the patterns, they are converted into penalty terms and added to the objective function. Otherwise, constraints are kept as is.

References

[1]: Fred Glover, et al. (2019),

A Tutorial on Formulating and Using QUBO Models, arXiv:1811.11538.

Parameters

penalty (Optional[float]) – Penalty factor to scale equality constraints that are added to objective. If None is passed, a penalty factor will be automatically calculated on every conversion.

Attributes

 penalty Returns the penalty factor used in conversion.

Methods

 convert(problem) Convert inequality constraints into penalty terms of the objective function. Convert the result of the converted problem back to that of the original problem