penalty function

A term added to the objective that charges an increasing cost for breaking a constraint.

For an inequality constraint $g(x)\le0$ , the quadratic penalty function is $P(x;\rho)=f(x)+\rho\,[\max(0,g(x))]^2$ , where $\rho>0$ is the penalty weight. It contributes nothing at a feasible point and grows with the square of the violation at an infeasible one, letting an ordinary unconstrained solver stand in for a constrained one.