Positive and negative penalty parameters in optimisation subjected to continuous constraints

In solving variational, differential and optimisation equations subject to constraints, the penalty method is a well known procedure. Recent publications show that with a combination of positive and negative penalty parameters, constraint violations can be determined and controlled. This relies on the monotonic nature of convergence of the penalised model towards the constrained system, which has been proven and demonstrated for systems with discrete constraints. This paper investigates the use of this approach for systems with continuous constraints. It addresses the questions of whether the number of critical penalty parameters for continuous constraints is finite, and whether the positive and negative penalty method result in monotonic and bounded convergence towards the constrained solution, and compares the use of penalty functions in integral form and discrete forms to represent continuous constraints.