Overall Optimization with Optimal Relaxations of the Constraints

Sometimes one or more constraints seriously affect the optimization of an objective function and therefore some relaxation of the constraints is desired if possible. It is assumed that constraints can be relaxed at the cost of introducing some penalty functions into the objective function. In some cases the optimization of the modified objective function (which includes penalty functions) subject to optimally relaxed constraints is preferred. This note deals with the optimal relaxation of the constraints with regard to the linear programming problem which consequently results in overall optimization.