  Penalty functions are widely used to enforce constraints in optimization
problems and reinforcement leaning algorithms. Softplus and algebraic penalty
functions are proposed to overcome the sensitivity of the Courant-Beltrami
method to strong objective function gradients. These penalty functions are
shown to converge in one fifth as many iterations on problems with fifty
optimized parameters and be less sensitive to objective function gradient and
scaling than the Courant-Beltrami penalty function. The softplus and algebraic
penalty functions developed in this work allow optimized solutions to be found
in fewer iterations with greater control over constraint accuracy.
