Analysis of bifurcation due to loss of linear independence and strict complementarity for penalty methods for solving constrained optimization problems, II

In this article we investigate the structure and persistence of critical point solutions of the nonlinear programming problem obtained from the quadratic penalty function, the logarithmic-barrier function, and the multiplier method. The analysis focuses on singularities arising from the loss of the linear independence constraint qualification and the loss of strict complementarity. The programming problem is first formulated as a system of equations using the Fritz John first order necessary conditions and a nonstandard normalization of the multipliers. The singularities of this system are then classified and solutions are investigated at each type of singularity of codimension zero and one in terms of the bifurcation behavior and persistence of minima of the critical point curves.