Bifurcation analysis for singularities on a tangent space for quadratic penalty-barrier and multipliers methods for solving constrained optimization problems, part I

The structure and persistence of critical point solutions obtained from solving constrained optimization problem by the quadratic penalty, the logarithmic-barrier functions, and the multiplier methods are analyzed. This analysis is mainly concerned with singularities due to the Hessian of the Lagrangian being singular on a tangent space. Formulation of the first order conditions of optimality into an algebraic system of equations is first established. The singularities of this system are then classified and solutions are investigated at singularities of codimensions zero and one in terms of the bifurcation behavior and persistence of the curve of critical points.