Coincidence of generic relative minimum singularities in problems with explicit and implicit constraints (Q5948397)

The authors consider smooth optimization problems under equality constraints which depend on a finite-dimensional parameter. They study the generic singularities of the optimal value function when \(n\), the dimension of the decision variable, and \(m\), the dimension of the parameter, satisfy \(n\geq m\). In a first section some known lists of generic singularities are reviewed for the case \(n=m\leq 4\), when the parameter enters the optimization problem only as a right-hand side perturbation of the constraints. Then the authors show that under mild assumptions the same singularities occur when either the objective or the constraint function is not parameter-dependent. Still the same singularities occur when the constraint is subject to right-hand side perturbations and the parameter enters additionally in the objective function. For arbitrary perturbations of the objective function and the constraint the list of singularities remains the same if a certain relation between \(m\), \(n\), and the dimension of the image space of the constraint mapping holds. Finally the authors show that for \(m=4\) and starting from \(n=6\) there do not appear any new generic singularities. Important additional references which are not mentioned in this highly interesting paper concern the work on generic singularities for one-parametric smooth optimization problems that started with the articles [\textit{H.Th. Jongen, P. Jonker, F. Twilt}, J. Optimization Theory Appl. 48, 141-161 (1986; Zbl 0556.90086) and Math. Program. 34, 333--353 (1986; Zbl 0599.90114)].