Optimization with Least Constraint Violation

A study about theory and algorithms for nonlinear programming usually assumes the feasibility of the problem. However, there are many important practical nonlinear programming problems whose feasible regions are not known to be nonempty or not. This leads to a class of problems called optimization with least constraint violation.
Firstly, the optimization problem with least constraint violation is proved to be a Lipschitz equality constrained optimization problem and an elegant necessary optimality condition, named as L-stationary condition, is established. Properties of the classical penalty method for this Lipschitz minimization problem are developed and the proximal gradient method for the penalized problem is studied.
Secondly, the optimization problem with least constraint violation is reformulated as an MPCC problem and a local minimizer of the MPCC problem is proved to an M-stationary point. The smoothing Fischer-Burmeister function method is constructed and analyzed for solving the related MPCC problem.
Thirdly, the solvability of the dual of the optimization problem with least constraint violation is investigated. The optimality conditions for the problem with least constraint violation are established in terms of the augmented Lagrangian. Moreover, it is proved that the augmented Lagrangian method can find an approximate solution to the optimization problem with least constraint violation and has a linear rate of convergence under an error-bound condition.
Finally, the constrained convex optimization problem with the least constraint violation is considered and analyzed under a general measure function. Several other related works on the optimization problem with least constraint violation will also be mentioned.