Optimization with Least Constraint Violation (CSIAM Transactions on Applied Mathematics, 2021)
Study about theory and algorithms for nonlinear programming usually assumes that the feasible region of the problem is nonempty. However, there are manyimportant practical nonlinear programming problems whose feasible regions are notknown to be nonempty or not, and optimizers of the objective function with the leastconstraint violation prefer to be found. A natural way for dealing with these problemsis to extend the nonlinear programming problem as the one optimizing the objectivefunction over the set of points with the least constraint violation. Firstly, the minimization problem with least constraint violation is proved to be a Lipschitz equality constrained optimization problem when the original problem is a convex nonlinear programming problem with possible inconsistent constraints, and it can be reformulatedas an MPCC problem; And the minimization problem with least constraint violation isrelaxed to an MPCC problem when the original problem is a nonlinear programmingproblem with possible inconsistent non-convex constraints. Secondly, for nonlinearprogramming problems with possible inconsistent constraints, it is proved that a localminimizer of the MPCC problem is an M-stationary point and an elegant necessaryoptimality condition, named as L-stationary condition, is established from the classical optimality theory of Lipschitz continuous optimization. Thirdly, properties of thepenalty method for the minimization problem with the least constraint violation aredeveloped and the proximal gradient method for the penalized problem is studied. Finally, the smoothing Fischer-Burmeister function method is constructed for solving theMPCC problem related to minimizing the objective function with the least constraintviolation. It is demonstrated that, when the positive smoothing parameter approachesto zero, any point in the outer limit of the KKT-point mapping is an L-stationary pointof the equivalent MPCC problem.