Study on convex optimization with least constraint violation under a general measure

Optimization problems with least constraint violation have many important practical backgrounds. The related works in the literature normally focus on the norm square measure function and the classical augmented Lagrangian method is proved to be suitable for finding an approximate solution. This paper considers the constrained convex optimization problem with the least constraint violation under a general measure function. The properties of the conjugate dual associated with the measure function of the shifted problem are discussed through the relations between the dual function and the optimal value function. The differentiability of the dual function associated with the measure function is proved. The properties of augmented Lagrangian method induced by the measure function are characterized in terms of the dual function. The optimality conditions for the problem with the least constraint violation are established in term of the augmented Lagrangian. It is shown that the augmented Lagrangian method has the properties that the sequence of shift measure values is decreasing, the sequence of multipliers is unbounded, and the sequence of shifts converges to the least violated shift under an extra assumption.