The Augmented Lagrangian Method Can Approximately Solve Convex Optimization with Least Constraint Violation

TL;DR

This paper investigates the augmented Lagrangian method's ability to approximately solve convex optimization problems where the feasible region may be empty, focusing on least constraint violation solutions and their properties.

Contribution

It introduces a framework for solving convex problems with potentially empty feasible sets using the augmented Lagrangian method, including convergence and optimality conditions.

Findings

The augmented Lagrangian sequence converges to the least violated shift.

The method can find approximate solutions even when the feasible region is empty.

Dual problem solutions can be unbounded under certain conditions.

Abstract

There are many important practical optimization problems whose feasible regions are not known to be nonempty or not, and optimizers of the objective function with the least constraint violation prefer to be found. A natural way for dealing with these problems is to extend the nonlinear optimization problem as the one optimizing the objective function over the set of points with the least constraint violation. This leads to the study of the shifted problem. This paper focuses on the constrained convex optimization problem. The sufficient condition for the closedness of the set of feasible shifts is presented and the continuity properties of the optimal value function and the solution mapping for the shifted problem are studied. Properties of the conjugate dual of the shifted problem are discussed through the relations between the dual function and the optimal value function. The solvability of the dual of the optimization problem with the least constraint violation is investigated. It is shown that, if the least violated shift is in the domain of the subdifferential of the optimal value function, then this dual problem has an unbounded solution set. Under this condition, 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 shifts converges to the least violated shift and the sequence of multipliers is unbounded. Moreover, it is proved that the augmented Lagrangian method is able to find an approximate solution to the problem with the least constraint violation.