# A dual approach to solving nonlinear programming problems by unconstrained optimization

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DOI: 10.1007/BF01580138

- Cite this article as:
- Rockafellar, R.T. Mathematical Programming (1973) 5: 354. doi:10.1007/BF01580138

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## Abstract

Several recent algorithms for solving nonlinear programming problems with equality constraints have made use of an augmented “penalty” Lagrangian function, where terms involving squares of the constraint functions are added to the ordinary Lagrangian. In this paper, the corresponding penalty Lagrangian for problems with inequality constraints is described, and its relationship with the theory of duality is examined. In the convex case, the modified dual problem consists of maximizing a differentiable concave function (indirectly defined) subject to no constraints at all. It is shown that any maximizing sequence for the dual can be made to yield, in a general way, an asymptotically minimizing sequence for the primal which typically converges at least as rapidly.