Revisiting Augmented Lagrangian Duals

For nonconvex optimization problems, possibly having mixed-integer variables, a convergent primal-dual solution algorithm is proposed. The approach applies a proximal bundle method to certain augmented Lagrangian dual that arises in the context of the so-called generalized augmented Lagrangians. We recast these Lagrangians into the framework of a classical Lagrangian, by means of a special reformulation of the original problem. Thanks to this insight, the methodology yields zero duality gap. Lagrangian subproblems can be solved inexactly without hindering the primal-dual convergence properties of the algorithm. Primal convergence is ensured even when the dual solution set is empty. The interest of the new method is assessed on several problems, including of unit-commitment, that arise in energy optimization. These problems are solved to optimality by solving separable Lagrangian subproblems.


Submitted to journal "Mathematical Programming, Series B", issue "Special Issue on Global Solution of Integer, Stochastic and Nonconvex Optimization Problems"



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