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Revisiting augmented Lagrangian duals

Abstract

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 unit-commitment, that arise in energy optimization. These problems are solved to optimality by solving separable Lagrangian subproblems.

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Correspondence to C. Sagastizábal.

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Appendix A: Tables

Appendix A: Tables

For comparison, the last column in Table 7 reports the solving time when tackling the UC instances directly with Gurobi (a direct solver should be faster in this case, because the considered instances are of small size).

Fig. 1
figure1

Optimality gap (left) and profile of CPU time (right) for unit-commitment problems

Table 4 Case 1—enumerative strategy versus solvers PDBM and MSM equipped with proximal Lagrangian             (\(a \,E-b\) stands for \(a\,10^{-b}\) and (\(500^*\)) that the maximum number of iterations was reached)
Table 5 Case 2—enumerative strategy versus solver PDBM and two starting points
Table 6 Comparison of 5 solvers to compute a solution of (20)
Table 7 Relative optimality gap and feasibility for unit-commitment problems (26)

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Cordova, M., Oliveira, W.d. & Sagastizábal, C. Revisiting augmented Lagrangian duals. Math. Program. (2021). https://doi.org/10.1007/s10107-021-01703-5

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Keywords

  • Bundle methods
  • Augmented Lagrangian duals
  • Unit commitment

Mathematics Subject Classification

  • 65K10
  • 90C06
  • 49J52