Geodesic
A Proximal Extension of the Column Generation Method to Nonconvex Conic Optimization Providing Bounds for the Duality Gap
Journal of convex analysis, Tome 17 (2010) no. 3, pp. 721-736.

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We consider nonconvex conic optimization that covers Standard Nonlinear Programming, Semidefinite Programming, Second Order Cone Programming. To the dual Lagrangian problem, we associate a relaxed primal convex problem, and give bounds for the duality gap. Then we propose a proximal extension of the column generation method of Dantzig-Wolfe algorithm (PECGM) which provides these bounds if we suppose in addition Slater's condition. Finally new applications are given in order to make implementable the step of PECGM for which a nonconvex program is supposed to be solved numerically.
Mots-clés : Standard nonlinear programming, semidefinite programming, second order cone programming, duality gap, generation column algorithm, proximal method 
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     author = {A. Auslender},
     title = {A {Proximal} {Extension} of the {Column} {Generation} {Method} to {Nonconvex} {Conic} {Optimization} {Providing} {Bounds} for the {Duality} {Gap}},
     journal = {Journal of convex analysis},
     pages = {721--736},
     publisher = {mathdoc},
     volume = {17},
     number = {3},
     year = {2010},
     url = {http://geodesic.mathdoc.fr/item/JCA_2010_17_3_JCA_2010_17_3_a2/}
}
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A. Auslender. A Proximal Extension of the Column Generation Method to Nonconvex Conic Optimization Providing Bounds for the Duality Gap. Journal of convex analysis, Tome 17 (2010) no. 3, pp. 721-736. http://geodesic.mathdoc.fr/item/JCA_2010_17_3_JCA_2010_17_3_a2/