Geodesic
A Bundle Interior Proximal Method for Solving Convex Minimization Problems
Journal of convex analysis, Tome 12 (2005) no. 1, pp. 95-111.

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We extend the standard bundle proximal method for finding the minimum of a convex not necessarily differentiable function on the nonnegative orthant. The strategy consists in approximating the objective function by a piecewise linear convex function and using distance-like functions based on second order homogeneous kernels. First we prove the convergence of this new bundle interior proximal method under the same asumptions as for the standard bundle method and then we report some preliminary numerical experiences for a particular distance function. 
@article{JCA_2005_12_1_JCA_2005_12_1_a5,
     author = {N. T. T. Van and V. H. Nguyen and J.-J. Strodiot},
     title = {A {Bundle} {Interior} {Proximal} {Method} for {Solving} {Convex} {Minimization} {Problems}},
     journal = {Journal of convex analysis},
     pages = {95--111},
     publisher = {mathdoc},
     volume = {12},
     number = {1},
     year = {2005},
     url = {http://geodesic.mathdoc.fr/item/JCA_2005_12_1_JCA_2005_12_1_a5/}
}
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N. T. T. Van; V. H. Nguyen; J.-J. Strodiot. A Bundle Interior Proximal Method for Solving Convex Minimization Problems. Journal of convex analysis, Tome 12 (2005) no. 1, pp. 95-111. http://geodesic.mathdoc.fr/item/JCA_2005_12_1_JCA_2005_12_1_a5/