Projections onto the canonical simplex with additional linear inequalities

References

• L. Adam, M. Červinka, and M. Pištěk, Normally admissible stratifications and calculation of normal cones to a finite union of polyhedral sets, Set-Valued Var. Anal. 24(2) (2016), pp. 207–229.
   Web of Science ®Google Scholar
• L. Adam, M. Hintermüller, and T. Surowiec, A semismooth Newton method with analytical path-following for the H1-projection onto the Gibbs simplex, IMA J. Numer. Anal. 39(3) (2018), 1276–1295.
   Web of Science ®Google Scholar
• G. Bayraksan and D.K. Love, Data-driven stochastic programming using phi-divergences. In The Operations Research Revolution, INFORMS, 2015, pp. 1–19.
   Google Scholar
• M. Bendsoe and O. Sigmund, Topology Optimization: Theory, Methods, and Applications, Springer, Berlin, 2013.
   Google Scholar
• M. Blondel, A. Fujino, and N. Ueda, Large-scale Multiclass Support Vector Machine Training via Euclidean Projection onto The Simplex, Proceedings of 22nd International Conference on Pattern Recognition, IEEE, 2014, pp. 1289–1294.
   Google Scholar
• L. Condat, Fast projection onto the simplex and the l1 ball, Math. Program. 158(1–2) (2016), pp. 575–585.
   Web of Science ®Google Scholar
• E. Delage and Y. Ye, Distributionally robust optimization under moment uncertainty with application to data-driven problems, Oper. Res. 58(3) (2010), pp. 595–612.
   Web of Science ®Google Scholar
• I. Dunning, J. Huchette, and M. Lubin, Jump: A modeling language for mathematical optimization, SIAM Rev. 59(2) (2017), pp. 295–320.
   Web of Science ®Google Scholar
• R. Gao and A.J. Kleywegt, Distributionally robust stochastic optimization with wasserstein distance, preprint (2016). Available at arXiv, math. 1604.02199.
   Google Scholar
• M. Held, P. Wolfe, and H.P. Crowder, Validation of subgradient optimization, Math. Program. 6(1) (1974), pp. 62–88.
   Google Scholar
• M.G. Kapteyn, K.E. Willcox, and A. Philpott, A Distributionally Robust Approach to Black-box Optimization, 2018 AIAA Non-Deterministic Approaches Conference, 2018, p. 0666.
   Google Scholar
• H. Kellerer, U. Pferschy, and D. Pisinger, Knapsack Problems, Springer, Berlin, 2013.
   Google Scholar
• K.C. Kiwiel, Breakpoint searching algorithms for the continuous quadratic knapsack problem, Math. Program. 112(2) (2008), pp. 473–491.
   Web of Science ®Google Scholar
• M. Lapin, M. Hein, and B. Schiele, Top-k multiclass SVM. In Advances in Neural Information Processing Systems, 2015, pp. 325–333.
   Google Scholar
• N. Li, R. Jin, and Z.-H. Zhou, Top Rank Optimization in Linear Time, Proceedings of the 27th International Conference on Neural Information Processing Systems Vol. 1, NIPS'14, MIT Press, Cambridge, MA, USA, 2014, pp. 1502–1510.
   Google Scholar
• J. Liu and J. Ye, Efficient Euclidean Projections in Linear Time, Proceedings of the 26th Annual International Conference on Machine Learning, ACM, 2009, pp. 657–664.
   Google Scholar
• N. Maculan and G.G. De Paula Jr, A linear-time median-finding algorithm for projecting a vector on the simplex of Rn, Oper. Res. Lett. 8(4) (1989), pp. 219–222.
   Web of Science ®Google Scholar
• H. Markowitz, Portfolio selection, J. Finance 7(1) (1952), pp. 77–91.
   Web of Science ®Google Scholar
• C. Michelot, A finite algorithm for finding the projection of a point onto the canonical simplex of Rn, J. Optim. Theory. Appl. 50(1) (1986), pp. 195–200.
   Web of Science ®Google Scholar
• J.A. Nelder and R. Mead, A simplex method for function minimization, Comput. J. 7(4) (1965), pp. 308–313.
   Web of Science ®Google Scholar
• A. Philpott, V. de Matos, and L. Kapelevich, Distributionally robust SDDP, Comput. Manag. Sci. 15(3-4) (2018), pp. 431–454.
   Web of Science ®Google Scholar
• H. Rahimian, G. Bayraksan, and T. Homem-de Mello, Identifying effective scenarios in distributionally robust stochastic programs with total variation distance, Math. Program. 173(1–2) (2019), pp. 393–430.
   Web of Science ®Google Scholar
• S. Shalev-Shwartz and Y. Singer, Efficient learning of label ranking by soft projections onto polyhedra, J. Mach. Learn. Res. 7 (2006), pp. 1567–1599.
   Web of Science ®Google Scholar
• E. Van Den Berg and M.P. Friedlander, Probing the Pareto frontier for basis pursuit solutions, SIAM. J. Sci. Comput. 31(2) (2008), pp. 890–912.
   Web of Science ®Google Scholar