A quasi-Newton algorithm for nonconvex, nonsmooth optimization with global convergence guarantees

Anstreicher, K.M., Lee, J.: A masked spectral bound for maximum-entropy sampling. In: MODA 7—Advances in Model-Oriented Design and Analysis, Berlin, pp. 1–10 (2004)

Ben-Tal, A., Nemirovski, A.: Robust optimization—methodology and applications. Math. Program. Ser. B 92(3), 453–480 (2002)

Bertsekas, D.P.: Convex Optimization Theory. Athena Scientific, Nashua (2009)

Bonnans, J.F., Gilbert, J.C., Lemaréchal, C., Sagastizábal, C.A.: Numerical Optimization: Theoretical and Practical Aspects. Springer, Berlin (2006)

Broyden, C.G.: The convergence of a class of double-rank minimization algorithms. J. Inst. Math. Appl. 6(1), 76–90 (1970)

Burke, J.V., Lewis, A.S., Overton, M.L.: Approximating subdifferentials by random sampling of gradients. Math. Oper. Res. 27(3), 567–584 (2002)

Burke, J.V., Lewis, A.S., Overton, M.L.: A robust gradient sampling algorithm for nonsmooth, nonconvex optimization. SIAM J. Optim. 15(3), 751–779 (2005)

Candès, E.J., Tao, T.: Near optimal signal recovery from random projections: universal encoding strategies. IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006)

Chartrand, R.: Fast algorithms for nonconvex compressive sensing: MRI reconstruction from very few data. In: IEEE International Symposium on Biomedical Imaging (ISBI’09), pp. 262–265 (2009)

Chen, X.: Smoothing methods for nonsmooth, nonconvex optimization. Math. Program. Ser. B 134(1), 71–99 (2012)

Chen, X., Niu, L., Yuan, Y.: Optimality conditions and a smoothing trust region newton method for non-Lipschitz optimization. SIAM J. Optim. 23(3), 1528–1552 (2013)

Clarke, F.H.: Optimization and nonsmooth analysis. In: Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1983)

Curtis, F.E., Overton, M.L.: A sequential quadratic programming algorithm for nonconvex, nonsmooth constrained optimization. SIAM J. Optim. 22(2), 474–500 (2012)

Curtis, F.E., Que, X.: An Adaptive gradient sampling algorithm for nonsmooth optimization. Optim. Methods Softw. (2012). doi: 10.1080/10556788.2012.714781

Dolan, E., Moré, J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)

Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)

Fletcher, R.: A new approach to variable metric algorithms. Comput. J. 13(3), 317–322 (1970)

Goldfarb, D.: A family of variable metric updates derived by variational means. Math. Comput. 24(109), 23–26 (1970)

Goldfarb, D., Iyengar, G.: Robust portfolio selection problems. Math. Oper. Res. 28(1), 1–38 (2003)

Goldstein, A.A.: Optimization of Lipschitz continuous functions. Math. Program. 13(1), 14–22 (1977)

Greif, C., Varah, J.: Minimizing the condition number for small rank modifications. SIAM J. Matrix Anal. Appl. 29(1), 82–97 (2006)

Guenter, S., Sempf, M., Merkel, P., Strumberger, E., Tichmann, C.: Robust control of resistive wall modes using pseudospectra. New J. Phys. 11(053015), 1–40 (2009)

Gumussoy, S., Henrion, D., Millstone, M., Overton, ML.: Multiobjective robust control with HIFOO 2.0. In: Proceedings of the IFAC Symposium on Robust Control Design, Haifa (2009)

Haarala, M.: Large-scale nonsmooth optimization: variable metric bundle method with limited memory. PhD thesis, University of Jyväskylä (2004)

Haarala, M., Miettinen, K., Mäkelä, M.M.: New limited memory bundle method for large-scale nonsmooth optimization. Optim. Methods Softw. 19(6), 673–692 (2004)

Hare, W., Macklem, M.: Derivative-free optimization methods for finite minimax problems. Optim. Methods Softw. 28(2), 300–312 (2013)

Hare, W., Nutini, J.: A derivative-free approximate gradient sampling algorithm for finite minimax problems. Comput. Optim. Appl. 56(1), 1–38 (2013)

Hiriart-Urruty, J.B., Lemaréchal, C.: Convex analysis and minimization algorithms II. In: A Series of Comprehensive Studies in Mathematics. Springer, New York (1993)

Karmitsa, N.: Limited memory bundle method (2014). http://napsu.karmitsa.fi/lmbm/ . Accessed 12 May 2014

Kiwiel, K.C.: Methods of descent for nondifferentiable optimization. In: Lecture Notes in Mathematics. Springer, New York (1985)

Kiwiel, K.C.: A method for solving certain quadratic programming problems arising in nonsmooth optimization. IMA J. Numer. Anal. 6(2), 137–152 (1986)

Kiwiel, K.C.: Convergence of the gradient sampling algorithm for nonsmooth nonconvex optimization. SIAM J. Optim. 18(2), 379–388 (2007)

Kiwiel, K.C.: A nonderivative version of the gradient sampling algorithm for nonsmooth nonconvex optimization. SIAM J. Optim. 20(4), 1983–1994 (2010)

Lemaréchal, C.: A view of line-searches. In: Optimization and Optimal Control: Proceedings of a Conference Held at Oberwolfach, March 16–22, 1980, Berlin, pp. 59–78 (1981)

Lewis, A.S., Overton, M.L.: Nonsmooth optimization via quasi-Newton methods. Math. Program. 141(1–2), 135–163 (2013)

Lukšan, L., Tůma, M., Šiška, M., Vlček, J., Ramešová, N.: UFO 2002: Interactive System for Universal Functional Optimization. Tech. Rep. 883, Institute of Computer Science, Academy of Sciences of the Czech Republic (2002)

Mifflin, R.: An algorithm for constrained optimization with semismooth functions. Math. Oper. Res. 2(2), 191–207 (1977)

Nocedal, J.: Updating quasi-Newton matrices with limited storage. Math. Comput. 35(151), 773–782 (1980)

Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, New York (2006)

Overton, M.L.: HANSO: hybrid algorithm for non-smooth optimization (2014). http://www.cs.nyu.edu/faculty/overton/software/hanso/ . Accessed 12 May 2014

Rockafellar, R.T.: Nonsmooth optimization. In: Birge, J.R., Murty, K.G. (eds.) Mathematical Programming: The State of the Art 1994, pp. 248–258. University of Michigan Press, Ann Arbor (1994)

Sanderson, C., Curtin, R.: Armadillo C $$++$$ + + linear algebra library (2014). http://arma.sourceforge.net/ . Accessed 12 May 2014

Shanno, D.F.: Conditioning of quasi-Newton methods for function minimization. Math. Comput. 24(111), 647–656 (1970)

Skajaa, A.: Limited memory BFGS for nonsmooth optimization. Master’s thesis, New York University, New York (2010)

Vanbiervliet, J., Verheyden, K., Michiels, W., Vandewalle, S.: A nonsmooth optimisation approach for the stabilisation of time-delay systems. ESAIM Control Optim. Calc. Var. 14(3), 478–493 (2008)

Vanbiervliet, J., Vandereycken, B., Michiels, W., Vandewalle, S., Diehl, M.: The smoothed spectral abscissa for robust stability optimization. SIAM J. Optim. 20(1), 156–171 (2009)

Watson, G.A.: Data fitting problems with bounded uncertainties in the data. SIAM J. Matrix Anal. Appl. 22(4), 1274–1293 (2001)

Wolfe, P.: A method of conjugate subgradients for minimizing nondifferentiable functions. Math. Program. Stud. 3, 145–173 (1975)