Steering exact penalty DCA for nonsmooth DC optimisation problems with equality and inequality constraints

References

• A. Bagirov, N. Karmitsa, and M.M. Mäkelä, Introduction to Nonsmooth Optimization. Theory, Practice and Software, Springer, Cahm, 2014.
   Google Scholar
• A.M. Bagirov, S. Taheri, K. Joki, N. Karmitsa, and M.M. Mäkelä, Aggregate subgradient method for nonsmooth DC optimization, Optim. Lett. 15 (2021), pp. 83–96.
   Web of Science ®Google Scholar
• A.M. Bagirov and J. Ugon, Codifferential method for minimizing nonsmooth DC functions, J. Glob. Optim. 50 (2011), pp. 3–22.
   Web of Science ®Google Scholar
• S. Burer and D. Vandenbussche, A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations, Math. Program. 113 (2008), pp. 259–282.
   Web of Science ®Google Scholar
• R.H. Byrd, G. Lopez-Calva, and J. Nocedal, A line search exact penalty method using steering rules, Math. Program. 133 (2012), pp. 39–73.
   Web of Science ®Google Scholar
• R.H. Byrd, J. Nocedal, and R.A. Waltz, Steering exact penalty methods for nonlinear programming, Optim. Methods Softw. 23 (2008), pp. 197–213.
   Web of Science ®Google Scholar
• R. Cambini and C. Sodini, Decomposition methods for solving nonconvex quadratic programs via branch and bound, J. Glob. Optim. 33 (2005), pp. 313–336.
   Web of Science ®Google Scholar
• Y. Carmon and J.C. Duchi, First-order methods for nonconvex quadratic minimization, SIAM Rev.62 (2020), pp. 395–436.
   Web of Science ®Google Scholar
• CVX Research, Inc., Matlab software for disciplined convex programming, version 2.2 (2020). Available at http://cvxr.com/cvx.
   Google Scholar
• W. de Oliveira, Proximal bundle methods for nonsmooth DC programming, J. Glob. Optim. 75 (2019), pp. 523–563.
   Web of Science ®Google Scholar
• W. de Oliveira and M.P. Tcheou, An inertial algorithm for DC programming, Set-Valued Var. Anal.27 (2019), pp. 895–919.
   Web of Science ®Google Scholar
• M.V. Dolgopolik, A unifying theory of exactness of linear penalty functions, Optimization 65 (2016), pp. 1167–1202.
   Web of Science ®Google Scholar
• M.V. Dolgopolik, Metric regularity of quasidifferentiable mappings and optimality conditions for nonsmooth mathematical programming problems, Set-Valued Var. Anal. 28 (2020), pp. 427–449.
   Web of Science ®Google Scholar
• M.V. Dolgopolik, A new constraint qualification and sharp optimality conditions for nonsmooth mathematical programming problems in terms of quasidifferentials, SIAM J. Optim. 30 (2020), pp. 2603–2627.
   Web of Science ®Google Scholar
• M.V. Dolgopolik, DC semidefinite programming and cone constrained DC optimization (2021). Available at arXiv, 2102.01481.
   Google Scholar
• R. Enkhbat, B. Barsbold, and M. Kamada, A numerical approach for solving some convex maximization problems, J. Glob. Optim. 35 (2006), pp. 85–101.
   Web of Science ®Google Scholar
• M. Gaudioso, G. Giallombardo, G. Miglionico, and A.M. Bagirov, Minimizing nonsmooth DC functions via successive DC piecewise-affine approximations, J. Glob. Optim. 71 (2018), pp. 37–55.
   Web of Science ®Google Scholar
• M. Grand and S. Boyd, Graph implementations for nonsmooth convex programs, in Recent Advances in Learning and Control, V. Blondel, S. Boyd, and H. Kimura, eds., Lecture Notes in Control and Information Sciences, Springer, London, 2008, pp. 95–110.
   Google Scholar
• R.J. Hillestad and S.E. Jacobsen, Reverse convex programming, Appl. Math. Optim. 6 (1980), pp. 63–78.
   Web of Science ®Google Scholar
• K. Joki and A.M. Bagirov, Bundle methods for nonsmooth DC optimization, in Numerical Nonsmooth Optimization, A. Bagirov, M. Gaudioso, N. Karmitsa, M. Mäkelä, and S. Taheri, eds., Springer, Cham, 2020, pp. 263–296.
   Google Scholar
• K. Joki, A.M. Bagirov, N. Karmitsa, and M.M. Mäkelä, A proximal bundle method for nonsmooth DC optimization utilizing nonconvex cutting planes, J. Glob. Optim. 68 (2017), pp. 501–535.
   Web of Science ®Google Scholar
• K. Joki, A.M. Bagirov, N. Karmitsa, M. Mäkelä, and S. Taheri, Double bundle method for finding Clarke stationary points in nonsmooth DC programming, SIAM J. Optim. 28 (2018), pp. 1892–1919.
   Web of Science ®Google Scholar
• H.A. Le Thi, V.N. Nuynh, and T. Pham Dinh, DC programming and DCA for general DC programs, in Advanced Computational Methods for Knowledge Engineering, T. van Do, H.A.L. Thi, and N.T. Nguyen, eds., Springer, Berlin, Heidelberg, 2014, pp. 15–35.
   Google Scholar
• H.A. Le Thi and T. Pham Dinh, A branch and bound method via DC optimization algorithms and ellipsoidal technique for box constrained nonconvex quadratic problems, J. Glob. Optim. 13 (1998), pp. 171–206.
   Web of Science ®Google Scholar
• H.A. Le Thi and T. Pham Dinh, The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems, Ann. Oper. Res. 133 (2005), pp. 23–46.
   Web of Science ®Google Scholar
• H.A. Le Thi and T. Pham Dinh, DC programming and DCA: Thirty years of developments, Math. Program. 169 (2018), pp. 5–68.
   Web of Science ®Google Scholar
• H.A. Le Thi, T. Pham Dinh, and L.D. Muu, Numerical solution for optimization over the efficient set by DC optimization algorithm, Oper. Res. Lett. 19 (1996), pp. 117–128.
   Web of Science ®Google Scholar
• T. Lipp and S. Boyd, Variations and extension of the convex-concave procedure, Optim. Eng. 17 (2016), pp. 263–287.
   Web of Science ®Google Scholar
• O. Montonen and K. Joki, Bundle-based descent method for nonsmooth multiobjective DC optimization with inequality constraints, J. Glob. Optim. 72 (2018), pp. 403–429.
   Web of Science ®Google Scholar
• J.V. Outrata, On a class of nonsmooth optimal control problems, Appl. Math. Optim. 10 (1983), pp. 287–306.
   Web of Science ®Google Scholar
• J.V. Outrata, On the usage of bundle methods in optimal control of nondifferentiable systems, in Trends in Mathematical Optimization. International Series on Numerical Mathematics, Vol. 84, K.H. Hoffmann, J. Zowe, J.B. Hiriart-Urruty, and C. Lemarechal, eds., Birkhäuser, Basel, 1988, pp. 233–245.
   Google Scholar
• P.M. Pardalos and J.B. Rosen, Methods for global concave minimization: A bibliographic survey, SIAM Rev. 28 (1986), pp. 367–379.
   Web of Science ®Google Scholar
• T. Pham Dinh and H.A. Le Thi, Convex analysis approach to DC programming: Theory, algorithms and applications, Acta Math. Vietnam. 22 (1997), pp. 289–355.
   Google Scholar
• T. Pham Dinh and H.A. Le Thi, DC optimization algorithms for solving the trust region subproblem, SIAM J. Optim. 8 (1998), pp. 476–505.
   Web of Science ®Google Scholar
• T. Pham Dinh and H.A. Le Thi, Recent advances in DC programming and DCA, in Transactions on Computational Intelligence XIII, N.T. Nguyen and H.A.L. Thi, eds., Springer, Berlin, Heidelberg, 2014, pp. 1–37.
   Google Scholar
• T. Pham Dinh and E.B. Souad, Algorithms for solving a class of nonconvex optimization problems. Methods of subgradients, in Fermat Days 85: Mathematics for Optimization. North-Holland Mathematics Studies. Vol. 129, J.B. Hiriart-Urruty, ed., Norht-Holland, Amsterdam, 1986, pp. 249–271.
   Google Scholar
• R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970.
   Google Scholar
• H.E. Romeijn, J. Geunes, and K. Taaffe, On a nonseparable convex maximization problem with continuous knapsack constraints, Oper. Res. Lett. 35 (2007), pp. 172–180.
   Web of Science ®Google Scholar
• A.S. Strekalovsky, On nonconvex optimization problems with DC equality and inequality constraints, IFAC-PapersOnLine 51 (2018), pp. 895–900.
   Google Scholar
• A.S. Strekalovsky, Local search for nonsmooth DC optimization with DC equality and inequality constraints, in Numerical Nonsmooth Optimization. State of the Art Algorithms, A.M. Bagirov, M. Gaudioso, N. Karmitsa, M.M. Mäkelä, and S. Taheri, eds., Springer, Cham, 2020, pp. 229–262.
   Google Scholar
• A.H. Tor, A. Bagirov, and B. Karasözen, Aggregate codifferential method for nonsmooth DC optimization, J. Comput. Appl. Math. 259 (2014), pp. 851–867.
   Web of Science ®Google Scholar
• W. van Ackooij and W. de Oliveira, Non-smooth DC-constrained optimization: constraint qualification and minimizing methodologies, Optim. Methods Softw. 34 (2019), pp. 890–920.
   Web of Science ®Google Scholar
• W. van Ackooij and W. de Oliveira, Nonsmooth and nonconvex optimization via approximate difference-of-convex decompositions, J. Optim. Theory Appl. 182 (2019), pp. 49–80.
   Web of Science ®Google Scholar
• W. van Ackooij, S. Demassey, P. Javal, H. Morais, W. de Oliveira, and B. Swaminathan, A bundle method for nonsmooth DC programming with application to chance-constrained problems, Comput. Optim. Appl. 78 (2021), pp. 451–490.
   Web of Science ®Google Scholar