Abstract
A smooth nonconvex optimization problem is considered, where the equality and inequality constraints and the objective function are given by DC functions. First, the original problem is reduced to an unconstrained problem with the help of I. I. Eremin’s exact penalty theory, and the objective function of the penalized problem also turns out to be a DC function. Necessary and sufficient conditions for minimizing sequences of the penalized problem are proved. On this basis, a “theoretical method” for constructing a minimizing sequence in the penalized problem with a fixed penalty parameter is proposed and the convergence of the method is proved. A well-known local search method and its properties are used for developing a new global search scheme based on global optimality conditions with a varying penalty parameter. The sequence constructed using the global search scheme turns out to be minimizing in the “limit” penalized problem, and each of its terms \(z^{k+1}\) turns out to be an approximately critical vector for the local search method and an approximate solution of the current penalized problem \((\mathcal{P}_{k})\triangleq(\mathcal{P}_{\sigma_{k}})\). Finally, under an additional condition of “approximate feasibility,” the constructed sequence turns out to be minimizing for the original problem with DC constraints.
REFERENCES
F. P. Vasil’ev, Optimization Methods (MTsNMO, Moscow, 2011), Vols. 1, 2 [in Russian].
V. V. Vasin and I. I. Eremin, Operators and Iterative Processes of Fejér Type: Theory and Applications (Inst. Komp. Issled., Moscow, 2005; De Gruyter, Berlin, 2009).
V. F. Dem’yanov, Extremum Conditions and Calculus of Variations (Vysshaya Shkola, Moscow, 2005) [in Russian].
Yu. G. Evtushenko, Numerical Optimization Techniques (Nauka, Moscow, 1982; Springer, New York, 1985).
I. I. Eremin, “The penalty method in convex programming,” Soviet Math. Dokl. 8, 459–462 (1967).
I. I. Eremin, “The penalty method in convex programming,” Cybernetics 3 (4), 53–56 (1967). https://doi.org/10.1007/bf01071708
I. I. Eremin and N. N. Astaf’ev, Introduction to the Theory of Linear and Convex Programming (Nauka, Moscow, 1976) [in Russian].
I. I. Eremin and V. D. Mazurov, Nonstationary Processes of Mathematical Programming (Nauka, Moscow, 1979) [in Russian].
I. I. Eremin, Contradictory Models of Optimal Planning (Nauka, Moscow, 1988) [in Russian].
V. G. Zhadan, Optimization Methods, Parts 1, 2 (MFTI, Moscow, 2015) [in Russian].
I. V. Konnov, Nonlinear Optimization and Variational Inequalities (Izd. Kazan. Univ., Kazan, 2013) [in Russian].
A. S. Strekalovsky, Elements of Nonconvex Optimization (Nauka, Novosibirsk, 2003) [in Russian].
A. S. Strekalovsky and A. V. Orlov, Bimatrix Games and Bilinear Programming (Fizmatlit, Moscow, 2007) [in Russian].
A. S. Strekalovsky and A. V. Orlov, Linear and Quadratic Linear Problems of Two-Level Optimization (Izd. SO RAN, Novosibirsk, 2019) [in Russian].
A. S. Strekalovsky, “New global optimality conditions in a problem with DC constraints,” Trudy Inst. Mat. Mekh. UrO RAN 25 (1), 245–261 (2019). https://doi.org/10.21538/0134-4889-2019-25-1-245-261
A. S. Strekalovsky, “Elements of global search in the general DC optimization problem,” Itogi Nauki Tekh. Sovrem. Mat. Prilozh. Temat. Obzory 196, 114–127 (2021). https://doi.org/10.36535/0233-6723-2021-196-114-127
A. G. Sukharev, A. V. Timokhov, and V. V. Fedorov, A Course in Optimization Methods, 2nd ed. (Fizmatlit, Moscow, 2011) [in Russian].
J.-F Bonnans, J. C. Gilbert, C. Lemaréchal, and C. A. Sagastizábal, Numerical Optimization: Theoretical and Practical Aspects (Springer, Berlin, 2006).
J. Burke, “An exact penalization viewpoint of constrained optimization,” SIAM J. Control Optim. 29 (4), 968–998 (1991). https://doi.org/10.1137/0329054
R. Byrd, G. Lopez-Calva, and J. Nocedal, “A line search exact penalty method using steering rules,” Math. Progr. 133 (1–2), 39–73 (2012). https://doi.org/10.1007/s10107-010-0408-0
G. Di Pillo, S. Lucidi, and F. Rinaldi, “An approach to constrained global optimization based on exact penalty functions,” J. Global Optim. 54 (2), 251–260 (2012). https://doi.org/10.1007/s10898-010-9582-0
M. Dur and J. B. Hiriart-Urruty, “Testing copositivity with the help of difference-of-convex optimization,” Math. Progr. 140 (1), 31–43 (2013). https://doi.org/10.1007/s10107-012-0625-9
Frontiers in Global Optimization, Ed. by C. A. Floudas and P. M. Pardalos (Kluwer Acad., Dordrecht, 2004).
A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques (Wiley, New York, 1968).
M. Gaudioso, T. V. Gruzdeva, and A. S. Strekalovsky, “On numerical solving the spherical separability problem,” J. Global Optim. 66 (1), 21–34 (2016). https://doi.org/10.1007/s10898-015-0319-y
J.-B. Hiriart-Urruty, “Generalized differentiability, duality and optimization for problems dealing with difference of convex functions,” in Convexity and Duality in Optimization, ed. by J. Ponstein (Springer, Berlin, 1985), Ser. Lecture Notes in Economics and Mathematical Systems, Vol. 256, pp. 37–69. https://doi.org/10.1007/978-3-642-45610-7_3
J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms (Springer, Berlin, 1993).
R. Horst and H. Tuy, Global Optimization: Deterministic Approaches (Springer, Berlin, 1993).
A. Kruger, “Error bounds and metric subregularity,” Optim. 64 (1), 49–79 (2015). https://doi.org/10.1080/02331934.2014.938074
H. A. Le Thi, T. Pham Dinh, and H. V. Ngai, “Exact penalty and error bounds in DC programming,” J. Global Optim. 52 (3), 509–535 (2012). https://doi.org/10.1007/s10898-011-9765-3
H. A. Le Thi, H. V. Ngai, and T. Pham Dinh, “DC Programming and DCA for general DC programs,” in Advanced Computational Methods for Knowledge Engineering, Ed. by D. Tien, A. L. T. Hoai, and T. N. Ngoc (Springer, Cham, 2014), Ser. Advances in Intelligent Systems and Computing, Vol. 282, pp. 15–35. https://doi.org/10.1007/978-3-319-06569-4_2
W. Mascarenhas, “The BFGS methods with exact line search fails for nonconvex objective functions,” Math. Progr. 99 (1), 49–61 (2004). https://doi.org/10.1007/s10107-003-0421-7
W. Mascarenhas, “On the divergence of line search methods,” Comput. Appl. Math. 26 (1), 129–169 (2007). https://doi.org/10.1590/S0101-82052007000100006
W. Mascarenhas, “Newton’s iterates can converge to non-stationary points,” Math. Progr. 112 (2), 327–334 (2008). https://doi.org/10.1007/s10107-006-0019-y
J. Nocedal and S. J. Wright, Numerical Optimization (Springer, Berlin, 2006).
J. S. Pang, “Three modelling paradigms in mathematical programming,” Math. Progr. 125 (2), 297–323 (2010). https://doi.org/10.1007/s10107-010-0395-1
J. S. Pang, M. Razaviyan, and A. Alvarado, “Computing B-stationary points of nonsmooth DC programs,” Math. Oper. Res. 42 (1), 95–118 (2016). https://doi.org/10.1287/moor.2016.0795
R. T. Rockafellar, Convex Analysis (Princeton Univ., Princeton, 1970).
A. S. Strekalovsky, “On solving optimization problems with hidden nonconvex structures,” in Optimization in Science and Engineering, Ed. by T. M. Rassias, C. A. Floudas, and S. Butenko (Springer, New York, 2014), pp. 465–502. https://doi.org/10.1007/978-1-4939-0808-0_23
A. S. Strekalovsky, “On local search in d.c. optimization problems,” Appl. Math. Comput. 255, 73–83 (2015). https://doi.org/10.1016/j.amc.2014.08.092
A. S. Strekalovsky, “Local search for nonsmooth DC optimization with DC equality and inequality constraints,” in Numerical Nonsmooth Optimization: State of the Art Algorithms, Ed. by A. M. Bagirov, M. Gaudioso, N. Karmitsa, M. M. Makela, and S. Taheri (Springer Internat., New York, 2020), pp. 229–261. https://doi.org/10.1007/978-3-030-34910-3_7
A. S. Strekalovsky and I. M. Minarchenko, “A local search method for optimization problem with d.c. inequality constraints,” Appl. Math. Model. 58, 229–244 (2018). https://doi.org/10.1016/j.apm.2017.07.031
A. S. Strekalovsky, “Global optimality conditions in nonconvex optimization,” J. Optim. Theory Appl. 173 (3), 770–792 (2017). https://doi.org/10.1007/s10957-016-0998-7
A. S. Strekalovsky, “Global optimality conditions and exact penalization,” Optim. Lett. 13 (3), 597–615 (2019). https://doi.org/10.1007/s11590-017-1214-x
A. S. Strekalovsky, “On global optimality conditions for D.C. minimization problems with D.C. constraints,” J. Appl. Numer. Optim. 3 (1), 175–196 (2021). https://doi.org/10.23952/jano.3.2021.1.10
A. S. Strekalovsky and M. V. Yanulevich, “On global search in nonconvex optimal control problems,” J. Global Optim. 65 (1), 119–135 (2016). https://doi.org/10.1007/s10898-015-0321-4
H. Tuy, “D.C. Optimization: Theory, methods and algorithms,” in Handbook of Global Optimization, Ed. by R. Horst and P. M. Pardalos (Kluwer Acad., Dordrecht, 1995), pp. 149–216.
W. Zangwill, “Non-linear programming via penalty functions,” Manag. Sci. 13 (5), 344–358 (1967). https://doi.org/10.1287/mnsc.13.5.344
W. I. Zangwill, Nonlinear Programming: A Unified Approach (Prentice-Hall, Englewood Cliffs, NJ, 1969).
A. J. Zaslavski, “Exact penalty property in optimization with mixed constraints via variational analysis,” SIAM J. Optim. 23 (1), 170–187 (2013). https://doi.org/10.1137/120870840
Funding
The research was funded by the Ministry of Science and Higher Education of the Russian Federation within the project “Theoretical foundations, methods, and high-performance algorithms for continuous and discrete optimization to support interdisciplinary research” (no. of state registration 121041300065-9, project code FWEW-2021-0003).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The author of this work declares that he has no conflicts of interest.
Additional information
Translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN, Vol. 29, No. 3, pp. 185 - 209, 2023 https://doi.org/10.21538/0134-4889-2023-29-3-185-209.
Translated by E. Vasil’eva
Publisher's Note Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations
Rights and permissions
About this article
Cite this article
Strekalovsky, A.S. Minimizing Sequences in a Constrained DC Optimization Problem. Proc. Steklov Inst. Math. 323 (Suppl 1), S255–S278 (2023). https://doi.org/10.1134/S0081543823060214
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543823060214