Abstract
We present a general theory of exact penalty functions with vectorial (multidimensional) penalty parameter for optimization problems in infinite dimensional spaces. In comparison with the scalar case, the use of vectorial penalty parameters provides much more flexibility, allows one to adaptively and independently take into account the violation of each constraint during optimization process, and often leads to a better overall performance of an optimization method using an exact penalty function. We obtain sufficient conditions for the local and global exactness of penalty functions with vectorial penalty parameters and study convergence of global exact penalty methods with several different penalty updating strategies. In particular, we present a new algorithmic approach to an analysis of the global exactness of penalty functions, which contains a novel characterisation of the global exactness property in terms of behaviour of sequences generated by certain optimization methods.
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References
Burachik, R.S., Kaya, C.Y., Price, C.J.: A primal-dual penalty method via rounded weighted-\(\ell _1\) Lagrangian duality. Optimization (2021). https://doi.org/10.1080/02331934.2021.1934680
Byrd, R.H., Nocedal, J., Waltz, R.A.: Steering exact penalty methods for nonlinear programming. Optim. Methods Softw. 23, 197–213 (2008)
Cominetti, R.: Metric regularity, tangent sets, and second-order optimality conditions. Appl. Math. Optim. 21, 265–287 (1990)
Conn, A.R., N. I. M. Gould, Toint, P.L.: Trust-Region Methods. SIAM, Philadelphia (2000)
Demyanov, V.F.: Nonsmooth optimization. In: Di Pillo, G., Schoen, F. (eds.) Nonlinear optimization. Lecture notes in mathematics, vol. 1989, pp. 55–163. Springer-Verlag, Berlin (2010)
Dolgopolik, M.V.: A unifying theory of exactness of linear penalty functions. Optim. 65, 1167–1202 (2016)
Dolgopolik, M.V.: A unifying theory of exactness of linear penalty functions II: parametric penalty functions. Optim. 66, 1577–1622 (2017)
Dolgopolik, M.V.: A unified approach to the global exactness of penalty and augmented Lagrangian functions I: parametric exactness. J. Optim. Theory Appl. 176, 728–744 (2018)
Dolgopolik, M.V.: A unified approach to the global exactness of penalty and augmented Lagrangian functions II: extended exactness. J. Optim. Theory Appl. 176, 745–762 (2018)
Dolgopolik, M.V.: Exact penalty functions for optimal control problems II: Exact penalization of terminal and pointwise state constraints. Optim. Control Appl. Methods 41, 898–947 (2020)
Dolgopolik, M.V., Fominyh, A.: Exact penalty functions for optimal control problems I: main theorem and free-endpoint problems. Optim. Control Appl. Meth. 40, 1018–1044 (2019)
Eremin, I.I.: Penalty method in convex programming. Soviet Math. Dokl. 8, 459–462 (1966)
Fernández, A., Naranjo, F.: Strictly positive linear functional and representation of Fréchet lattices with the Lebesgue property. Indagationes Mathematicae 10, 383–391 (1999)
Han, S.P., Mangasarian, O.L.: Exact penalty functions in nonlinear programming. Math. Program. 17, 251–269 (1979)
Le Thi, H.A., Pham Dinh, T., Van Ngai, H.: Exact penalty and error bounds in DC programming. J. Glob. Optim. 52, 509–535 (2012)
Lipp, T., Boyd, S.: Variations and extension of the convex-concave procedure. Optim. Eng. 17, 263–287 (2016)
Lucidi, S., Rinaldi, F.: An exact penalty global optimization approach for mixed-integer programming problems. Optim. Lett. 7, 297–307 (2013)
Moreau, J.J.: Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires. Competes rendus hebdomadaires des séances de l’Académie des sciences 255, 238–240 (1962)
Di Pillo, G., Grippo, L.: Exact penalty functions in constrained optimization. SIAM J. Control Optim. 27, 1333–1360 (1989)
Di Pillo, G., Lucidi, S., Rinaldi, F.: An approach to constrained global optimization based on exact penalty functions. J. Glob. Optim. 54, 251–260 (2012)
van Casteren, J.A.: Strictly positive functionals on vector lattices. Proc. London Math. Soc. 39, 51–72 (1979)
Rubinov, A., Yang, X.: Lagrange-Type Functions in Constrained Non-Convex Optimization. Kluwer Academic Publishers, Boston (2003)
Soltan, V.: Moreau-type characterizations of polar cones. Linear Algebra Appl. 567, 45–62 (2019)
Strekalovsky, A.S.: Global optimality conditions and exact penalization. Optim. Lett. 13, 597–615 (2019)
Zangwill, W.I.: Nonlinear programming via penalty functions. Manag. Sci. 13, 344–358 (1967)
Zaslavski, A.J.: Optimization on Metric and Normed Spaces. Springer, New York (2010)
Zaslavski, A.J.: Exact penalty property in optimization with mixed constraints via variational analysis. SIAM J. Optim. 23, 170–187 (2013)
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Dolgopolik, M.V. Exact penalty functions with multidimensional penalty parameter and adaptive penalty updates. Optim Lett 16, 1281–1300 (2022). https://doi.org/10.1007/s11590-021-01777-2
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DOI: https://doi.org/10.1007/s11590-021-01777-2