Publication Cover
Optimization
A Journal of Mathematical Programming and Operations Research
Volume 71, 2022 - Issue 13
201
Views
3
CrossRef citations to date
0
Altmetric
Research Article

A primal–dual penalty method via rounded weighted-ℓ1 Lagrangian duality

ORCID Icon, ORCID Icon & ORCID Icon
Pages 3981-4017 | Received 18 Sep 2020, Accepted 02 May 2021, Published online: 01 Jun 2021

References

  • Costa MFP, Rocha AMAC, Fernandes EMGP. Filter-based DIRECT method for constrained global optimization. J Global Optim. 2018;71:517–536.
  • G. Macêdo MJF, Karas EW, Costa MFP, et al. Filter-based stochastic algorithm for global optimization. J Global Optim. 2020;77:777–805.
  • Price CJ, Reale M, Robertson BL. Stochastic filter methods for generally constrained global optimization. J Glob Optim. 2016;65:441–456.
  • Burachik RS, Iusem AN, Melo JG. The exact penalty map for nonsmooth and nonconvex optimization. Optimization. 2015;64(4):717–738.
  • Burachik RS, Iusem AN, Melo JG. An inexact modified subgradient algorithm for primal–dual problems via augmented Lagrangians. J Optim Theory Appl. 2013;157(1):108–131.
  • Burachik RS, Iusem AN, Melo JG. Duality and exact penalization for general augmented Lagrangians. J Optim Theor Appl. 2010;147(1):125–140.
  • Burachik RS, Rubinov AM. Abstract convexity and augmented Lagrangians. SIAM J Optim. 2007;18(2):413–436.
  • Dolgopolik MV. A unifying theory of exactness of linear penalty functions II: parametric penalty functions. Optimization. 2017;66(10):1577–1622.
  • Dolgopolik MV. A unified approach to the global exactness of penalty and augmented Lagrangian functions I: parametric exactness. J Optim Theor Appl. 2018;176(3):728–744.
  • Dolgopolik MV. A unified approach to the global exactness of penalty and augmented Lagrangian functions II: extended exactness. J Optim Theor Appl. 2018;176(3):745–762.
  • Wang CY, Yang XQ, Yang XM. Nonlinear augmented Lagrangian and duality theory. Math Oper Res. 2013;38(4):740–760.
  • Wang CY, Yang XQ, Yang XM. A unified nonlinear Lagrangian approach to duality and optimal paths. J Optim Theory Appl. 2007;135:85–100.
  • Huang XX, Teo KL, Yang XQ. Approximate augmented Lagrangian functions and nonlinear semidefinite programs. Acta Math Sinica. 2006;22:1283–1296. English Series.
  • Burachik RS, Yang XQ. Asymptotic strong duality. Numer Algebra Control Optim. 2011;1(3):539–548.
  • Burachik RS. On asymptotic Lagrangian duality for nonsmooth optimization. ANZIAM J. 2017;58:C93–C123.
  • Price CJ. Nonsmooth constrained optimization via rounded ℓ1 penalty functions. Optim Methods Softw. 2020; appeared online: Available from: https://doi.org/10.1080/10556788.2020.1746961
  • Lian S, Duan Y. Smoothing of the lower-order exact penalty function for inequality constrained optimization. J Inequal Appl. 2016;2016:344.
  • Lian S, Niu N. Smoothing approximation to the lower order exact penalty function for inequality constrained optimization. J Inequal Appl. 2018;2018:283.
  • Xu X, Dang C, Chan F, et al. On smoothing ℓ1 exact penalty function for constrained optimization problems. Numer Funct Anal Optim. 2019;40:1–18.
  • Wu ZY, Bai FS, Yang XQ, et al. An exact lower order penalty function and its smoothing in nonlinear programming. Optimization. 2004;53:51–68.
  • Rockafellar RT, Wets RJ-B. Variational analysis. Berlin: Springer-Verlag; 1998.
  • Burachik RS, Freire WP, Kaya CY. Interior epigraph directions method for nonsmooth and nonconvex optimization via generalized augmented Lagrangian duality. J Glob Optim. 2014;60(3):501–529.
  • Burachik RS, Gasimov RN, Ismayilova NA, et al. On a modified subgradient algorithm for dual problems via sharp augmented Lagrangian. J Glob Optim. 2006;34(1):55–78.
  • Burachik RS, Kaya CY. An update rule and a convergence result for a penalty function method. J Indust Man Optim. 2007;3(2):381–398.
  • Burachik RS, Kaya CY. An augmented penalty function method with penalty parameter updates for nonconvex optimization. Nonlin Anal Theor Methods Appl. 2012;75(3):1158–1167.
  • Burachik RS, Kaya CY, Mammadov M. An inexact modified subgradient algorithm for nonconvex optimization. Comput Optim Appl. 2010;45(1):1–24.
  • Gasimov RN. Augmented Lagrangian duality and nondifferentiable optimization methods in nonconvex programming. J Glob Optim. 2002;24(2):187–203.
  • Gasimov RN, Rubinov AM. On augmented Lagrangians for optimization problems with a single constraint. J Glob Optim. 2004;28(2):153–173.
  • Andreani R, Birgin EG, Martínez JM, et al. On augmented Lagrangian methods with general lower-level constraints. SIAM J Optim. 2007;18(4):1286–1309.
  • Artelys Knitro. Nonlinear optimization solver. Available from: https://www.artelys.com/knitro
  • Birgin EG, Martínez JM. Practical augmented Lagrangian methods for constrained optimization. Philadelphia: SIAM Publications; 2014.
  • Gill PE, Murray W, Saunders MA. SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev. 2005;47:99–131.
  • Shanno DF, Vanderbei RJ. Interior-point methods for nonconvex nonlinear programming: orderings and higher-order methods. Math Prog. 2000;87:303–316.
  • Vanderbei RJ, Shanno DF. An interior-point algorithm for nonconvex nonlinear programming. Comput Optim Appl. 1999;13:231–252.
  • Wächter A, Biegler LT. On the implementation of a primal–dual interior point filter line search algorithm for large-scale nonlinear programming. Math Program. 2006;106:25–57.
  • Hock W, Schittkowski K. Test examples for nonlinear programming codes. Springer-Verlag: Berlin Heidelberg, Germany; 1981. (Lecture Notes in Economics and Mathematical Systems; 187).
  • Helou ES, Santos SA, Simoes LE. A new sequential optimality condition for constrained nonsmooth optimization. SIAM J Optim. 2020;30(2):1610–1637.
  • Krejić N, Martínez JM, Mello M, et al. Validation of an augmented Lagrangian algorithm with a Gauss-Newton Hessian approximation using a set of hard-spheres problems. Comput Optim Appl. 2000;16:247–263.
  • Kaya CY. Markov–Dubins path via optimal control theory. Comput Optim Appl. 2017;68(3):719–747.
  • Fourer R, Gay DM, Kernighan BW. AMPL: a modeling language for mathematical programming. 2nd ed. California: Brooks/Cole Publishing Company/Cengage Learning; 2003.
  • Huber PJ. Robust estimation of a location parameter. Ann Math Stat. 1964;35(1):73–101.
  • Burachik RS, Iusem AN, Melo JG. A primal dual modified subgradient algorithm with sharp Lagrangian. J Glob Optim. 2010;46(3):347–361.
  • Kaya CY. Markov–Dubins interpolating curves. Comput Optim Appl. 2019;73(2):647–677.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.