Publication Cover
Optimization
A Journal of Mathematical Programming and Operations Research
Volume 65, 2016 - Issue 6
307
Views
19
CrossRef citations to date
0
Altmetric
Articles

A unifying theory of exactness of linear penalty functions

Pages 1167-1202 | Received 12 May 2015, Accepted 22 Oct 2015, Published online: 22 Dec 2015

References

  • Eremin II. Penalty method in convex programming. Soviet Math. Dokl. 1966;8:459–462.
  • Zangwill WI. Nonlinear programming via penalty functions. Manage. Sci. 1967;13:344–358.
  • Han SP, Mangasarian OL. Exact penalty functions in nonlinear programming. Math. Program. 1979;17:251–269.
  • Ioffe AD. Necessary and sufficient conditions for a local minimum. I: a reduction theorem and first order conditions. SIAM J. Control Optim. 1979;17:245–250.
  • Mangasarian OL. Sufficiency of exact penalty minimization. SIAM J. Control Optim. 1985;23:30–37.
  • Di Pillo G, Grippo L. On the exactness of a class of nondifferentiable penalty functions. J. Optim. Theory Appl. 1988;57:399–410.
  • Di Pillo G, Grippo L. Exact penalty functions in constrained optimization. SIAM J. Control Optim. 1989;27:1333–1360.
  • Burke JV. An exact penalization viewpoint on constrained optimization. SIAM J. Control Optim. 1991;29:968–998.
  • Demyanov VF. Exact penalty function in problems of nonsmooth optimization. Vestn. St. Peterb. Univ. Math. 1994;27:16–22.
  • Di Pillo G, Facchinei F. Exact barrier function methods for Lipschitz programs. Appl. Math. Optim. 1995;32:1–31.
  • Demyanov VF, Di Pillo G, Facchinei F. Exact penalization via Dini and Hadamard conditional derivatives. Optim. Methods Softw. 1998;9:19–36.
  • Polyakova LN. On the method of exact quasidifferentiable penalty functions. Comput. Math. Math. Phys. 2001;41:205–218.
  • 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.
  • Demyanov VF. Nonsmooth optimization. In: Di Pillo G, Schoen F, editors. Nonlinear optimization. Vol. 1989, Lecture notes in mathematics. Berlin: Springer-Verlag; 2010. p. 55–164.
  • Zaslavski AJ. Optimization on metric and normed spaces. New York (NY): Springer Science+Business Media; 2010.
  • Antczak TA. Lower bound for the penalty parameter in the exact minimax penalty function method for solving nondifferentiable extremum problems. J. Optim. Theory Appl. 2013;159:437–453.
  • Rubinov AM, Yang XQ. Lagrange-type functions in constrained non-convex optimization. Dordrecht: Kluwer Academic; 2003.
  • Meng KW, Yang XQ. Optimality conditions via exact penalty functions. SIAM J. Optim. 2010;20:3208–3231.
  • Meng KW, Yang XQ. First- and second-order necessary conditions via exact penalty functions. J. Optim. Theory Appl. 2015;165:720–752.
  • Burke JV. Calmness and exact penalization. SIAM J. Control Optim. 1991;29:493–497.
  • Pang JS. Error bounds in mathematical programming. Math. Program. 1997;79:299–332.
  • Ng KF, Zheng XY. Error bounds for lower semicontinuous functions in normed spaces. SIAM J. Optim. 2001;12:1–17.
  • Wu Z, Ye JJ. On error bounds for lower semicontinuous functions. Math. Program. 2002;92:301–314.
  • Zălinescu CA. Nonlinear extension of Hoffman’s error bounds for linear inequalities. Math. Oper. Res. 2003;28:524–532.
  • Bosch P, Jourani A, Henrion R. Sufficient conditions for error bounds and applications. Appl. Math. Optim. 2004;50:161–181.
  • Penot JP. Error bounds, calmness and their applications in nonsmooth analysis. In: Leizarowitz A, Mordukhovich BS, Shafrir I, Zaslavski AJ, editors. Nonlinear analysis and optimization II: optimization. Vol. 514, Contemporary mathematics. Providence (RI): American Mathematical Society; 2010. p. 225–248.
  • Bednarczuk EM, Kruger AY. Error bounds for vector-valued functions: necessary and sufficient conditions. Nonlinear Anal. 2012;75:1124–1140.
  • Ioffe AD. Metric regularity and subdifferential calculus. Russ. Math. Surv. 2000;55:501–558.
  • Azé DA. Unified theory for metric regularity of multifunctions. J. Convex Anal. 2006;13:225–252.
  • Li G, Mordukhovich BS. Hölder metric subregularity with applications to proximal point method. SIAM J. Optim. 2012;22:1655–1684.
  • Kruger AY. Error bounds and metric subregularity. Optimization. 2015;64:49–79.
  • Dedieu JP. Penalty functions in subanalytic optimization. Optimization. 1992;26:27–32.
  • Luo ZQ, Pang JS, Ralph D, et al. Exact penalization and stationarity conditions of mathematical programs with equilibrium constraints. Math. Program. 1996;75:19–76.
  • Luo ZQ, Pang JS, Ralph D. Mathematical programs with equilibrium constraints. Cambridge: Cambridge University Press; 1996.
  • Lin GH, Fukushima M. Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 2003;118:67–80.
  • Huyer W, Neumaier A. A new exact penalty function. SIAM J. Optim. 2003;13:1141–1158.
  • Wang C, Ma C, Zhou J. A new class of exact penalty functions and penalty algorithms. J. Glob. Optim. 2014;58:51–73.
  • Li B, Yu CJ, Teo KL, et al. An exact penalty function method for continuous inequality constrained optimal control problem. J. Optim. Theory Appl. 2011;151:260–291.
  • Jiang C, Lin Q, Yu C, et al. An exact penalty method for free terminal time optimal control problem with continuous inequality constraints. J. Optim. Theory Appl. 2012;154:30–53.
  • Lin Q, Loxton R, Teo KL, et al. Optimal feedback control for dynamic systems with state constraints: an exact penalty approach. Optim. Lett. 2014;8:1535–1551.
  • Clarke FH. A new approach to Lagrange multipliers. Math. Oper. Res. 1976;1:165–174.
  • Clarke FH. Optimization and nonsmooth analysis. New York (NY): Wiley; 1983.
  • Yang XQ, Ralph D. Characterization for perturbed exact penalty functions. Nonlinear Anal. 2005;62:101–106.
  • Huang XX, Teo KL, Yang XQ. Calmness and exact penalization in vector optimization with cone constraints. Comput. Optim. Appl. 2006;35:47–67.
  • Zhai J, Huang XX. Calmness and exact penalization in vector optimization under nonlinear perturbations. J. Optim. Theory Appl. 2014;162:856–872.
  • Henrion R, Outrata JV. Calmness of constraint systems with applications. Math. Program. 2005;104:437–464.
  • Wen S. Calmness and error bounds for convex constraint systems. SIAM J. Optim. 2006;17:353–371.
  • Penot JP. Calmness and stability properties of marginal and performance functions. Numer. Funct. Anal. Optim. 2004;25:287–308.
  • Uderzo A. Exact penalty functions and calmness for mathematical programming under nonlinear perturbations. Nonlinear Anal. 2010;73:1596–1609.
  • Bai FS, Wu ZY, Zhu DL. Lower order calmness and exact penalty function. Optim. Methods Softw. 2006;21:515–525.
  • Kelley JL. General topology. New York (NY): Springer-Verlag; 1975.
  • Mawhin J, Willem M. Origin and Evolution of the Palais--Smale condition in critical point theory. J. Fixed Point Theory Appl. 2010;7:265–290.
  • Demyanov VF. Conditions for an extremum in metric spaces. J. Glob. Optim. 2000;17:55–63.
  • Huang XX, Yang XQ. A unified augmented Lagrangian approach to duality and exact penalization. Math. Oper. Res. 2003;28:533–552.
  • Ye JJ. The exact penalty principle. Nonlinear Anal. 2012;75:1642–1654.
  • Giannessi F. Semidifferentiable functions and necessary optimality conditions. J. Optim. Theory Appl. 1989;60:191–240.

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.