Skip to main content
×
Home
    • Aa
    • Aa

Lipschitz modulus in convex semi-infinite optimization via d.c. functions

  • María J. Cánovas (a1), Abderrahim Hantoute (a2), Marco A. López (a3) and Juan Parra (a1)
Abstract

We are concerned with the Lipschitz modulus of the optimal set mapping associated with canonically perturbed convex semi-infinite optimization problems. Specifically, the paper provides a lower and an upper bound for this modulus, both of them given exclusively in terms of the problem's data. Moreover, the upper bound is shown to be the exact modulus when the number of constraints is finite. In the particular case of linear problems the upper bound (or exact modulus) adopts a notably simplified expression. Our approach is based on variational techniques applied to certain difference of convex functions related to the model. Some results of [M.J. Cánovas et al., J. Optim. Theory Appl. (2008) Online First] (which go back to [M.J. Cánovas, J. Global Optim.41 (2008) 1–13] and [Ioffe, Math. Surveys55 (2000) 501–558; Control Cybern.32 (2003) 543–554]) constitute the starting point of the present work.

We are concerned with the Lipschitz modulus of the optimal set mapping associated with canonically perturbed convex semi-infinite optimization problems. Specifically, the paper provides a lower and an upper bound for this modulus, both of them given exclusively in terms of the problem's data. Moreover, the upper bound is shown to be the exact modulus when the number of constraints is finite. In the particular case of linear problems the upper bound (or exact modulus) adopts a notably simplified expression. Our approach is based on variational techniques applied to certain difference of convex functions related to the model. Some results of [M.J. Cánovas et al., J. Optim. Theory Appl. (2008) Online First] (which go back to [M.J. Cánovas, J. Global Optim.41 (2008) 1–13] and [Ioffe, Math. Surveys55 (2000) 501–558; Control Cybern.32 (2003) 543–554]) constitute the starting point of the present work.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Lipschitz modulus in convex semi-infinite optimization via d.c. functions
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      Lipschitz modulus in convex semi-infinite optimization via d.c. functions
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      Lipschitz modulus in convex semi-infinite optimization via d.c. functions
      Available formats
      ×
Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

J.V. Burke and M.C. Ferris , Weak sharp minima in mathematical programming. SIAM J. Contr. Opt. 31 (1993) 13401359.

M.J. Cánovas , D. Klatte , M.A. López and J. Parra , Metric regularity in convex semi-infinite optimization under canonical perturbations. SIAM J. Optim. 18 (2007) 717732.

A.D. Ioffe , Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk 55 (2000) 103162; English translation in Math. Surveys 55 (2000) 501–558.

D. Klatte and B. Kummer , Strong Lipschitz stability of stationary solutions for nonlinear programs and variational inequalities. SIAM J. Optim. 16 (2005) 96119.

D. Klatte and G. Thiere , A note of Lipschitz constants for solutions of linear inequalities and equations. Linear Algebra Appl. 244 (1996) 365374.

S.M. Robinson , Bounds for error in the solution set of a perturbed linear program. Linear Algebra Appl. 6 (1973) 6981.

M. Studniarski and D.E. Ward , Weak sharp minima: Characterizations and sufficient conditions. SIAM J. Contr. Opt. 38 (1999) 219236.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

ESAIM: Control, Optimisation and Calculus of Variations
  • ISSN: 1292-8119
  • EISSN: 1262-3377
  • URL: /core/journals/esaim-control-optimisation-and-calculus-of-variations
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 6 *
Loading metrics...

Abstract views

Total abstract views: 7 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 24th June 2017. This data will be updated every 24 hours.