Skip to main content
×
×
Home

Facial reduction for a cone-convex programming problem

  • Jon M. Borwein (a1) and Henry Wolkowicz (a2)
Abstract

In this paper we study the abstract convex program

where S is an arbitrary convex cone in a finite dimensional space, Ω is a convex set and p and g are respectively convex and S (on Ω). We use the concept of a minimal cone for (P) to correct and strengthen a previous characterization of optimality for (P), see Theorem 3.2. The results presented here are used in a sequel to provide a Lagrange multiplier theorem for (P) which holds without any constraint qualification.

    • 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. 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.

      Facial reduction for a cone-convex programming problem
      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 <service> account. Find out more about sending content to Dropbox.

      Facial reduction for a cone-convex programming problem
      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 <service> account. Find out more about sending content to Google Drive.

      Facial reduction for a cone-convex programming problem
      Available formats
      ×
Copyright
References
Hide All
Abrams, R. A., and Kerzner, L. (1978), ‘A simplified test for optimality’, J. Optimization Theory Appl. 25, 161170.
Barker, G. P. (1973), ‘The lattice of faces of a finite dimensional cone’, Linear Algebra and Appl. 7, 7182.
Barker, G. and Carlson, D. (1975), ‘Cones of diagonally dominant matrices’, Pacific J. Math. 57, 1532.
Ben-Israel, A., Ben-Tal, A. and Zlobec, S. (1976), ‘Optimality conditions in convex programming’, The IX International Symposium of Mathematical Programming (Budapest, Hungary, 08).
Ben-Tal, A. and Ben-Israel, A. (1979), ‘Characterization of optimality in convex programming: the nondifferentiable case’, Applicable Anal. 9, 137156.
Berman, A. and Ben-Israel, A. (1969), ‘Linear equations over cones with interior: a solvability theorem with applications to matrix theory’, (Report No. 69–1, Series in Applied Math., Northwestern University).
Borwein, J. (1978), ‘Weak tangent cones and optimization in a Banach space’, SIAM J. Control Optimization 16, 512522.
Borwein, J. (1980), ‘Continuity and differentiability of convex operators’, Proc. London Math. Soc., to appear.
Borwein, J. M. and Wolkowicz, H. (1979a), ‘Regularizing the abstract convex program’, J. Math. Anal. Appl., to appear.
Borwein, J. and Wolkowicz, H. (1979b), ‘Characterizations of optimality without constraint qualification for the abstract convex program’, (Research Report No. 14, Dalhousie University, Canada).
Borwein, J. M. and Wolkowicz, H. (1980), ‘Characterization of optimality for the abstract convex program with finite dimensional range space’, J. Austral. Math. Soc., to appear.
Craven, B. D. and Zlobec, S. (1980), ‘Complete characterization of optimality for convex programming in Banach spaces’, Applicable Anal. 11, 6178.
Gould, F. J. and Tolle, J. W. (1972), ‘Geometry of optimality conditions and constraint qualifications’, Math. Programming 2, 118.
Holmes, R. B. (1975), Geometric functional analysis and its applications (Springer-Verlag).
Massam, H. (1979), ‘Optimality conditions for a cone-convex programming problem’, J. Austral. Math. Soc. Ser. A. 27, 141162.
Peressini, A. L. (1967), Ordered topological vector spaces (Harper and Row).
Robertson, A. P. and Robertson, W. J. (1964), Topological vector spaces (Cambridge University Press).
Rockafellar, R. T. (1970), Convex analysis (Princeton University Press).
Zowe, J. (1974), ‘Subdifferentiability of convex functions with values in an ordered vector space’, Math Scand. 34, 6983.
Zowe, J. (1975), ‘Linear maps majorized by a sublinear map’, Arch. Math. (Basel), 26, 637645.
Recommend this journal

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

Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 122 *
Loading metrics...

Abstract views

Total abstract views: 223 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 18th August 2018. This data will be updated every 24 hours.