Hostname: page-component-89b8bd64d-5bvrz Total loading time: 0 Render date: 2026-05-07T09:47:41.304Z Has data issue: false hasContentIssue false
  • English
  • Français

Apex duality for constrained optimization

Published online by Cambridge University Press:  17 February 2009

R. J. Duffin ,
D. F. Karney  and
E. Z. Prisman
R. J. Duffin
Affiliation:
Department of Mathematics, Carnegie-Mellon University, Pittsburgh, PA 15213
D. F. Karney
Affiliation:
School of Business, Univeristy of Kansas, Lawrence, KS 66045
E. Z. Prisman
Affiliation:
College of Business Administration, Department of Finance, Arizona State University, Tempe, AZ 85287
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In this paper, we develop a duality theory for the Apex dual in the case of primal constraints. As suggested by Duffin in [4], the objective function in this framework is a weighted average of the Legendre-Lagrangian function evaluated at key points. We show that whenever this new dual is feasible there is no duality gap for this dual, and moreover, no duality gap for both the Lagrangian and Wolfe duals too. We conclude with an outline of an algorithm to solve constrained minimization problems in the Apex framework.

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 28 , Issue 1 , July 1986 , pp. 134 - 146
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Duffin, R. J., “Lagrange multiplier method for convex programs”, Proc. Nat. Acad. Sci. USA 72 (1975), 1778–1781.Google Scholar
[2]Duffin, R. J., “Convex programs having some linear constraints”, Proc. Nat. Acad. Sci. USA 74 (1977), 26–28.CrossRefGoogle ScholarPubMed
[3]Duffin, R. J., “Transformations that aid numerical solution of nonlinear programs”, Opsearch 18 (1981), 158–166.Google Scholar
[4]Duffin, R. J., “Numerical estimation of optima by use of dual inequalities”, in Semi-infinite programming and applications (eds. Fiacco, A. V. and Kortanek, K. O.), (Springer-Verlag, Berlin, 1983), 118–127.CrossRefGoogle Scholar
[5]Duffin, R. J., “Duality inequalities as a numerical aid”, Oper. Res. Lett. 3 (1984), 65–68.CrossRefGoogle Scholar
[6]Falk, J. E., “Lagrange multipliers and nonlinear programming”, J. Math. Anal. Appl. 19 (1967). 141–159.CrossRefGoogle Scholar
[7]Karney, D. F., “Asymptotic convex programming”, Georgia Institute of Technology, Management Science Technical Report MS-81–9 (1981).Google Scholar
[8]Luenberger, D. G., Linear and nonlinear programming, 2nd ed., (Addison-Wesley, Reading, MA, 1984).Google Scholar
[9]Mangasarian, O. L., Nonlinear programming (McGraw-Hill, New York, 1969).Google Scholar