Lagrange Multiplier Conditions Characterizing the Optimal Solution Sets of Cone-Constrained Convex Programs

V. Jeyakumar, G. M. Lee and N. Dinh
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V. Jeyakumar: University of New South Wales
G. M. Lee: Pukyong National University
N. Dinh: Pedagogical Institute

Journal of Optimization Theory and Applications, 2004, vol. 123, issue 1, No 4, 83-103

Abstract: Abstract Various characterizations of optimal solution sets of cone-constrained convex optimization problems are given. The results are expressed in terms of subgradients and Lagrange multipliers. We establish first that the Lagrangian function of a convex program is constant on the optimal solution set. This elementary property is then used to derive various simple Lagrange multiplier-based characterizations of the solution set. For a finite-dimensional convex program with inequality constraints, the characterizations illustrate that the active constraints with positive Lagrange multipliers at an optimal solution remain active at all optimal solutions of the program. The results are applied to derive corresponding Lagrange multiplier characterizations of the solution sets of semidefinite programs and fractional programs. Specific examples are given to illustrate the nature of the results.

Keywords: Solution sets; abstract convex programs; semidefinite programs; Lagrange multipliers; fractional programs (search for similar items in EconPapers)
Date: 2004
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Citations: View citations in EconPapers (13)

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DOI: 10.1023/B:JOTA.0000043992.38554.c8

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