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

Author

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  • V. Jeyakumar

    (University of New South Wales)

  • G. M. Lee

    (Pukyong National University)

  • N. Dinh

    (Pedagogical Institute)

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.

Suggested Citation

  • V. Jeyakumar & G. M. Lee & N. Dinh, 2004. "Lagrange Multiplier Conditions Characterizing the Optimal Solution Sets of Cone-Constrained Convex Programs," Journal of Optimization Theory and Applications, Springer, vol. 123(1), pages 83-103, October.
    • Handle: RePEc:spr:joptap:v:123:y:2004:i:1:d:10.1023_b:jota.0000043992.38554.c8
    DOI: 10.1023/B:JOTA.0000043992.38554.c8
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    References listed on IDEAS

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    1. S. Deng, 1998. "Characterizations of the Nonemptiness and Compactness of Solution Sets in Convex Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 96(1), pages 123-131, January.
    2. A. Fischer & V. Jeyakumar & D. T. Luc, 2001. "Solution Point Characterizations and Convergence Analysis of a Descent Algorithm for Nonsmooth Continuous Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 110(3), pages 493-513, September.
    3. V. Jeyakumar, 1997. "Asymptotic Dual Conditions Characterizing Optimality for Infinite Convex Programs," Journal of Optimization Theory and Applications, Springer, vol. 93(1), pages 153-165, April.
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    1. Jeyakumar, V. & Lee, G.M. & Dinh, N., 2006. "Characterizations of solution sets of convex vector minimization problems," European Journal of Operational Research, Elsevier, vol. 174(3), pages 1380-1395, November.
    2. S. K. Mishra & B. B. Upadhyay & Le Thi Hoai An, 2014. "Lagrange Multiplier Characterizations of Solution Sets of Constrained Nonsmooth Pseudolinear Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 160(3), pages 763-777, March.
    3. H. Luo & X. Huang & J. Peng, 2012. "Generalized weak sharp minima in cone-constrained convex optimization with applications," Computational Optimization and Applications, Springer, vol. 53(3), pages 807-821, December.
    4. Kin Keung Lai & Shashi Kant Mishra & Sanjeev Kumar Singh & Mohd Hassan, 2022. "Stationary Conditions and Characterizations of Solution Sets for Interval-Valued Tightened Nonlinear Problems," Mathematics, MDPI, vol. 10(15), pages 1-16, August.
    5. Vsevolod I. Ivanov, 2019. "Characterizations of Solution Sets of Differentiable Quasiconvex Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 181(1), pages 144-162, April.
    6. N. V. Tuyen & C.-F. Wen & T. Q. Son, 2022. "An approach to characterizing $$\epsilon $$ ϵ -solution sets of convex programs," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(2), pages 249-269, July.
    7. Satoshi Suzuki & Daishi Kuroiwa, 2015. "Characterizations of the solution set for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential," Journal of Global Optimization, Springer, vol. 62(3), pages 431-441, July.
    8. X. M. Yang, 2009. "On Characterizing the Solution Sets of Pseudoinvex Extremum Problems," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 537-542, March.
    9. Joydeep Dutta, 2005. "Generalized derivatives and nonsmooth optimization, a finite dimensional tour," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 13(2), pages 185-279, December.
    10. Vsevolod I. Ivanov, 2013. "Optimality Conditions and Characterizations of the Solution Sets in Generalized Convex Problems and Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 65-84, July.
    11. D. H. Fang & Y. Zhang, 2018. "Extended Farkas’s Lemmas and Strong Dualities for Conic Programming Involving Composite Functions," Journal of Optimization Theory and Applications, Springer, vol. 176(2), pages 351-376, February.
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    13. S. Deng, 2009. "Characterizations of the Nonemptiness and Boundedness of Weakly Efficient Solution Sets of Convex Vector Optimization Problems in Real Reflexive Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 140(1), pages 1-7, January.

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