Abstract
We establish necessary and sufficient conditions for a stable Farkas’ lemma. We then derive necessary and sufficient conditions for a stable duality of a cone-convex optimization problem, where strong duality holds for each linear perturbation of a given convex objective function. As an application, we obtain stable duality results for convex semi-definite programs and convex second-order cone programs.
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Alizadeh F., Goldfarb D. (2003). Second-order cone programming. Math. Program. Ser. B 95: 3–51
Andersen E.D., Roos C., Terlaky T. (2002). Notes on duality in second order and p-order cone optimization. Optimization 51: 627–643
Barbu, V., Precupanu, Th.: Convexity and Optimization in Banach Spaces, D. Reidel Publishing Company, Durdrecht (1986)
Bot R.I., Wanka G. (2005). Farkas-type results with conjugate functions. SIAM J. Optim. 15: 540–554
Bot R.I., Wanka G. (2006). An alternative formulation for a new closed cone constraint qualification. Nonlinear Anal. Theory Methods Appl. 64(6): 1367–1381
Burachik R.S., Jeyakumar V. (2005). Dual condition for the convex subdifferential sum formula with applications. J. Convex Anal. 12: 279–290
Craven B.D. (1995). Control and Optimization. Champman and Hall, London
Dinh N., Jeyakumar V., Lee G.M. (2005). Sequential Lagrangian conditions for convex programs with applications to semidefinite programming. J. Optim. Theory Appl. 125: 85–112
Dinh N., Goberna M.A., López M.A. (2006). From linear to convex systems: Consistency, Farkas’ Lemma and application. J. Convex Anal. 13: 1–21
Glover B.M. (1982). A generalized Farkas lemma with applications to quasidifferentiable programming. Z. Oper. Res. Ser. A-B 26: 125–141
Goberna M.A., López M.A. (1998). Linear Semi-infinite optimization. Wiley, Chichester
Gwinner J., Pomerol J.-C. (1989). On weak* closedness, coerciveness and inf-sup theorems. Arch. Math., 52: 159–167
Gwinner, J., Jeyakumar, V.: Stable minimax on noncompact sets, fixed point theory and applications (Marseille 1989), pp. 215–220 Pitman Res. Notes Math. Ser. 255 Longman Sci. Tech. Harlow (1991)
Gwinner J. (1987). Results of Farkas type. Numer. Funct. Anal. Optim. 9: 471–520
Gwinner J. (1987). Corrigendum and addendum to Results of Farkas type. Numer. Funct. Anal. Optim. 10: 415–418
Jeyakumar, V.: Farkas Lemma: Generalizations, Encyclopedia of Optimization, vol. 2, pp. 87–91. Kluwer, Boston (2001)
Jeyakumar V. (2006). The strong conical hull intersection property for convex programming. Math. Program. Ser. A 106: 81–92
Jeyakumar V., Glover B.M. (1995). Nonlinear extensions of Farkas lemma with applications to global optimization and least squares. Math. Oper. Res. 20(4): 818–837
Jeyakumar V., Lee G.M., Dinh N. (2006). Characterization of solution sets of convex vector minimization problems. Eur. J. Oper. Res. 174: 1380–1395
Jeyakumar V., Rubinov A.M., Glover B.M., Ishizuka Y. (1996). Inequality systems and global optimization. J. Math. Anal. Appl. 202: 900–919
Lobo M.S., Vandenberghe L., Boyd S., Lebret H. (1998). Applications of second-order cone programming. Linear Algebra Appl. 284: 193–228
Precupanu T. (1984). Closedness conditions for the optimality of a family of non-convex optimization problems. Math. Operationsforsch. Statist. Ser. Optim. 15: 339–346
Zalinescu C. (2002). Convex Analysis in General Vector Space. World Scientific Publishing, Singapore
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The authors are grateful to the referees for their valuable suggestions and helpful detailed comments which have contributed to the final preparation of the paper. The first author was supported by the Australian Research Council Linkage Program. The second author was supported by the Basic Research Program of KOSEF (Grant No. R01-2006-000-10211-0).
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Jeyakumar, V., Lee, G.M. Complete characterizations of stable Farkas’ lemma and cone-convex programming duality. Math. Program. 114, 335–347 (2008). https://doi.org/10.1007/s10107-007-0104-x
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DOI: https://doi.org/10.1007/s10107-007-0104-x
Keywords
- Stable Farkas lemma
- Stable duality
- Convex optimization
- Semi-definite programming
- Second-order cone programming