Abstract
We present some Farkas-type results for inequality systems involving finitely many DC functions. To this end we use the so-called Fenchel-Lagrange duality approach applied to an optimization problem with DC objective function and DC inequality constraints. Some recently obtained Farkas-type results are rediscovered as special cases of our main result.
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Boţ R.I., Kassay G. and Wanka G. (2005). Strong duality for generalized convex optimization problems. J. Optimization Theory Appl. 127: 45–70
Boţ R.I. and Wanka G. (2005). Duality for multiobjective optimization problems with convex objective functions and D.C. constraints. J. Math. Anal. Appl. 315: 526–543
Boţ R.I. and Wanka G. (2005). Farkas-type results with conjugate functions. SIAM J. Optimization 15: 540–554
Boţ, R.I., Wanka, G.: Farkas-type results for max-functions and applications. Positivity. 10(4):761–777
Glover B.M., Jeyakumar V. and Oettli W. (1994). Solvability theorems for classes of difference convex functions. Nonlinear Analysis. Theory. Methods Appl. 22: 1191–1200
Hiriart-Urruty J.B. (1991). How to regularize a difference of convex functions. J. Math. Anal. Appl. 162: 196–209
Horst R. and Thoai N.V. (1993). DC programming: overwiew. J. Optimization Theory Appl. 103: 1–43
Jeyakumar V. (2003). Characterizing set containments involving infinite convex constraints and reverse-convex constraints. SIAM J. Optimization 13: 947–959
Jeyakumar V. and Glover B.M. (1996). Characterizing global optimality for DC optimization problems under convex inequality constraints. J. Glob. Optimization 8: 171–187
Jeyakumar V., Rubinov A.M., Glover B.M. and Ishizuka Y. (1996). Inequality systems and global optimization. J. Math. Anal. Appl. 202: 900–919
Lemaire B. (1998). Duality in reverse convex optimization. SIAM J. Optimization 8: 1029–1037
Lemaire B. and Volle M. (1998). A general duality scheme for nonconvex minimization problems with a strict inequality constraint. J. Glob. Optimization 13: 317–327
Martínez-Legaz J.E. and Volle M. (1999). Duality in D. C. programming: The case of several D. C. constraints. J. Math. Anal. Appl. 237: 657–671
Pham D.T. and El Bernoussi S. (1988). Duality in d. c. (difference of convex functions) optimization. Subgradient methods. Int. Ser. Numerical Math. 84: 277–293
Rockafellar R.T. (1970). Convex Analysis. Princeton University Press, Princeton
Rubinov A.M., Glover B.M. and Jeyakumar V. (1995). A general approach to dual characterizations of solvability of inequality systems with applications. J. Convex Anal. 2: 309–344
Tuy H. (1987). Global minimization of a difference of two convex functions. Math. Program. Study 30: 150–182
Tuy H. (1995). D.C. optimization: theory, methods and algorithms. Nonconvex Optimization Appl. 2: 149–216
Volle M. (1988). Concave duality: application to problems dealing with difference of functions. Math. Program. 41: 261–278
Volle M. (2002). Duality principles for optimization problems dealing with the difference of vector-valued convex mappings. J. Optimization Theory Appl. 114: 223–241
Wanka, G., Boţ, R.I.: On the relations between different dual problems in convex mathematical programming. In: Chamoni, P., Leisten, R., Martin, A., Minnermann, J., Stadtler, H. (eds.) Operations Research Proceedings 2001. Springer Verlag, Berlin (2002)
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Boţ, R.I., Hodrea, I.B. & Wanka, G. Some new Farkas-type results for inequality systems with DC functions. J Glob Optim 39, 595–608 (2007). https://doi.org/10.1007/s10898-007-9159-8
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DOI: https://doi.org/10.1007/s10898-007-9159-8