Article Outline
Keywords
Infinite-Dimensional Optimization
Nonsmooth Optimization
Global Nonlinear Optimization
Nonconvex Optimization
Semidefinite Programming
See also
References
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ben-Israel A (1969) Linear inequalities and inequalities on finite dimensional real or complex vector spaces: A unified theory. J Math Anal Appl 27:367–389
Borwein JM (1983) Adjoint process duality. Math Oper Res 8:403–437
Borwein JM (1983) A note on the Farkas lemma. Utilitas Math 24:235–241
Borwein JM, Wolkowicz H (1982) Characterizations of optimality without constraint qualification for the abstract convex program. Math Program Stud 19:77–100
Craven BD (1978) Mathematical programming and control theory. Chapman and Hall, London
Craven BD, Koliha JJ (1977) Generalizations of Farkas' theorem. SIAM J Math Anal 8:983–997
Farkas J (1901) Theorie der einfachen Ungleichungen. J Reine Angew Math 124:1–27
Glover BM (1982) A generalized Farkas lemma with applications to quasidifferentiable programming. Z Oper Res 26:125–141
Glover BM, Ishizuka Y, Jeyakumar V, Tuan HD (1996) Complete characterization of global optimality for problems involving the pointwise minimum of sublinear functions. SIAM J Optim 6:362–372
Glover BM, Jeyakumar V, Oettli W (1994) Farkas lemma for difference sublinear systems and quasidifferentiable programming. Math Program 63:333–349
Glover BM, Jeyakumar V, Rubinov AM (1999) Dual conditions characterizing optimality for convex multi-objective programs. Math Program 84:201–217
Goberna MA, Lopez MA, Pastor J (1981) Farkas-Minkowski systems in semi-infinite programming. Appl Math Optim 7:295–308
Gwinner J (1987) Corrigendum and addendum to ‘Results of Farkas type’. Numer Funct Anal Optim 10:415–418
Gwinner J (1987) Results of Farkas type. Numer Funct Anal Optim 9:471–520
Ha Ch-W (1979) On systems of convex inequalities. J Math Anal Appl 68:25–34
Ills T, Kassay G (1994) Farkas type theorems for generalized convexities. Pure Math Appl 5:225–239
Jeyakumar V (1987) A general Farkas lemma and characterization of optimality for a nonsmooth program involving convex processes. J Optim Th Appl 55:449–461
Jeyakumar V (1990) Duality and infinite dimensional optimization. Nonlinear Anal Th Methods Appl 15:1111–1122
Jeyakumar V, Glover BM (1993) A new version of Farkas' lemma and global convex maximization. Appl Math Lett 6(5):39–43
Jeyakumar V, Glover BM (1995) Nonlinear extensions of Farkas' lemma with applications to global optimization and least squares. Math Oper Res 20:818–837
Jeyakumar V, Gwinner J (1991) Inequality systems and optimization. J Math Anal Appl 159:51–71
Jeyakumar V, Rubinov AM, Glover BM, Ishizuka Y (1996) Inequality systems and global optimization. J Math Anal Appl 202:900–919
Kuhn HW, Tucker AW Nonlinear programming, Proc. Second Berkeley Symp. Math. Statist. and Probab., Univ. Calif. Press, Berkeley, CA, pp 481–492
Lasserre JB (1997) A Farkas lemma without a standard closure condition. SIAM J Control Optim 35:265–272
Mangasarian OL (1969) Nonlinear programming. McGraw-Hill, New York
Prékopa A (1980) On the development of optimization theory. Amer Math Monthly 87:527–542
Pshenichnyi BN (1971) Necessary conditions for an extremum. M. Dekker, New York
Ramana MV (1977) An exact duality theory for semidefinite programming and its complexity implications. Math Program 77:129–162
Rubinov AM, Glover BM, Jeyakumar V (1995) A general approach to dual characterizations of solvability of inequality systems with applications. J Convex Anal 2(2):309–344
Schirotzek W (1985) On a theorem of Ky Fan and its application to nondifferentiable optimization. Optim 16:353–366
Zalinescu C (1978) A generalization of the Farkas lemma applications to convex programming. J Math Anal Appl 66:651–678
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag
About this entry
Cite this entry
Jeyakumar, V. (2008). Farkas Lemma: Generalizations . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_176
Download citation
DOI: https://doi.org/10.1007/978-0-387-74759-0_176
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-74758-3
Online ISBN: 978-0-387-74759-0
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering