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A simple constraint qualification in infinite dimensional programming.(English)Zbl 0597.90056

For the abstract mathematical programming problem of minimizing a functional f subject to equality constraints and a sign-restriction \(x\in S\) (S a positivity cone) a constant qualification is introduced which, in finite dimensions, is equivalent to Slater’s condition. Further, it is shown that together with some closure conditions for linear functionals f duality theorems can be proved. Some applications are discussed including constrained \(L_ 2\)-approximation.
Reviewer: R.Hettich

MSC:

90C05 Linear programming
90C34 Semi-infinite programming

Cite

References:

[1]A. Ben-Israel, ”Linear equations and inequalities on finite dimensional, real or complex, vector spaces: A unified theory”,Journal of Mathematical Analysis and Applications 27 (1969) 367–389. ·Zbl 0174.31502 ·doi:10.1016/0022-247X(69)90054-7
[2]A. Ben-Israel, A. Ben-Tal and S. Zlobec,Optimality in non-linear programming: A feasible directions approach (Wiley, New York, 1981). ·Zbl 0454.90043
[3]J.M. Borwein and H. Wolkowicz, ”Characterizations of optimality for the abstract convex program with finite dimensional range”,Journal of the Australian Mathematical Society 30 (1981) 390–411. ·Zbl 0469.90088 ·doi:10.1017/S1446788700017882
[4]J.M. Borwein and H. Wolkowicz, ”Characterizations of optimality without constraint qualification for the abstract convex program”,Mathematical Programming Study 19 (1982) 77–100. ·Zbl 0495.90085
[5]B.D. Craven and J.J. Koliha, ”Generalizations of Farkas’ theorem”,SIAM Journal on Mathematical Analysis 8 (1977) 938–997. ·Zbl 0408.52006 ·doi:10.1137/0508076
[6]R.B. Holmes,Geometric functional analysis and its applications (Springer-Verlag, Berlin, 1975). ·Zbl 0336.46001
[7]C. Kallina and A.C. Williams, ”Linear programming in reflexive spaces”,SIAM Review 13 (1971) 350–376. ·Zbl 0224.90042 ·doi:10.1137/1013065
[8]C.A. Micchelli, P.W. Smith, J. Swetits, and J.D. Ward, ”ConstrainedL p approximation”,Journal of Constructive Approximation 1 (1985) 93–102. ·Zbl 0582.41002 ·doi:10.1007/BF01890024
[9]H. Wolkowicz, ”Some applications of optimization in matrix theory”,Linear Algebra and Its Applications 40 (1981) 101–118. ·Zbl 0472.90078 ·doi:10.1016/0024-3795(81)90143-9
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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