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Characterization of optimality for the abstract convex program with finite dimensional range

  • Jon M. Borwein (a1) (a2)

This paper presents characterizations of optimality for the abstract convex program

when S is an arbitrary convex cone in a finite dimensional space, Ω is a convex set and p and g are respectively convex and S-convex (on Ω). These characterizations, which include a Lagrange multiplier theorem and do not presume any a priori constraint qualification, subsume those presently in the literature.

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Abrams, R. A. and Kerzner, L. (1978), “A simplified test for optimality”, J. Optimization Theory Appl. 25, 161170.
Barker, G. P. and Carlson, D. (1975), “Cones of diagonally dominant matrices”, Pacific J. Math. 57, 1532.
Ben-Israel, A., Ben-Tal, A. and Zlobec, S. (1976), “Optimality conditions in convex programming”, The IX International Symposium on Mathematical Programming (Budapest, Hungary, August).
Ben-Israel, A. and Greville, T. N. E. (1973), Generalized inverse theory and applications (Wiley-Interscience, New York).
Ben-Tal, A. and Ben-Israel, A. (1979), “Characterizations of optimality in convex programming: the nondifferentiable case”, Applicable Anal. 9, 137156.
Ben-Tal, A., Ben-Israel, A. and Zlobec, S. (1976), “Characterixations of optimality in convex programming without a constraint qualification”, J. Optimization Theory Appl. 20, 417437.
Berman, A. and Ben-Israel, A. (1969), “Linear equations over cones with interior: a solvability theorem with applications to matrix theory”, (Report No. 69–1 Series in Applied Math., Northwestern University).
Borwein, J. (1977a), “Proper efficient points for maximizations with respect to cones”, SIAM J. Control Optimization 15, 5763.
Borwein, J. (1977b), “Multivalued convexity and optimization: a unified approach to inequality and equality constraints”, Math. Programming 13, 183199.
Borwein, J. (1978), “Weak tangent cones and optimization in a Banach space”, SIAM J. Control Optimization 16, 512522.
Borwein, J. (1980a), “The geometry of Pareto efficiency over cones”, Math. Operationsforsch. Statist., 11, 235248.
Borwein, J. (1980b), “A Lagrange multiplier theorem and a sandwich theorerr for convex relations” (Report No. 80-1, Dalhousie University, Canada) Math. Scand., to appear.
Borwein, J. and Wolkowicz, H. (1979), “Characterizations of optimality without constraint qualification for the abstract convex program”, (Research Report No. 14, Dalhousie University, Halifax, N. S., Canada).
Borwein, J. and Wolkowicz, H. (1981), “Facial reduction for a cone-convex programming problem”, J. Austral. Math. Soc. Ser. A. 30, 000–000.
Craven, B. D. and Zlobec, S., (1981), “Complete characterization of optimality for convex programming in Banach spaces”, Applicable Anal. 11, 6178.
Gauvin, J. (1977), “A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming”, Math. Programming 12, 136138.
Gauvin, J. and Tolle, J. (1977), “Differential stability in nonlinear programming, SIAM J. Control Optimization 15, 294311.
Geoffrion, A. M. (1968), “Proper efficiency and the theory of vector maximization”, J. Math. Anal. Appl. 22, 618630.
Holmes, R. B. (1975), Geometric functional analysis and its applications, (Springer-Verlag).
Luenberger, D. G. (1968), “Quasi-convex programming”, SIAM J. Appl. Math. 16, 10901095.
Luenberger, D. G. (1969), Optimization by vector space methods (John Wiley & Sons, Inc.).
Massam, H. (1979), “Optimality conditions for a cone-convex programming problem”, J. Austral. Math. Soc. Ser. A. 27, 141162.
Penot, J. P. (1978), “L'optimisation ά la Pareto: deux ou trois choses que je sais d'elle” Communication au Colloque “Structure Economiques et Econometric”.
Peressini, A. L. (1967), Ordered topological vector spaces (Harper and Row).
Robertson, A. P. & Robertson, J. W. (1964), Topological vector spaces (Cambridge University Press).
Rockafellar, R. T. (1970a), Convex analysis (Princeton University Press).
Rockafellar, R. T. (1970b), ‘Some convex programs whose duals are linearly constrained”, Nonlinear Programming, edited by Rosen, J. B., Mansasarian, O. L. and Ritter, K., pp. 293322 (Academic Press, N. Y.).
Wolkowicz, H. (1978), “Calculating the cone of directions of constancy”, J. Optimization Theory Appl. 25, 451457.
Wolkowicz, H. (1980), “Geometry of optimality conditions and constraint qualifications: the convex case”, Math. Programming 19, 3260.
Zowe, J. (1974), “Subdifferentiability of convex functions with values in an ordered vector space”, Math. Scand. 34, 6983.
Zowe, J. (1975a), “Linear maps majorized by a sublinear map”, Arch. Math. (Basel) 26, 637645.
Zowe, J. (1975b), “A duality theorem for a convex programming problem in order complete vector lattices”, J. Math. Anal. Appl. 50, 273287.
Zowe, J. (1978), “Regularity and stability for the mathematical programming problem in Banach spaces”, Preprint No. 37.
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