# zbMATH — the first resource for mathematics

Support separation theorems and their applications to vector surrogate reverse duality. (English) Zbl 0770.49004
Separation theorems play a crucial role in optimization theory. One of the best known separation theorems is the following: Given a nonempty closed convex subset $$A$$ of a Hausdorff locally convex space $$X$$ and a point $$x\in X\backslash A$$, there always exists a nonzero continuous functional $$\varphi\in X^*$$ such that (1) $$\varphi(x)\geq\sup\varphi(A)$$. The author considers the question when the inequality in (1) could be replaced with equality. This problem was first studied by I. Singer. He has proved the existence of such a functional under the assumptions that the space $$X$$ is a normable linear space and $$A$$ is a bounded set.
In this paper a satisfying answer to this question for the general case when $$X$$ is a Hausdorff locally convex space and no constraining assumptions are imposed is given. The obtained results are then used to establish some strong duality principles concerning the surrogate reverse duality in vector optimization.
##### MSC:
 49J27 Existence theories for problems in abstract spaces 49N15 Duality theory (optimization)
##### Keywords:
surrogate reverse duality; vector optimization
Full Text:
##### References:
 [1] R. B: Holmes: Geometrical Functional Analysis and Its Applications. Springer-Verlag, New York - Heidelberg - Berlin 1975. [2] I. Ekeland, R. Temam: Analyse Convexe et Problemes Variationelles. Dunod, Paris 1974. [3] J. Bair: On the convex programming problem in an ordered vector space. Bull. Soc. Royal Science LIEGE 46 (1977), 234 - 240. · Zbl 0378.90079 [4] E. G. Golstein: Duality Theory in Mathematical Programming and Its Applications. Nauka, Moscow 1971. In Russian. [5] V. V. Podinovskij, V.D. Nogin: Pareto optimal solutions in multiobjective problems. Nauka, Moscow 1982. In Russian. · Zbl 0496.90053 [6] G.S. Rubinstein: Duality in mathematical programming and some questions of convex analysis. Uspekhi mat. nauk 25 (1970), 5, 155, 171 - 201. In Russian. [7] M. Vlach: On necessary conditions of optimality in linear spaces. Comment. Math. Univ. Carol. 11 (1970), 3, 501 - 503. · Zbl 0206.12005 · eudml:16381 [8] M. Vlach: A separation theorem for finite families. Comment. Math. Univ. Carol. 12 (1971), 4, 655 - 670. · Zbl 0229.52008 · eudml:16456 [9] M. Vlach: A note on separation by linear mappings. Comment. Math. Univ. Carol. 18 (1977), 1, 167 - 168. · Zbl 0345.52001 · eudml:16814 [10] Tran Quoc Chien: Duality in vector optimization. Part I: Abstract duality scheme. Kybernetika 20 (1984), 4, 304-313. · Zbl 0556.49010 · eudml:27982 [11] Tran Quoc Chien: Duality in vector optimization. Part II: Vector quasiconcave programming. Kybernetika 20 (1984), 5, 386 - 404. · Zbl 0575.49006 · eudml:27641 [12] Tran Quoc Chien: Duality in vector optimization. Part III: Vector partially quasiconcave programming and vector fractional programming. Kybernetika 20 (1984), 6, 458 - 472. · Zbl 0575.49007 · eudml:27759 [13] Tran Quoc Chien: Fenchel-Lagrange duality in vector fractional programming via abstract duality scheme. Kybernetika 23 (1986), 4, 299 - 319. · Zbl 0616.90081 · eudml:27529 [14] Tran Quoc Chien: Parturbation theory of duality in vector optimization via abstract duality scheme. Kybernetika 23 (1987), 1, 67 - 81. · Zbl 0615.49007 · eudml:27925 [15] I. Singer: Optimization by level set methods VI: Generalization of surrogate type reverse convex duality. Optimization 18 (1987), 4, 485 - 499. · Zbl 0638.49006 · doi:10.1080/02331938708843264 [16] I. Singer: Maximization of lower semicontinuous convex functionals on bounded subsets of locally convex spaces I: Hyperplane theorems. Appl. Math. Optim. 5 (1979), 349 - 362. · Zbl 0421.90077 · doi:10.1007/BF01442563 [17] I. Singer: A general theory of surrogate dual and perturbational extended surrogate dual optimization problems. J. Math. Anal. Appl. 104 (1984), 351 - 389. · Zbl 0607.90089 · doi:10.1016/0022-247X(84)90002-7 [18] I. Singer: Surrogate dual problems and surrogate Lagrangians. J. Math. Anal. Appl. 98 (1984), 31 - 71. · Zbl 0584.49006 · doi:10.1016/0022-247X(84)90277-4 [19] J.-E. Martinez Legaz, I. Singer: Surrogate duality for vector optimization. Numer. Funct. Anal. Optim. 9 (1987), 5-6, 544-568. · Zbl 0609.49012 · doi:10.1080/01630568708816247 [20] I. Singer: Minimization of continuous convex functionals on complements of convex subsets of locally convex space. Math. Operationsforsch. Statist. Ser. Optim. 11 (1980), 221 - 234. · Zbl 0449.90100 · doi:10.1080/02331938008842649 [21] I. Singer: Extension with larger norm and separation with double support in normed linear spaces. Bull. Austral. Math. Soc. 21 (1980), 93 - 105. · Zbl 0412.46004 · doi:10.1017/S0004972700011321 [22] I. Singer: Optimization and best approximation. Nonlinear Analysis, Theory and Applications (R. Kluge, Abh. Akad. Wiss. DDR, Berlin 1981, pp. 275 - 285. · Zbl 0472.41023
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.