×

zbMATH — the first resource for mathematics

A general theory of surrogate dual and perturbational extended surrogate dual optimization problems. (English) Zbl 0607.90089
Many duality concepts exist in optimization theory, each having its own advantage. However, for a good general duality theory it is desirable to have either a system of dependences between the different dualities or even a (or some) unified hypertheory. The author studies two completely different general concepts of dual problems to a given primal problem in locally convex spaces and also discusses a series of realizations of his concepts. The first one works with a one-parameter family of surrogate constraint sets, the second one at first uses a perturbational functional (and then also some surrogate constraint sets). No conjugation is involved. At the end of his paper the author announces a paper in which he has constructed a unified theory of optimization problems, which encompasses many of the known duality concepts as particular cases.

MSC:
90C48 Programming in abstract spaces
49N15 Duality theory (optimization)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Crouzeix, J.-P, Contributions à l’étude des fonctions quasi-convexes, ()
[2] Day, M.M, Normed linear spaces, (1973), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0268.46013
[3] Dolecki, S; Kurcyusz, S, On φ-convexity in extremal problems, SIAM J. control. optim., 16, 277-300, (1978) · Zbl 0397.46013
[4] Ekeland, I; Temam, R, Analyse convexe et problèmes variationnels, (1974), Dunod/Gauthier Villars Paris · Zbl 0281.49001
[5] Glover, F, A multiphase-dual algorithm for the zero-one integer programming problem, Oper. res., 13, 879-919, (1965) · Zbl 0163.41301
[6] Greenberg, H.J; Pierskalla, W.P, Surrogate mathematical programming, Oper. res., 18, 924-939, (1970) · Zbl 0232.90059
[7] Greenberg, H.J; Pierskalla, W.P, Quasi-conjugate functions and surrogate duality, Cahiers centre études rech. opér., 15, 437-448, (1973) · Zbl 0276.90051
[8] Luenberger, D.G, Quasi-convex programming, SIAM J. appl. math., 16, 1090-1095, (1968) · Zbl 0212.23905
[9] Luenberger, D.G, Optimization by vector space methods, (1969), Wiley New York/London/Sydney/Toronto · Zbl 0176.12701
[10] Singer, I, Generalizations of methods of best approximation to convex optimization in locally convex spaces. II. hyperplane theorems, J. math. anal. appl., 69, 571-584, (1979) · Zbl 0466.41013
[11] Singer, I, Duality theorems for linear systems and convex systems, J. math. anal. appl., 76, 339-368, (1980) · Zbl 0445.49007
[12] Singer, I, Duality theorems for constrained convex optimization, Control. cybernet., 9, 37-45, (1980) · Zbl 0526.49009
[13] Singer, I, Duality theorems for perturbed convex optimization, J. math. anal. appl., 81, 437-452, (1981) · Zbl 0462.90091
[14] Singer, I, On the perturbation and Lagrangian duality theories of rockafellar and kurcyusz, (), 153-156 · Zbl 0467.90077
[15] Singer, I, Pseudo-conjugate functionals and pseudo-duality, (), 115-134
[16] Singer, I, Optimization by level set methods. I. duality formulae, (), 13-43
[17] Singer, I, Optimization by level set methods. II. further duality formulae in the case of essential constraints, (), 383-411
[18] Singer, I, Optimization by level set methods. III. characterizations of solutions in the presence of duality, Numer. funct. anal. optim., 4, 151-170, (1981-1982) · Zbl 0496.49014
[19] Singer, I, Optimization by level set methods. IV. generalizations and complements, Numer. funct. anal. optim., 4, 279-310, (1981-1982) · Zbl 0497.49022
[20] Singer, I, Surrogate dual problems and surrogate Lagrangians, J. math. anal. appl., 98, 31-71, (1984) · Zbl 0584.49006
[21] Singer, I, The lower semi-continuous quasi-convex hull as a normalized second conjugate, Nonlinear anal. theory, math. appl., 7, 1115-1121, (1983) · Zbl 0528.49007
[22] Singer, I, Optimization by level set methods. V. duality theorems for perturbed optimization problems, Math. operationsforsch. statist. ser. optim., 15, 3-36, (1984) · Zbl 0538.49004
[23] Singer, I, Surrogate conjugate functionals and surrogate convexity, Applicable anal., 16, 291-327, (1983) · Zbl 0526.90097
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.