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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.

90C48 Programming in abstract spaces
49N15 Duality theory (optimization)
Full Text: DOI
[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
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