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On symmetric and asymmetric duality. II. (English) Zbl 0745.90064
[For part I see the author, Lucrările Conferinţei de matematică aplicată şi mecanică, 20–23 oct. 1988, Cluj- Napoca, 377–386 (1990).]
Given a constrained minimum problem (primal), the Lagrange multipliers theory allows us to formulate an unconstrained maximum problem (dual) satisfying the so-called weak duality theorem, i.e. the dual optimum is less than or equal to the primal one. Such a Lagrangian dual is asymmetric in the sense that the dual feasible region is generally contained in a space whose dimension is greater than that of the primal one.
In 1965 G. Dantzig, E. Eisenberg and R. W. Cottle [Pac. J. Math. 15, 809–812 (1965; Zbl 0136.14001)] formulated a symmetric dual scheme, in the sense that the feasible regions of the two problems, primal and dual, are contained in spaces having the same dimension and the same variables. M. Pappalardo [“Symmetric and asymmetric duality”, Sect. Appl. Math., Group Optim. Oper. Res., Univ. Pisa/Italy (1985)] showed that the above duality schemes, under the hypotheses of convexity and differentiability for the considered problems, are equivalent.
In this paper the equivalence between symmetric and asymmetric duality without the hypothesis of differentiability when the primal problem and its dual are formulated as extremum problems over nonempty convex cones is shown.
Reviewer: L.Blaga
90C30 Nonlinear programming
49N15 Duality theory (optimization)