Abstract
The notion of approximate solutions or ε-solutions emerged early in the development of modern convex analysis. An analogue of the well-known statement concerning the minimum of a convex function and its subgradient also holds in the approximate case: a convex function f has an ε-approximate minimum at x if and only if 0 ∈ ∂ 0 f(x), where ∂ 0 f (x) is the ε-subdifferential of f at x. Particular attention has been paid to ε-subdifferentials (see Hiriart-Urruty, 1982; Demyanov, 1981). This has resulted in the construction of a new class of optimization procedures, the ε-subgradient methods. The virtually complete set of calculation rules derived for the ε-subdifferential has made possible the study and characterization of constrained convex optimization problems in both the real-valued and vector-valued cases, as in Strodiot et al. (1983), or for ordered vector spaces (Kutateladze, 1978).
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Vályi, I. (1985). On Duality Theory Related to Approximate Solutions of Vector-Valued Optimization Problems. In: Demyanov, V.F., Pallaschke, D. (eds) Nondifferentiable Optimization: Motivations and Applications. Lecture Notes in Economics and Mathematical Systems, vol 255. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-12603-5_14
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