Abstract
In the present paper, the Polyak’s principle, concerning convexity of the images of small balls through C1, 1 mappings, is employed in the study of vector optimization problems. This leads to extend to such a context achievements of local programming, an approach to nonlinear optimization, due to B.T. Polyak, which consists in exploiting the benefits of the convex local behaviour of certain nonconvex problems. In doing so, solution existence and optimality conditions are established for localizations of vector optimization problems, whose data satisfy proper assumptions. Such results are subsequently applied in the analysis of welfare economics, in the case of an exchange economy model with infinite-dimensional commodity space. In such a setting, the localization of an economy yields existence of Pareto optimal allocations, which, under certain additional assumptions, lead to competitive equilibria.
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“This paper is dedicated to Boris Mordukhovich, guide and friend, on the occasion of his 65th birthday...for never giving smoothness and convexity for granted”.
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Uderzo, A. Localizing Vector Optimization Problems with Application to Welfare Economics. Set-Valued Var. Anal 22, 483–501 (2014). https://doi.org/10.1007/s11228-013-0267-y
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DOI: https://doi.org/10.1007/s11228-013-0267-y
Keywords
- Modulus of convexity
- Polyak’s convexity principle
- Openness at a linear rate
- Lagrangian function
- Vector optimization
- 𝜖-localization of a problem
- exchange economy
- Regular feasible allocation
- Pareto optimality
- Competitive equilibrium