اصل اکسترم انتزاعی با برنامه کاربردی برای اقتصاد رفاه

In this paper we introduce general prenormal and normal structures in Banach spaces that cover conventional concepts of normals to arbitrary closed sets under minimal requirements. Based on these structures, we establish new abstract versions of the extremal principle in variational analysis, which plays a fundamental role in many applications. The main applications of this paper concern necessary conditions for Pareto optimality in nonconvex models of welfare economics. We obtain new results in this direction that extend approximate and exact versions of the generalized second welfare theorem for Pareto, weak Pareto, and strong Pareto optimal allocations.

It has been well recognized that the convex separation principle plays a crucial role in many aspects of nonlinear analysis, optimization, and their applications. In particular, a conventional approach to derive necessary optimality conditions in various optimization, optimal control, and equilibrium problems consists of applying convex separation theorems to either the convex sets in question or their tangential convex approximations. This paper develops another approach to optimal solutions and related aspects of variational analysis that does not involve any convex approxima- 1 Research was partly supported by the National Science Foundation under Grants DMS- 9704751 and DMS-0072179 and also by the Distinguished Faculty Fellowship at Wayne State University.tions and convex separation arguments. Instead, it is based on a different principle to study extremality of set systems using Žgenerally nonconvex. normal cones in dual spaces that are not generated by primal tangential approximations. This approach, unified under the name of the extremal principle 27, goes back to the beginning of dual-spaced methods in nonsmooth variational analysis; see 26, 28 for more details, references, and discussions. Results obtained in this direction can be treated as variational extensions of the classical separation theorems to systems of nonconvex sets. The primary goal of this paper is to obtain general versions of the extremal principle in terms of abstract prenormal and normal structures in Banach spaces. Then we apply these results to the study of Pareto optimal allocations in nonconvex models of welfare economics. Discussions of the results obtained and their comparison with the literature are presented in the subsequent sessions. Our notation is basically standard. Let us mention that BX and B*X* stand, respectively, for the unit closed balls in the Banach space w* in question and its dual;
signifies the weak* convergence in X*, and cl* denotes the weak* topological closure. Depending on the context, we use the notation Limsup for either the topological Painlev´e
Kuratowski upper Žouter. limit
w* Limsup FŽ x . cl*½ x*X* sequences x
x, x
x* k k x
x with x
FŽ x . , k
5 Ž1.1. k k of a set-valued mapping F: XX*, or for its sequential counterpart when cl* is omitted in Ž1.1..