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Equality Constraints

  • Mohamed Ali El-Hodiri

Abstract

By way of introducing the problem we deal with in this chapter, consider the problem of maximizing a smooth function f(x1, x2) subject to g(x1, x2) = 0, where g is also smooth and where f and g are real valued. Suppose x̂ provides a local solution to this problem. If (ĝx1,ĝx2) ≠ 0 then we can apply the implicit function theorem to solve for, say, x2 uniquely in terms of x1. Thus we have g(x1,ξ(x1)) ≡ 0 in a neighborhood of x̂1. So the constraint is always satisfied in that neighborhood. Our problem now is to maximize φ(x1)= f(xl, ξ(x1)) locally and with no constraints in a sense to be made precise presently. By the 1st order necessary condition of Chapter 1 we have: f̂1 + f̂2ξ̂′ = 0. But g(x1, ξ(x1)) is a constant function around x̂1. Thus ĝ1 + ĝ2ξ̂′ = 0. Solving for ξ̂′ we get: ξ̂′ = -ĝ12.

Keywords

Quadratic Form Order Condition Local Solution Implicit Function Theorem Constraint Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin · Heidelberg 1991

Authors and Affiliations

  • Mohamed Ali El-Hodiri
    • 1
  1. 1.Department of EconomicsUniversity of KansasLawrenceUSA

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