## Abstract

By way of introducing the problem we deal with in this chapter, consider the problem of maximizing a smooth function f(x^{1}, x^{2}) subject to g(x^{1}, x^{2}) = 0, where g is also smooth and where f and g are real valued. Suppose x̂ provides a local solution to this problem. If (ĝ_{x}1,ĝ_{x}2) ≠ 0 then we can apply the implicit function theorem to solve for, say, x^{2} uniquely in terms of x^{1}. Thus we have g(x^{1},ξ(x^{1})) ≡ 0 in a neighborhood of x̂^{1}. So the constraint is always satisfied in that neighborhood. Our problem now is to maximize φ(x^{1})= f(x^{l}, ξ(x^{1})) locally and with no constraints in a sense to be made precise presently. By the 1st order necessary condition of Chapter 1 we have: f̂_{1} + f̂_{2}ξ̂′ = 0. But g(x1, ξ(x^{1})) is a constant function around x̂^{1}. Thus ĝ_{1} + ĝ_{2}ξ̂′ = 0. Solving for ξ̂′ we get: ξ̂′ = -ĝ_{1}/ĝ_{2}.

## Keywords

Quadratic Form Order Condition Local Solution Implicit Function Theorem Constraint Manifold## Preview

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