Equality Constraints

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 (ĝ_x1,ĝ_x2) ≠ 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′ = -ĝ₁/ĝ₂.