By Vinogradov V.

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Example text

Xn , λ1 , . . , λm ) = f (x1 , . . , xn ) + j=1 where λ1 , λ2 , . . , λm are constant (Lagrange multipliers). Recipe 13 – What are the Necessary Conditions for the Solution of (4)? Equate all partials of L with respect to x1 , . . , xn , λ1 , . . , λm to zero: m ∂L(x1 , . . , xn , λ1 , . . , λm ) ∂f (x1 , . . , xn ) ∂g j (x1 , . . , xn ) = − λj = 0, ∂xi ∂xi ∂xi j=1 ∂L(x1 , . . , xn , λ1 , . . , λm ) = bj − g j (x1 , . . , xn ) = 0, ∂λj i = 1, 2, . . , n, j = 1, 2, . . , m. Solve these equations for x1 , .

X1 ≥ 0, . . , xn ≥ 0 (6) i Similarly, the minimization problem is min f (x1 , . . , xn ) subject to g (x1 , . . , xn ) ≥ bi , i = 1, 2, . . , m. x1 ≥ 0, . . , xn ≥ 0 i First, note that there are no restrictions on the relative size of m and n, unlike the case of equality constraints. Second, note that the direction of the inequalities (≤ or ≥) at the constraints is only a convention, because the inequality g i ≤ bi can be easily converted to the ≥ inequality by multiplying it by –1, yielding −g i ≥ −bi .

M ) which satisfies Kuhn-Tucker necessary conditions and for which L(x, λ∗ ) ≤ L(x∗ , λ∗ ) ≤ L(x∗ , λ) for all x = (x1 , . . , xn ) ≥ 0, λ = (λ1 , . . , λm ) ≥ 0 is known as the saddle point problem. Proposition 31 If (x∗ , λ∗ ) solves the saddle point problem then (x∗ , λ∗ ) solves the problem (6). 51 Economics Application 7 (Economic Interpretation of a Nonlinear Program and KuhnTucker Conditions) A nonlinear program can be interpreted much like a linear program. A maximization program in the general form, for example, is the production problem facing a firm which has to produce n goods such that it maximizes its revenue subject to m resource (factor) constraints.

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