7Concave Programming

7Concave Programming - f is concave, g is convex, and...

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Concave Programming Let f : R n R for some positive integer n , and let g : R n R m for some positive integer m be continuously differentiable functions. Consider the problem max x R n + f ( x ) (1) subject to the constraint that g ( x ) 0. Let L ( x,λ ) = f ( x ) - λ · g ( x ). The Kuhn-Tucker conditions : there is a point ( x * * ) R n × R m such that L x ( x * * ) 0 x * ·L x ( x * * ) = 0 x * 0 L λ ( x * * ) 0 λ * ·L λ ( x * * ) = 0 λ * 0 (2) The Saddlepoint condition : there is a ( x * * ) R m + × R n + such that L ( x,λ * ) ≤ L ( x * * ) ≤ L ( x * ) for all ( x,λ ) R n + × R m + . (3) Slater’s constraint qualification : There is a point ˆ x R n ++ such that g x ) << 0. The following facts are useful. 1 1. If x * solves problem (1), f is concave, g is convex, and Slater’s constraint qualification holds, then there is a number λ * such that the Kuhn-Tucker conditions are satisfied. 2. If f is concave, g is convex and there is a λ * R m + such that the Kuhn-Tucker conditions holds, then x * solves problem (1). 3. If the Saddlepoint Condition holds, then x * solves problem (1). 4. If x * solves problem (1),
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Unformatted text preview: f is concave, g is convex, and Slaters constraint qualication holds, then the Saddlepoint condition holds. Note that facts 1. and 3. give properties so that the Kuhn-Tucker conditions are necessary for a solution; 2. and 4. give properties for the Kuhn-Tucker conditions to be sucient for a solution. I used fact 2. in the proof I gave for the First Welfare Theorem for the 2-good quasilinear economy (under convexity assumptions on preferences and technologies). 1 You will nd a proof of facts 1. and 2. in R. Sundarams A First Course in Optimization , pages 194-198. The two facts are summarized in his Theorem 7.16 on page 187. Note however that he subsumes nonnegativity constraints into his h function, whereas I list them separately. 1...
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