By Vinogradov V.
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Additional info for A cookbook of mathematics
With respect to l, we have: pf11 (l, k0 ) ∂l = 1 or ∂w ∂l 1 = . ) For another economic application see the section ‘Constrained Optimization’. 9 Concavity, Convexity, Quasiconcavity and Quasiconvexity Definition 42 A function z = f (x1 , . . , xn ) is concave if and only if for any pair of distinct points u = (u1 , . . , un ) and v = (v1 , . . , vn ) in the domain of f and for any θ ∈ (0, 1) θf (u) + (1 − θ)f (v) ≤ f (θu + (1 − θ)v) and convex if and only if θf (u) + (1 − θ)f (v) ≥ f (θu + (1 − θ)v).
Nonclassical optimization, also known as mathematical programming, tackles problems with inequality constraints like g(x1 , . . , xn ) ≤ c. Mathematical programming includes linear programming and nonlinear programming. In linear programming, the objective function and all inequality constraints are linear. When either the objective function or an inequality constraint is nonlinear, we face a problem of nonlinear programming. In the following, we restrict our attention to non-linear programming.
The conditions for concavity and convexity are necessary and sufficient, while those for strict concavity (strict convexity) are only sufficient. Definition 43 A set S in Rn is convex if for any x, y ∈ S and any θ ∈ [0, 1] the linear combination θx + (1 − θ)y ∈ S. Definition 44 A function z = f (x1 , . . , xn ) is quasiconcave if and only if for any pair of distinct points u = (u1 , . . , un ) and v = (v1 , . . , vn ) in the convex domain of f and for any θ ∈ (0, 1) f (v) ≥ f (u) =⇒ f (θu + (1 − θ)v) ≥ f (u) and quasiconvex if and only if f (v) ≥ f (u) =⇒ f (θu + (1 − θ)v) ≤ f (v).