Convex and concave functions
2006; Springer Science+Business Media; Linguagem: Inglês
10.1016/b978-813120376-7/50010-5
ISSN1436-4646
Autores Tópico(s)Advanced Optimization Algorithms Research
ResumoThis chapter introduces convex and concave functions defined on convex sets in Rn and gives some of their basic properties and provides some fundamental theorems involving these functions. These theorems are very important in deriving optimality conditions for nonlinear programming problems and developing suitable computational schemes. A function of f defined on a convex set S in Rn, is said to be a convex function on S, if it satisfies the equation f [(X1 + (1 – (,)X2] ≤ (f(X1) + (1 - () f(X2). The function f is said to be strictly convex on S if the above inequality is strict for X1≠ X 2, and 0 < ( < 1. A function f is said to be concave (strictly concave) if –f is convex (strictly convex). It is clear that a linear function is convex as well as concave but neither strictly convex nor strictly concave. Alternatively, a function f defined on a convex set S in Rn is convex (concave) if linear interpolation between the values of the function never underestimates the actual value at the interpolated point.
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