By Mordecai Avriel, Erwin Diewert, Siegfried Schaible, Israel Zang

Initially released in 1988, this enduring textual content is still the main accomplished publication on generalized convexity and concavity. The authors current generalized concave services in a unified framework, exploring them essentially from the domain names of optimization and economics.

Concavity of a functionality is a standard estate utilized in lots of the very important theorems referring to houses of optimization difficulties in mathematical economics, operations learn, mathematical programming, engineering, and administration technological know-how. Generalized concavity offers with the various nonconcave capabilities that experience homes just like these of concave features.

particular issues coated during this e-book include:a overview of concavity and the fundamentals of generalized concavity; purposes of generalized concavity to economics; particular functionality kinds akin to composite kinds, items, ratios, and quadratic capabilities; fractional programming; and concave transformable capabilities.

**Audience:** Mathematicians, economists, computational mathematicians, administration scientists, operations researchers, and optimization theorists will locate this ebook invaluable.

**Contents:** Preface to the Classics version; Preface; bankruptcy 1: advent; bankruptcy 2: Concavity; bankruptcy three: Generalized Concavity; bankruptcy four: program of Generalized Concavity to Economics; bankruptcy five: certain practical types I: Composite services, items, and Ratios; bankruptcy 6: specified useful varieties II: Quadratic features; bankruptcy 7: Fractional Programming; bankruptcy eight: Concave Transformable capabilities; bankruptcy nine: extra Generalizations of Concavity; Supplementary Bibliography; writer Index; topic Index.

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W. V. (1906), Sur les Fonctions Convexes et les Inegalities Entre les Valeurs Moyennes, Acta Math. 30,175-193. JORGENSON, D. , and LAU, L. J. (1974), Duality and differentiability in production, J. Econ. Theory 9, 23-42. KARLIN, S. (1959), Mathematical Methods and Theory in Games, Programming and Economics, Vol. 1, Addison-Wesley, Reading, Massachusetts. KARUSH, W. S. dissertation, Department of Mathematics, University of Chicago. KUHN, H. , and TUCKER, A. W. (1951), Nonlinear programming, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, Edited by J.

23. Let f be a differentiable concave (strictly concave) function on the convex set C. 108) C, then f attains its maximum (unique maximum) at x*. Proof. 109) and the inequality above is strict if f is strictly concave. 110) inequality is strict if f is strictly concave. Thus x* is a maximum maximum) of f. 0 us turn our attention to minima of concave functions. The first as follows. 24 (Fenchel, 1951). If f is a concave function defined on the convex set C and attains its global minimum over C at an interior point XO of C, then f is constant on C.

We can also characterize differentiable concave functions by the monotonic behavior of their derivatives. 4. The "graph below the tangent'· property of concave functions. 3 (Fenchel, 1951). Letfbe a differentiable function on the open convex set C c R. It is concave (strictly concave) if and only if l' is a nonincreasing (decreasing) function. Proof. 26) and l' is nonincreasing. 26) are strict if f is strictly concave. Conversely, let Xl E C, x 2 E C, Xl < x 2 and x 3 = AXI + (1 - A )x 2 for some 0 < A < 1.