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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The writer want to recognize his legal responsibility to all his (;Olleagues and buddies on the Institute of Mathematical Sciences of recent York collage for his or her stimulation and feedback that have contributed to the writing of this tract. the writer additionally needs to thank Aughtum S. Howard for permission to incorporate effects from her unpublished dissertation, Larkin Joyner for drawing the figures, Interscience Publishers for his or her cooperation and aid, and especially Lipman Bers, who prompt the booklet in its current shape.
This booklet is designed to be an simply readable, intimidation-free advisor to complex calculus. rules and techniques of evidence construct upon one another and are defined completely. this can be the 1st e-book to hide either unmarried and multivariable research in this kind of transparent, reader-friendly surroundings. bankruptcy issues hide sequences, limits of features, continuity, differentiation, integration, limitless sequence, sequences and sequence of features, vector calculus, features of 2 variables, and a number of integration.
This booklet, meant as a realistic operating consultant for calculus scholars, contains 450 workouts. it's designed for undergraduate scholars in Engineering, arithmetic, Physics, or the other box the place rigorous calculus is required, and should enormously profit an individual looking a problem-solving method of calculus.
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Additional resources for Generalized Concavity
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.