Download PDF by Sam Efromovich: Nonlinear Integer Programming

By Sam Efromovich

The methodological improvement of integer programming has grown by means of leaps and limits long ago 4 many years, with its major specialize in linear integer programming. notwithstanding, the prior few years have additionally witnessed definite promising theoretical and methodological achievements in nonlinear integer programming. those fresh advancements have produced functions of nonlinear (mixed) integer programming throughout quite a few quite a few components of medical computing, engineering, administration technological know-how and operations learn. Its fashionable functions comprise, for examples, portfolio choice, capital budgeting, creation making plans, source allocation, laptop networks, reliability networks and chemical engineering.

In attractiveness of nonlinearity's educational value in optimization and its value in actual international functions, NONLINEAR INTEGER PROGRAMMING is a complete and systematic remedy of the method. The book's target is to deliver the state of the art of the theoretical starting place and answer tools for nonlinear integer programming to scholars and researchers in optimization, operations study, and machine technological know-how. This booklet systemically investigates thought and answer methodologies for basic nonlinear integer programming, and while, presents a well timed and accomplished precis of the theoretical and algorithmic improvement within the final 30 years in this subject. the next are a few positive factors of the book:

Duality conception for nonlinear integer programming is carefully discussed.

Convergent Lagrangian and slicing tools for separable nonlinear integer programming are defined and demonstrated.

Convexification scheme and the relation among the monotonicity and convexity is explored and illustrated.

A answer framework is supplied utilizing worldwide descent.

Computational implementations for large-scale nonlinear integer programming difficulties are tested for a number of effective resolution algorithms awarded within the book.

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Extra resources for Nonlinear Integer Programming

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This often results in nonlinear mixed integer programming formulations at the design stage. 2 from [53] presents a superstructure of a chemical process in which all competitive alternative process configurations have been embedded. A zero-one variable yi is attached to each process unit, while its final value will determine whether or not a process unit is in the final optimal configuration. The continuous variable Xij represents a process parameter such as the flow rate of materials. The objective is to minimize a summation of fixed-charge costs and operation costs, while the constraints correspond to design specifications, topological considerations and physical conservation laws.

5)^ which is also the global minimizer. An integers G X is defined Introduction 9 here as a discrete local minimizer of / over X if its function value is less than or equal to that of its 4 neighboring points x ± (1,0)-^ and x ± (0,1)"^ (if they are included in X). 1 lists the values of / for all integer points on X, where discrete local minima are labelled by "*" and the global minimum by "V. It is clear that even for a convex function, there may exist multiple minima on an integer domain. 1. Multiple discrete local minimizers of a convex function.

Let vi = {cos 6^ sin 9)^, V2 = {—sin 6^ cos 6)^. Then vi and V2 are orthonormal and the angle between vi and xi-axis is 9. ^2' Ai cos^ ^ + A2 sin^ 9 (Ai — A2) sin 9 cos 9 \ (Ai - A2) sin 9 cos 9 Ai sin^ ^ + A2 cos^ 9 J ' ^ ^ It follows that P is a 2 X 2 symmetric positive definite matrix and it has eigenvalues Ai and A2 with corresponding eigenvectors vi and V2, respectively. 27), XQ = (0,1/2)^ and A2 G (0,1/4). For any positive integer m > 0 and A2 E (0,1/4), we can determine the values of 9 and Ai such that axis 0:2 = 0 supports ellipsoid E{xo^ P~^) = {x e 36 NONLINEAR INTEGER PROGRAMMING R'^ \ {x — xo)^P~^{x — xo) < 1} at ( - m , 0 ) .

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