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Download Nonlinear Integer Programming by Sam Efromovich PDF

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By Sam Efromovich

The methodological improvement of integer programming has grown via leaps and boundaries long ago 4 many years, with its major specialise in linear integer programming. despite the fact that, the earlier few years have additionally witnessed convinced promising theoretical and methodological achievements in nonlinear integer programming. those fresh advancements have produced functions of nonlinear (mixed) integer programming throughout a number of a number of components of clinical computing, engineering, administration technology and operations examine. Its widespread purposes contain, for examples, portfolio choice, capital budgeting, construction making plans, source allocation, laptop networks, reliability networks and chemical engineering.

In popularity of nonlinearity's educational importance in optimization and its significance in actual international purposes, NONLINEAR INTEGER PROGRAMMING is a entire and systematic remedy of the technique. The book's objective is to carry the cutting-edge of the theoretical starting place and resolution tools for nonlinear integer programming to scholars and researchers in optimization, operations study, and computing device technological know-how. This publication systemically investigates idea and resolution methodologies for normal nonlinear integer programming, and while, presents a well timed and complete precis of the theoretical and algorithmic improvement within the final 30 years in this subject. the next are a few good points of the book:

Duality idea 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 resolution framework is supplied utilizing international descent.

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

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Extra resources for Nonlinear Integer Programming

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10) gives rise to ^x* < b. 8) that D c^x* — c^x. 2 31 Linearly constrained separable convex integer program The proximity results in the previous subsection can be extended to separable convex programming problems. t. t. , TI, are all convex functions on M, A is an integer m X n matrix and b 6 R^. 12) both exist. 11) such that THEOREM < nA{A). 12) such that \x — z\ < n/\{A). 14) Proof. 11), respectively. 12) with the minimal box that contains x and z. 15) Then S'* can be expressed as {x E M^ | A*x < 6*}.

X f } . Remove {P{Xi)) from L^ and set L:= LUL^U Lf. Go to Step 1. Step 5 (Fathoming). Remove {P{Xi)) from L^. If i < k, SQii := i + 1 and return to Step 2. Otherwise, go to Step 4. 1. 1 stops at an optimal solution to (P) within a finite number of iterations. Proof. Note that the fathoming procedure, either in Step 3 or Step 5 of the algorithm, will not remove any feasible solution of (P) better than the incumbent. Notice that X is finite. Thus only a finite number of branching steps can be executed.

K), where K = 2^. 4) 28 NONLINEAR INTEGER PROGRAMMING where A == {7 G R^ | J^f^i ji = 1, -fi > 0, i = 1,... ,K}. It is clear that 0 is a piecewise Hnear convex function on conv{X). 5) and f{x) = (/)(x) for all x e Q. Recall that f{x) and (/)(x) have the same global optimal value over conv{X) (see [182]). Notice that a concave function always achieves its minimum over a polyhedron at one of the extreme points. Also, the extreme points of conv{X) are integer points. p,/i€M^ {4>{x) I Ax < 6, Bx = d} xEconv{X) < min {/(x) \ Ax

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