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Parametric integer optimization. (English) Zbl 0643.90043
Mathematical Research, 39. Berlin (GDR): Akademie-Verlag. 138 p. DM 20.00 (1988).
This volume is an excellent survey on parametric integer optimization and includes a large variety of important and most recent results especially on the following type of nonlinear parametric (mixed-) integer programs: \[ P(p,b): \inf \{f(x)+p^ Tx| x\in M(b),x_ 1,...,x_ s\quad integer\}, \] with \(p\in {\mathbb{R}}^ n\), \(b\in {\mathbb{R}}^ m\), \(f: {\mathbb{R}}^ n\to {\mathbb{R}}\) quadratic or convex polynomial, \(M(b):=C(b)+V\) as constraints set with a compact-valued upper semi-continuous multifunction C and a fixed convex polyhedral cone V.
The following three aspects of a stability theory for P(p,b) are treated in a comprehensive way: existence and stability of feasible points, semi- continuity properties of the value functions of P(p,b), their level-sets, especially optimal sets, existence of optimal points.
The volume is well organized and gives a lot of important results and interesting details (even in the linear (mixed-) integer case too) - for example it contains a short survey on integer points in certain subsets of \({\mathbb{R}}^ n\) and related distribution properties.
Summarizing, this book has to be very well recommended as it almost closes the gap between the knowledge on continuous parametric optimization problems and on (mixed-) integer analoga.
Reviewer: H.Noltemeier

MSC:
90C10 Integer programming
90C31 Sensitivity, stability, parametric optimization
90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming
90C11 Mixed integer programming
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