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Quadratic programming and affine variational inequalities. A qualitative study. (English) Zbl 1092.90033

Nonconvex Optimization and Its Applications 78. New York, NY: Springer (ISBN 0-387-24277-5/hbk; 978-1-4419-3713-1/pbk; 978-1-4419-3713-1/ebook). xiii, 345 p. (2005).
This book presents a detailed exposition of qualitative results for quadratic programming (QP) and affine variational inequalities (AVI). Both topics are developed into a unifying approach.
It is divided into 18 chapters. In the first four of them the quadratic programming model and its basic properties are presented. The results given are, among others: existence and characterization of solutions, simple properties of the solution set (closedness, boundedness, etc). In the next three chapters the affine variational problem is defined and two main properties are studied: the existence of solutions (under monotonicity and copositivity) and the upper-Lipschitz continuity of the solution map. The chapters 8 and 9 are devoted to two concrete models: Lineal fractional vector optimization and the traffic equilibrium problem.
The next two chapters study the upper and the lower semicontinuity of the (Karush-Kuhn Tucker) KKT point set mapping of QP problems. In the chapter 12 the same semicontinuity properties are studied but for the solution map of QP. The optimal value function of QP problems is then analysed in two chapters. One chapter is devoted to the continuity and another one to the directional differentiablility.
The case of quadratic programming under linear perturbations is treated along three chapters. The first one is devoted to the continuity of the solution map and the second one to the optimal value function. The chapter 17 contains the convex case with applications. The book is concluded with a chapter studying the lower and upper semicontinuity of the solution map of affine variational inequalities.

MSC:

90C20 Quadratic programming
90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
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