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New and old bounds for standard quadratic optimization: dominance, equivalence and incomparability. (English) Zbl 1171.90007
For the problem of minimizing a quadratic indefinite form over a simplex known and new bounds for the optimal objective function value are derived. Main tools for this are semidefinite programming and the decomposition of the objective function into the difference of two convex functions. The different bounds are completely compared and a new bound is developed which dominates all the other bounds, especially also Shrijver’s improvement of Lovasz’s \(\theta\) function. Interesting results are also obtained with respect to the Lagrangean duality, where tight dual formulations are found. Special topics in this interesting and comprehensive paper are convex underestimation bounds, Nowak’s bound and copositive as well as decomposition bounds.

90C20 Quadratic programming
90C26 Nonconvex programming, global optimization
90C22 Semidefinite programming
Full Text: DOI
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