Extensions of the Constraint Sets

Abstract

AbstractIn this chapter, we shall extend the approximation methods for polynomial optimization discussed in Chap. 2. The extensions are focused on the constraint sets of the polynomial optimization models, including binary hypercube, hypercube, the Euclidean sphere, intersection of co-centered ellipsoids, a general convex compact set, and even a mixture of binary hypercube and the Euclidean sphere. These extensions are not straightforward generalizations of the approximation methods proposed before. Rather, they entail specifications to account for the different structures of the constraint sets at hand. The most noticeable novelty is in the decomposition routines, which play an instrumental role in designing approximation algorithms for multilinear form optimization models, as Sect. 2.1 already shows. Along with the approximation methods discussed in Chap. 2, we hope these extended techniques will be helpful in designing approximation methods when new models are encountered.KeywordsApproximation AlgorithmApproximation RatioPolynomial OptimizationEuclidean SphereMultilinear FormThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.