Projection methods in conic optimization

There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace. Those conic projections are in turn the work-horse of a range of algorithms in conic optimization, having a variety of applications in science, finance and engineering. This chapter reviews some of these algorithms, emphasizing the so-called regularization algorithms for linear conic optimization, and applications in polynomial optimization. This is a presentation of the material of several recent research articles; we aim here at clarifying the ideas, presenting them in a general framework, and pointing out important techniques

Primal and dual conic representable sets: a fresh view on multiparametric analysis

This paper introduces the concepts of the primal and dual conic (linear inequality) representable sets and applies them to explore a novel kind of duality in multiparametric conic linear optimization. Such a kind of duality may be described by the set-valued mappings between the primal and dual conic representable sets, which allows us to generalize as well as treat previous results for mulitparametric analysis in a unified framework. In particular, it leads to the invariant region decomposition of a conic representable set that is more general than the known results in the literatures. We develop the classical duality theory in conic linear optimization and obtain the multiparametric KKT conditions. As their applications, we then discuss the behaviour of the optimal partition of a conic representable set and investigate the multiparametric analysis of conic linear optimization problems. All results are corroborated by examples having correlation.Comment: 38 pages, 3 figur