On Semidefinite Representations of Non-closed Sets
نویسنده
چکیده
Spectrahedra are sets defined by linear matrix inequalities. Projections of spectrahedra are called semidefinite representable sets. Both kinds of sets are of practical use in polynomial optimization, since they occur as feasible sets in semidefinite programming. There are several recent results on the question which sets are semidefinite representable. So far, all results focus on the case of closed sets. In this work we develop a new method to prove semidefinite representability of sets which are not closed. For example, the interior of a semidefinite representable set is shown to be semidefinite representable. More general, one can remove faces of a semidefinite representable set and preserve semidefinite representability, as long as the faces are parametrized in a suitable way.
منابع مشابه
On semidefinite representations of plane quartics
This note focuses on the problem of representing convex sets as projections of the cone of positive semidefinite matrices, in the particular case of sets generated by bivariate polynomials of degree four. Conditions are given for the convex hull of a plane quartic to be exactly semidefinite representable with at most 12 lifting variables. If the quartic is rationally parametrizable, an exact se...
متن کاملOn Regular Generalized $delta$-closed Sets in Topological Spaces
In this paper a new class of sets called regular generalized $delta$-closed set (briefly rg$delta$-closed set)is introduced and its properties are studied. Several examples are provided to illustrate the behaviour of these new class of sets.
متن کاملExposed Faces of Semidefinite Representable Sets
A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite. Motivated by the fact that diagonal LMIs define polyhedra, the solution set of an LMI is called a spectrahedron. Linear images of spectrahedra are called semidefinite representable sets. Part of the interest in spectrahedra and semid...
متن کاملSome results on functionally convex sets in real Banach spaces
We use of two notions functionally convex (briefly, F--convex) and functionally closed (briefly, F--closed) in functional analysis and obtain more results. We show that if $lbrace A_{alpha} rbrace _{alpha in I}$ is a family $F$--convex subsets with non empty intersection of a Banach space $X$, then $bigcup_{alphain I}A_{alpha}$ is F--convex. Moreover, we introduce new definition o...
متن کاملA New Look at Nonnegativity on Closed Sets and Polynomial Optimization
We first show that a continuous function f is nonnegative on a closed set K ⊆ Rn if and only if (countably many) moment matrices of some signed measure dν = fdμ with suppμ = K, are all positive semidefinite (if K is compact μ is an arbitrary finite Borel measure with suppμ = K). In particular, we obtain a convergent explicit hierarchy of semidefinite (outer) approximations with no lifting, of t...
متن کامل