499 research outputs found
Conditions for Existence of Dual Certificates in Rank-One Semidefinite Problems
Several signal recovery tasks can be relaxed into semidefinite programs with
rank-one minimizers. A common technique for proving these programs succeed is
to construct a dual certificate. Unfortunately, dual certificates may not exist
under some formulations of semidefinite programs. In order to put problems into
a form where dual certificate arguments are possible, it is important to
develop conditions under which the certificates exist. In this paper, we
provide an example where dual certificates do not exist. We then present a
completeness condition under which they are guaranteed to exist. For programs
that do not satisfy the completeness condition, we present a completion process
which produces an equivalent program that does satisfy the condition. The
important message of this paper is that dual certificates may not exist for
semidefinite programs that involve orthogonal measurements with respect to
positive-semidefinite matrices. Such measurements can interact with the
positive-semidefinite constraint in a way that implies additional linear
measurements. If these additional measurements are not included in the problem
formulation, then dual certificates may fail to exist. As an illustration, we
present a semidefinite relaxation for the task of finding the sparsest element
in a subspace. One formulation of this program does not admit dual
certificates. The completion process produces an equivalent formulation which
does admit dual certificates
Set optimization - a rather short introduction
Recent developments in set optimization are surveyed and extended including
various set relations as well as fundamental constructions of a convex analysis
for set- and vector-valued functions, and duality for set optimization
problems. Extensive sections with bibliographical comments summarize the state
of the art. Applications to vector optimization and financial risk measures are
discussed along with algorithmic approaches to set optimization problems
Tangential Extremal Principles for Finite and Infinite Systems of Sets, II: Applications to Semi-infinite and Multiobjective Optimization
This paper contains selected applications of the new tangential extremal
principles and related results developed in Part I to calculus rules for
infinite intersections of sets and optimality conditions for problems of
semi-infinite programming and multiobjective optimization with countable
constraint
Strong duality of a conic optimization problem with a single hyperplane and two cone constraints
Strong (Lagrangian) duality of general conic optimization problems (COPs) has
long been studied and its profound and complicated results appear in different
forms in a wide range of literatures. As a result, characterizing the known and
unknown results can sometimes be difficult. The aim of this article is to
provide a unified and geometric view of strong duality of COPs for the known
results. For our framework, we employ a COP minimizing a linear function in a
vector variable subject to a single hyperplane constraint and two
cone constraints , . It can be identically reformulated
as a simpler COP with the single hyperplane constraint and the single
cone constraint . This simple COP and its dual as well as
their duality relation can be represented geometrically, and they have no
duality gap without any constraint qualification. The dual of the original
target COP is equivalent to the dual of the reformulated COP if the Minkowski
sum of the duals of the two cones and is closed or if the dual of
the reformulated COP satisfies a certain Slater condition. Thus, these two
conditions make it possible to transfer all duality results, including the
existence and/or boundedness of optimal solutions, on the reformulated COP to
the ones on the original target COP, and further to the ones on a standard
primal-dual pair of COPs with symmetry
- …