1,782 research outputs found
A Semidefinite Approach for Truncated K-Moment Problems
A truncated moment sequence (tms) of degree d is a vector indexed by
monomials whose degree is at most d. Let K be a semialgebraic set.The truncated
K-moment problem (TKMP) is: when does a tms y admit a positive Borel measure
supported? This paper proposes a semidefinite programming (SDP) approach for
solving TKMP. When K is compact, we get the following results: whether a tms y
of degree d admits a K-measure or notcan be checked via solving a sequence of
SDP problems; when y admits no K-measure, a certificate will be given; when y
admits a K-measure, a representing measure for y would be obtained from solving
the SDP under some necessary and some sufficient conditions. Moreover, we also
propose a practical SDP method for finding flat extensions, which in our
numerical experiments always finds a finitely atomic representing measure for a
tms when it admits one
Convex optimization over intersection of simple sets: improved convergence rate guarantees via an exact penalty approach
We consider the problem of minimizing a convex function over the intersection
of finitely many simple sets which are easy to project onto. This is an
important problem arising in various domains such as machine learning. The main
difficulty lies in finding the projection of a point in the intersection of
many sets. Existing approaches yield an infeasible point with an
iteration-complexity of for nonsmooth problems with no
guarantees on the in-feasibility. By reformulating the problem through exact
penalty functions, we derive first-order algorithms which not only guarantees
that the distance to the intersection is small but also improve the complexity
to and for smooth functions. For
composite and smooth problems, this is achieved through a saddle-point
reformulation where the proximal operators required by the primal-dual
algorithms can be computed in closed form. We illustrate the benefits of our
approach on a graph transduction problem and on graph matching
Lower bounds on the size of semidefinite programming relaxations
We introduce a method for proving lower bounds on the efficacy of
semidefinite programming (SDP) relaxations for combinatorial problems. In
particular, we show that the cut, TSP, and stable set polytopes on -vertex
graphs are not the linear image of the feasible region of any SDP (i.e., any
spectrahedron) of dimension less than , for some constant .
This result yields the first super-polynomial lower bounds on the semidefinite
extension complexity of any explicit family of polytopes.
Our results follow from a general technique for proving lower bounds on the
positive semidefinite rank of a matrix. To this end, we establish a close
connection between arbitrary SDPs and those arising from the sum-of-squares SDP
hierarchy. For approximating maximum constraint satisfaction problems, we prove
that SDPs of polynomial-size are equivalent in power to those arising from
degree- sum-of-squares relaxations. This result implies, for instance,
that no family of polynomial-size SDP relaxations can achieve better than a
7/8-approximation for MAX-3-SAT
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