16,822 research outputs found
Relative Entropy Relaxations for Signomial Optimization
Signomial programs (SPs) are optimization problems specified in terms of
signomials, which are weighted sums of exponentials composed with linear
functionals of a decision variable. SPs are non-convex optimization problems in
general, and families of NP-hard problems can be reduced to SPs. In this paper
we describe a hierarchy of convex relaxations to obtain successively tighter
lower bounds of the optimal value of SPs. This sequence of lower bounds is
computed by solving increasingly larger-sized relative entropy optimization
problems, which are convex programs specified in terms of linear and relative
entropy functions. Our approach relies crucially on the observation that the
relative entropy function -- by virtue of its joint convexity with respect to
both arguments -- provides a convex parametrization of certain sets of globally
nonnegative signomials with efficiently computable nonnegativity certificates
via the arithmetic-geometric-mean inequality. By appealing to representation
theorems from real algebraic geometry, we show that our sequences of lower
bounds converge to the global optima for broad classes of SPs. Finally, we also
demonstrate the effectiveness of our methods via numerical experiments
Finiteness theorems in stochastic integer programming
We study Graver test sets for families of linear multi-stage stochastic
integer programs with varying number of scenarios. We show that these test sets
can be decomposed into finitely many ``building blocks'', independent of the
number of scenarios, and we give an effective procedure to compute these
building blocks. The paper includes an introduction to Nash-Williams' theory of
better-quasi-orderings, which is used to show termination of our algorithm. We
also apply this theory to finiteness results for Hilbert functions.Comment: 36 p
Chain Decomposition Theorems for Ordered Sets (and Other Musings)
A brief introduction to the theory of ordered sets and lattice theory is
given. To illustrate proof techniques in the theory of ordered sets, a
generalization of a conjecture of Daykin and Daykin, concerning the structure
of posets that can be partitioned into chains in a ``strong'' way, is proved.
The result is motivated by a conjecture of Graham's concerning probability
correlation inequalities for linear extensions of finite posets
Chordal Decomposition in Rank Minimized Semidefinite Programs with Applications to Subspace Clustering
Semidefinite programs (SDPs) often arise in relaxations of some NP-hard
problems, and if the solution of the SDP obeys certain rank constraints, the
relaxation will be tight. Decomposition methods based on chordal sparsity have
already been applied to speed up the solution of sparse SDPs, but methods for
dealing with rank constraints are underdeveloped. This paper leverages a
minimum rank completion result to decompose the rank constraint on a single
large matrix into multiple rank constraints on a set of smaller matrices. The
re-weighted heuristic is used as a proxy for rank, and the specific form of the
heuristic preserves the sparsity pattern between iterations. Implementations of
rank-minimized SDPs through interior-point and first-order algorithms are
discussed. The problem of subspace clustering is used to demonstrate the
computational improvement of the proposed method.Comment: 6 pages, 6 figure
Exploiting symmetries in SDP-relaxations for polynomial optimization
In this paper we study various approaches for exploiting symmetries in
polynomial optimization problems within the framework of semi definite
programming relaxations. Our special focus is on constrained problems
especially when the symmetric group is acting on the variables. In particular,
we investigate the concept of block decomposition within the framework of
constrained polynomial optimization problems, show how the degree principle for
the symmetric group can be computationally exploited and also propose some
methods to efficiently compute in the geometric quotient.Comment: (v3) Minor revision. To appear in Math. of Operations Researc
The power of symmetric extensions for entanglement detection
In this paper, we present new progress on the study of the symmetric
extension criterion for separability. First, we show that a perturbation of
order O(1/N) is sufficient and, in general, necessary to destroy the
entanglement of any state admitting an N Bose symmetric extension. On the other
hand, the minimum amount of local noise necessary to induce separability on
states arising from N Bose symmetric extensions with Positive Partial Transpose
(PPT) decreases at least as fast as O(1/N^2). From these results, we derive
upper bounds on the time and space complexity of the weak membership problem of
separability when attacked via algorithms that search for PPT symmetric
extensions. Finally, we show how to estimate the error we incur when we
approximate the set of separable states by the set of (PPT) N -extendable
quantum states in order to compute the maximum average fidelity in pure state
estimation problems, the maximal output purity of quantum channels, and the
geometric measure of entanglement.Comment: see Video Abstract at
http://www.quantiki.org/video_abstracts/0906273
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