36 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
H_2-Optimal Decentralized Control over Posets: A State-Space Solution for State-Feedback
We develop a complete state-space solution to H_2-optimal decentralized
control of poset-causal systems with state-feedback. Our solution is based on
the exploitation of a key separability property of the problem, that enables an
efficient computation of the optimal controller by solving a small number of
uncoupled standard Riccati equations. Our approach gives important insight into
the structure of optimal controllers, such as controller degree bounds that
depend on the structure of the poset. A novel element in our state-space
characterization of the controller is a remarkable pair of transfer functions,
that belong to the incidence algebra of the poset, are inverses of each other,
and are intimately related to prediction of the state along the different paths
on the poset. The results are illustrated by a numerical example.Comment: 39 pages, 2 figures, submitted to IEEE Transactions on Automatic
Contro
False Discovery and Its Control in Low Rank Estimation
Models specified by low-rank matrices are ubiquitous in contemporary
applications. In many of these problem domains, the row/column space structure
of a low-rank matrix carries information about some underlying phenomenon, and
it is of interest in inferential settings to evaluate the extent to which the
row/column spaces of an estimated low-rank matrix signify discoveries about the
phenomenon. However, in contrast to variable selection, we lack a formal
framework to assess true/false discoveries in low-rank estimation; in
particular, the key source of difficulty is that the standard notion of a
discovery is a discrete one that is ill-suited to the smooth structure
underlying low-rank matrices. We address this challenge via a geometric
reformulation of the concept of a discovery, which then enables a natural
definition in the low-rank case. We describe and analyze a generalization of
the Stability Selection method of Meinshausen and B\"uhlmann to control for
false discoveries in low-rank estimation, and we demonstrate its utility
compared to previous approaches via numerical experiments
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 nonconvex 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
An Optimal Controller Architecture for Poset-Causal Systems
We propose a novel and natural architecture for decentralized control that is
applicable whenever the underlying system has the structure of a partially
ordered set (poset). This controller architecture is based on the concept of
Moebius inversion for posets, and enjoys simple and appealing separation
properties, since the closed-loop dynamics can be analyzed in terms of
decoupled subsystems. The controller structure provides rich and interesting
connections between concepts from order theory such as Moebius inversion and
control-theoretic concepts such as state prediction, correction, and
separability. In addition, using our earlier results on H_2-optimal
decentralized control for arbitrary posets, we prove that the H_2-optimal
controller in fact possesses the proposed structure, thereby establishing the
optimality of the new controller architecture.Comment: 32 pages, 9 figures, submitted to IEEE Transactions on Automatic
Contro
A partial order approach to decentralized control
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 173-177).In this thesis we consider the problem of decentralized control of linear systems. We employ the theory of partially ordered sets (posets) to model and analyze a class of decentralized control problems. Posets have attractive combinatorial and algebraic properties; the combinatorial structure enables us to model a rich class of communication structures in systems, and the algebraic structure allows us to reparametrize optimal control problems to convex problems. Building on this approach, we develop a state-space solution to the problem of designing Hâ‚‚-optimal controllers. Our solution is based on the exploitation of a key separability property of the problem that enables an efficient computation of the optimal controller by solving a small number of uncoupled standard Riccati equations. Our approach gives important insight into the structure of optimal controllers, such as controller degree bounds that depend on the structure of the poset. A novel element in our state-space characterization of the controller is a pair of transfer functions, that belong to the incidence algebra of the poset, are inverses of each other, and are intimately related to estimation of the state along the different paths in the poset. We then view the control design problem from an architectural viewpoint. We propose a natural architecture for poset-causal controllers. In the process, we establish interesting connections between concepts from order theory such as Mobius inversion and control-theoretic concepts such as state estimation, innovation, and separability principles. Finally, we prove that the Hâ‚‚-optimal controller in fact posseses the proposed controller structure, thereby proving the optimality of the architecture.by Parikshit Shah.Ph.D