<span class=strong>IMA Reception and Poster Session</span>
Tuesday, January 16, 2007 - 4:00pm - 6:30pm
- Obstacle-sensitive Gain scheduling Using Semidefinite Programming
Eric Feron (Georgia Institute of Technology)
Joint work with Mazen Farhood.
We present an application of semidefinite programming techniques to the regulation of vehicle trajectories in the vicinity of obstacles. We design control laws, together with Lyapunov functions that guarantee closed-loop stability and performance of the vehicle's regulation loop. These control laws are easy to implement and automatically relax the system's gains when away from the obstacles, while tightening them when obstacle proximity is detected.
- Distributed Optimization in an Energy-constrained Network
Seid Alireza Razavi Majomard (University of Minnesota, Twin Cities)
We consider a distributed optimization problem whereby two nodes S1 and S2 wish to jointly minimize a common convex quadratic cost function f(x1; x2), subject to separate local constraints on x1 and x2, respectively. Suppose that node S1 has control of variable x1 only and node S2 has control of variable x2 only. The two nodes locally update their respective variables and periodically exchange their values over a noisy channel. Previous studies of this problem have mainly focused on the convergence issue and the analysis of convergence rate. In this work, we focus on the communication energy and study its impact on convergence. In particular, we consider a class of distributed stochastic gradient type algorithms implemented using certain linear analog messaging schemes. We study the minimum amount of communication energy required for the two nodes to compute an epsilon-minimizer of f(x1; x2) in the mean square sense. Our analysis shows that the communication energy must grow at least at the rate of (1/epsilon). We also derive specific designs, which attain this minimum energy bound, and provide simulation results that confirm our theoretical analysis. Extension to the multiple node case is described.
- Application of Semidefinite Programming to Eigenvalue Problems for Elliptic Linear Partial Differential Equations
Carlos Handy (Texas Southern University)
The calculation of eigenvalues for stiff elliptic linear
partial differential equations (LPDEs) can be plagued with
significant inaccuracies depending on the estimation methods
used (i.e. variational, finite differencing, asymptotic
analysis, perturbative, Galerkin, etc.). A preferred approach
is to be able to generate tight, converging lower and upper
bounds to the eigenvalues, thereby removing any uncertainties
in the reliability of the generated results. Twenty years ago
one such method was developed by Handy, Bessis, and co-workers
[1-3]. This general approach is referred to as the Eigenvalue
Moment Method (EMM) and involves a Semidefinite Programming
formalism coupled with a Linear Programming based “Cutting
Algorithm.” It makes use of well known nonnegativity properties
of Sturm-Liouville type systems combined with important
theorems from the classic Moment Problem. The EMM procedure has
been used to solve a variety of LPDEs on various support spaces
(i.e. unbounded, semi-bounded, bounded, periodic, discrete).
Equivalent gradient search variational reformulations,
exploiting higher levels of convexity, have also been developed
leading to the Volcano Function representation . It is also
possible to extend EMM to certain non-hermitian systems of
importance in forefront areas in mathematical physics. Here
too, the EMM approach can yield converging lower and upper
bounds to the real and imaginary parts of the complex
eigenvalues (or other physical parameters) . More recently
EMM was broadened (exploiting certain quasi-convexity
properties and the generalized eigenvlaue problem) in order to
convexify a multi-extrema plagued procedure in mathematical
physics . We outline the important EMM results achieved over
the last two decades.
1. C. R. Handy and D. Bessis, Rapidly Convergent Lower Bounds
for the Schrodinger Equation Ground State Energy, Phys. Rev.
Lett. 55, 931 (1985).
2. C. R. Handy, D. Bessis, and T. R. Morley, Generating
Quantum Energy Bounds by the Moment Method: A Linear
Programming Approach, Phys. Rev. A 37, 4557 (1988).
3. C. R. Handy, D. Bessis, G. Sigismondi, and T. D. Morley,
Rapidly Converging Bounds for the Ground State Energy of
Hydrogenic Atoms in Superstrong Magnetic Fields, Phys. Rev.
Lett. 60, 253 (1988).
4. C. R. Handy, K. Appiah, and D. Bessis Moment-Problem
Formulation of a Minimax Quantization Procedure, Phys. Rev. A
50, 988 (1994).
5. C. R. Handy, Generating Converging Bounds to the (Complex)
Discrete States of the P2 + iX3 + iaX Hamiltonian, J. Phys.
A: Math. Gen. 34, 5065 (2001).
6. C. R. Handy “(Quasi)-convexification of Barta’s
(multi-extrema) bounding theorem, J. Phys. A: Math. Gen.
39, 3425 (2006)
- SparsePOP and Numerical Results
Sunyoung Kim (Ewha Women's University)
SparesPOP is MATLAB implementation of a sparse semidefinite programming (SDP)
relaxation method proposed for polynomial optimization problems (POPs)
in the recent paper by Waki, Kim, Kojima and Muramatsu. The sparse SDP relaxation
is based on a hierarchy of LMI relaxations of increasing dimensions
by Lasserre, and exploits a sparsity structure of polynomials in POPs.
The efficiency of SparsePOP to compute bounds for optimal values of POPs
is increased and larger scale POPs can be handled. Numerical results are shown
to illustrate the perfomance of SparsePOP.
- Semidefinite Characterization and Computation of Real Radical Ideals
Philipp Rostalski (Eidgenössische TH Zürich-Hönggerberg)
Joint work with J.-B. Lasserre and M. Laurent.
For an ideal
given by a set of generators, h_1...h_m in R[x] a new
characterization of its real radical ideal I(V_R(I))is
provided it is zero-dimensional (even if I is not). Moreover
an algorithm using numerical linear algebra and semidefinite
optimization techniques, to compute all (finitely many)
points of the
real variety V_R=V(I) subset R^n as well as generators of
radical ideal. The latter are obtained in the form of border
bases. The algorithm is based on moment relaxations and, in
contrast to other existing methods, it exploits the real algebraic nature
problem right from the beginning and avoids the computation
of complex components.
- Recent Progress in Applying Semidefinite Optimization to the Satisfiability Problem
Miguel Anjos (University of Waterloo)
Extending the work of de Klerk, Warners and van Maaren, we propose new
semidefinite programming (SDP) relaxations for the satisfiability (SAT)
problem. The SDP relaxations are partial liftings motivated by the
Lasserre hierarchy of SDP relaxations for 0-1 optimization problems.
Theoretical and computational results show that these relaxations have a
number of favourable properties, particularly as a means to prove that a
given SAT formula is unsatisfiable, and that this approach compares
favourably with existing techniques for SAT.
- SeDuMi: A Package for Conic Optimization
Yuriy Zinchenko (McMaster University)
- A PARALLEL Conic Interior Point Decomposition Approach for BLOCK ANGULAR Semidefinite Programs
Kartik Sivaramakrishnan (North Carolina State University)
Semidefinite programs (SDPs) with a BLOCK-ANGULAR structure occur
routinely in practice. In some cases, it is also possible to exploit the
SPARSITY and SYMMETRY in an unstructured SDP,
and preprocess it into an equivalent SDP with a block-angular structure.
We present a PARALLEL CONIC INTERIOR POINT DECOMPOSITION approach to
solve block-angular SDPs. Our aim is to solve such a SDP in an iterative
fashion between a master problem (a quadratic conic program); and
decomposed and distributed subproblems (smaller SDPs) in a parallel
computing environment. We present our computational results with the
algorithm on several test instances; our computations were performed on
the distributed HENRY2 cluster at North Carolina State University.
- Solving Polynomial Systems via LMIs
Graziano Chesi (University of Hong Kong)
Joint work with Y.S. Hung.
The problem of computing the solution of systems of polynomial equalities and inequalities is considered. First, it is shown that the solutions of these systems can be found by looking for vectors with polynomial structure in linear spaces obtained via a convex LMI optimization. Then, it is shown that an upper bound to the dimension of the linear spaces where the sought solutions are looked for can be imposed in a non-conservative way by imposing suitable linear matricial constraints. This allows one to obtain the linear spaces with the smallest dimension, which is important because the solutions can be extracted only if the dimension of the linear spaces is smaller than a certain threshold. Moreover, the proposed approach allows one to improve the numerical accuracy of the extraction mechanism.
- Numerical Optimization Assisted by Noncommutative Symbolic Algebra
Mauricio de Oliveira (University of California, San Diego)
We describe how a symbolic computer algebra tool (NCAlgebra) that can handle symbolic matrix (noncommutative) products is used to numerically solve systems and control problems. Our current focus is on semidefinite programs arising from control theory, where matrix variables appear naturally. Our tools keep matrix variables aggregated at all steps of a primal-dual interior-point algorithm. Symbolic matrix expressions are automatically generated and used on iterative numerical procedures for the determination of search directions, showing promising results.
- Inverse Dynamical Analysis of Gene Networks Using Sparsity-promoting Regularization
James Lu (Johann Radon Institute for Computational and Applied Mathematics )
Given an ODE model for a biological system, the forward problem consists of
determining its solution behavior. However, many biological questions are of
the inverse type: what are the possible dynamics that can arise out of the
model? how is the control mechanism encoded in the topology of the regulatory
We propose inverse dynamical analysis as a methodology for addressing various
questions that arise in studying biological systems, from the initial
modelling to the proposal of new experiments. In addition, once a
satisfactory model has been developed, the method can be used to design
various bifurcation phenotypes that exhibit certain dynamical properties. To
summarize, the proposed methodology consists of the following two inverse
- inverse eigenvalue analysis, to probe and characterize parameter
combinations leading to changes in the qualitative dynamics;
- inverse bifurcation analysis, to identify and design mechanisms that can
give rise to a set of bifurcation phenotypes.
In our work, we use sparsity-promoting regularization functional to 'sparsely'
map dynamical behaviors to parameter sets. In combination with hierarchical
identification strategies, the underlying mechanisms can be elucidated. We
demonstrate some applications, from exploring possible evolutionary scenarios
to analyzing influential interactions in a cell phase transition model.
- inverse eigenvalue analysis, to probe and characterize parameter
- Graphs of Transportation Polytopes
Edward Kim (University of California)
Joint work with Jesus A. de Loera (University of California, Davis).
Transportation polytopes are well-known objects in operations reseach and mathematical programming. These polytopes have very quick
tests for feasiblity, coordinates of a vertex can be quickly determined, and they have nice embedding properties: every polytope can
be viewed as a certain kind of transportation polytope. Using the notion of chamber complex, Gale diagrams, and the theory of secondary
polytopes we are able to exhaustively and systematically enumerate all combinatorial types of nondegenerate transportation polytopes of
small sizes. These generic polytopes (those of maximal dimension whose vertices are simple) will have the largest graph diameters and
vertex counts in their class. Using our exhaustive lists, we give results on some of the conjectures of Yemelichev, Kovalev, and
Kratsov. In particular, this poster focuses on questions related to the 1-skeleton graph of these polyhedra. The study of
1-skeleta of these polytopes are fundamental in attempting to consider the complexity of the simplex method of linear programming.
- Experiments with Linear and Semidefinite Relaxations for Solving the Minimum Graph Bisection Problem
Christoph Helmberg (Technische Universität Chemnitz-Zwickau)
Given a graph with node weights, the convex hull of the incidence
vectors of all cuts satisfying a weight restricition on each side
is called the bisection cut polytope. We study the facial structure
of this polytope which shows up in many graph partitioning problems
with applications in VLSI-design or frequency assignment. We give
necessary and in some cases sufficient conditions for the knapsack
tree inequalities introduced in Ferreira et al. 1996 to be
facet-defining. We extend these inequalities to a richer class by
exploiting that each cut intersects each cycle in an even number of
edges. Furthermore, we present a new class of inequalities that are
based on non-connected substructures yielding non-linear right-hand
sides. We show that the supporting hyperplanes of the convex
envelope of this non-linear function correspond to the faces of the
so-called cluster weight polytope, for which we give a complete
description under certain conditions. Finally, we incorporate
cutting planes algorithms based on the presened inequalities in a
branch-and-cut framework and discuss their interaction with the
linear and semidefinite relaxation.
- Sensor Network Localization, Euclidean Distance Matrix Completions, and Graph Realization
Henry Wolkowicz (University of Waterloo)
We study Semidefinite Programming, SDP, relaxations for Sensor Network
Localization, SNL, with anchors and with noisy distance information. The
main point of the paper is to view SNL as a (nearest) Euclidean Distance
Matrix, EDM, completion problem and to show the advantages for using
this latter, well studied model. We first show that the current popular
SDP relaxation is equivalent to known relaxations in the literature for
EDM completions. The existence of anchors in the problem is not special.
The set of anchors simply corresponds to a given fixed clique for the
graph of the EDM problem. We next propose a method of projection when a
large clique or a dense subgraph is identified in the underlying graph.
This projection reduces the size, and improves the stability, of the
In addition, viewing the problem as an EDM completion problem yields
better low rank approximations for the low dimensional realizations.
And, the projection/reduction procedure can be repeated for other given
cliques of sensors or for sets of sensors, where many distances are
known. Thus, further size reduction can be obtained.
Optimality/duality conditions and a primal-dual interior-exterior path
following algorithm are derived for the SDP relaxations We discuss the
relative stability and strength of two formulations and the
corresponding algorithms that are used. In particular, we show that the
quadratic formulation arising from the SDP relaxation is better
conditioned than the linearized form, that is used in the literature and
that arises from applying a Schur complement.
- Stability Region Analysis Using Simulations and Sum-of-Squares Programming
Ufuk Topcu (University of California, Berkeley)
The problem of computing bounds on the region-of-attraction for systems
with polynomial vector fields is considered. Invariant subsets of the
region-of-attraction are characterized as sublevel sets of Lyapunov
functions. Finite dimensional polynomial parameterizations for Lyapunov
functions are used. A methodology utilizing information from simulations
to generate Lyapunov function candidates satisfying necessary conditions
for bilinear constraints is proposed. The suitability of Lyapunov function
candidates are assessed solving linear sum-of-squares optimization
problems. Qualified candidates are used to compute invariant subsets of
the region-of-attraction and to initialize various bilinear search
strategies for further optimization. We illustrate the method on several
small examples from the literature and a controlled aircraft dynamics
- Computing the Best Low Rank Approximation of a Matrix
Kenneth Driessel (Iowa State University)
Consider the following Problem: Given an m-by-n real matrix
A and a positive integer k, find the m-by-n matrix with rank k
that is closest to A. (I use the Frobenius inner product.)
I shall present a rank-preserving differential equation (d.e.)
in the space of m-by-n real matrices which solves this
problem. In particular, I shall show that if X(t) is a solution
of this d.e. then the distance between X(t) and A decreases;
in other words, this distance function is a Lyapunov function
for the d.e. I shall also show that (generically) this d.e. converges
to a unique stable equilibrium point. This point is the
solution of the given problem.
- An Exact Characterization of Bad Semidefinite Programs
Gabor Pataki (University of North Carolina, Chapel Hill)
SDP's duality theory has been somewhat less well studied than its algorithmic aspects. Strong
duality, — expected in linear programming fails in many cases, and the variety of how
things can go wrong is bewildering: one can have nonattainment in either one of the primal
and the dual problems, attainment on both sides, but a finite duality gap, etc.
The main result we present in this talk is a surprisingly simple, exact,
excluded minor type characterization of all semidefinite systems that have a badly behaved
dual for some objective function.
The characterization is based on some new, fundamental results in convex analysis on the closedness of the linear image of a closed convex cone. In particular, our result is a necessary
condition for the closedness of the linear image
— as opposed to the usual sufficient conditions, such as the existence
of a Slater-point, or polyhedrality. Our conditions are
necessary and sufficient, when the cone belongs to a large class,
called nice cones.
- Advances on the BMV Trace Conjecture
Christopher Hillar (Texas A & M University)
We discuss some progress on a long-standing conjecture in
mathematical physics due to Bessis, Moussa, and Villani (1975). The
statement is enticingly simple (thanks to a reformulation by Elliot Lieb
and Robert Seiringer): For every positive integer m and every pair of
positive semidefinite matrices A and B, the polynomial
p(t) = Tr[(A+tB)m]
has nonnegative coefficients. Our approach allows for several reductions
to this difficult conjecture. For instance, it would be enough to show
that a nonzero (matrix) coefficient (A+tB)m has at least 1 positive
eigenvalue. Additionally, if the conjecture is true for infinitely many
m, then it is true for all m. Finally, two challenges to the SOS
community are proposed: Prove the conjecture in dimension 3 for m = 6
(known) and m = 7 (unknown).