31 research outputs found
Lagrangian Duality in 3D SLAM: Verification Techniques and Optimal Solutions
State-of-the-art techniques for simultaneous localization and mapping (SLAM)
employ iterative nonlinear optimization methods to compute an estimate for
robot poses. While these techniques often work well in practice, they do not
provide guarantees on the quality of the estimate. This paper shows that
Lagrangian duality is a powerful tool to assess the quality of a given
candidate solution. Our contribution is threefold. First, we discuss a revised
formulation of the SLAM inference problem. We show that this formulation is
probabilistically grounded and has the advantage of leading to an optimization
problem with quadratic objective. The second contribution is the derivation of
the corresponding Lagrangian dual problem. The SLAM dual problem is a (convex)
semidefinite program, which can be solved reliably and globally by
off-the-shelf solvers. The third contribution is to discuss the relation
between the original SLAM problem and its dual. We show that from the dual
problem, one can evaluate the quality (i.e., the suboptimality gap) of a
candidate SLAM solution, and ultimately provide a certificate of optimality.
Moreover, when the duality gap is zero, one can compute a guaranteed optimal
SLAM solution from the dual problem, circumventing non-convex optimization. We
present extensive (real and simulated) experiments supporting our claims and
discuss practical relevance and open problems.Comment: 10 pages, 4 figure
A Semi-Definite Programming Approach to Stability Analysis of Linear Partial Differential Equations
We consider the stability analysis of a large class of linear 1-D PDEs with
polynomial data. This class of PDEs contains, as examples, parabolic and
hyperbolic PDEs, PDEs with boundary feedback and systems of in-domain/boundary
coupled PDEs. Our approach is Lyapunov based which allows us to reduce the
stability problem to the verification of integral inequalities on the subspaces
of Hilbert spaces. Then, using fundamental theorem of calculus and Green's
theorem, we construct a polynomial problem to verify the integral inequalities.
Constraining the solution of the polynomial problem to belong to the set of
sum-of-squares polynomials subject to affine constraints allows us to use
semi-definite programming to algorithmically construct Lyapunov certificates of
stability for the systems under consideration. We also provide numerical
results of the application of the proposed method on different types of PDEs
Scaling the semidefinite program solver SDPB
We present enhancements to SDPB, an open source, parallelized, arbitrary
precision semidefinite program solver designed for the conformal bootstrap. The
main enhancement is significantly improved performance and scalability using
the Elemental library and MPI. The result is a new version of SDPB that runs on
multiple nodes with hundreds of cores with excellent scaling, making it
practical to solve larger problems. We demonstrate performance on a
moderate-size problem in the 3d Ising CFT and a much larger problem in the
Model.Comment: 13 pages plus references, 2 figure
SDPNAL+: A Matlab software for semidefinite programming with bound constraints (version 1.0)
SDPNAL+ is a {\sc Matlab} software package that implements an augmented
Lagrangian based method to solve large scale semidefinite programming problems
with bound constraints. The implementation was initially based on a majorized
semismooth Newton-CG augmented Lagrangian method, here we designed it within an
inexact symmetric Gauss-Seidel based semi-proximal ADMM/ALM (alternating
direction method of multipliers/augmented Lagrangian method) framework for the
purpose of deriving simpler stopping conditions and closing the gap between the
practical implementation of the algorithm and the theoretical algorithm. The
basic code is written in {\sc Matlab}, but some subroutines in C language are
incorporated via Mex files. We also design a convenient interface for users to
input their SDP models into the solver. Numerous problems arising from
combinatorial optimization and binary integer quadratic programming problems
have been tested to evaluate the performance of the solver. Extensive numerical
experiments conducted in [Yang, Sun, and Toh, Mathematical Programming
Computation, 7 (2015), pp. 331--366] show that the proposed method is quite
efficient and robust, in that it is able to solve 98.9\% of the 745 test
instances of SDP problems arising from various applications to the accuracy of
in the relative KKT residual