641 research outputs found
A Convex Feasibility Approach to Anytime Model Predictive Control
This paper proposes to decouple performance optimization and enforcement of
asymptotic convergence in Model Predictive Control (MPC) so that convergence to
a given terminal set is achieved independently of how much performance is
optimized at each sampling step. By embedding an explicit decreasing condition
in the MPC constraints and thanks to a novel and very easy-to-implement convex
feasibility solver proposed in the paper, it is possible to run an outer
performance optimization algorithm on top of the feasibility solver and
optimize for an amount of time that depends on the available CPU resources
within the current sampling step (possibly going open-loop at a given sampling
step in the extreme case no resources are available) and still guarantee
convergence to the terminal set. While the MPC setup and the solver proposed in
the paper can deal with quite general classes of functions, we highlight the
synthesis method and show numerical results in case of linear MPC and
ellipsoidal and polyhedral terminal sets.Comment: 8 page
Oracle-Based Robust Optimization via Online Learning
Robust optimization is a common framework in optimization under uncertainty
when the problem parameters are not known, but it is rather known that the
parameters belong to some given uncertainty set. In the robust optimization
framework the problem solved is a min-max problem where a solution is judged
according to its performance on the worst possible realization of the
parameters. In many cases, a straightforward solution of the robust
optimization problem of a certain type requires solving an optimization problem
of a more complicated type, and in some cases even NP-hard. For example,
solving a robust conic quadratic program, such as those arising in robust SVM,
ellipsoidal uncertainty leads in general to a semidefinite program. In this
paper we develop a method for approximately solving a robust optimization
problem using tools from online convex optimization, where in every stage a
standard (non-robust) optimization program is solved. Our algorithms find an
approximate robust solution using a number of calls to an oracle that solves
the original (non-robust) problem that is inversely proportional to the square
of the target accuracy
Large-scale Binary Quadratic Optimization Using Semidefinite Relaxation and Applications
In computer vision, many problems such as image segmentation, pixel
labelling, and scene parsing can be formulated as binary quadratic programs
(BQPs). For submodular problems, cuts based methods can be employed to
efficiently solve large-scale problems. However, general nonsubmodular problems
are significantly more challenging to solve. Finding a solution when the
problem is of large size to be of practical interest, however, typically
requires relaxation. Two standard relaxation methods are widely used for
solving general BQPs--spectral methods and semidefinite programming (SDP), each
with their own advantages and disadvantages. Spectral relaxation is simple and
easy to implement, but its bound is loose. Semidefinite relaxation has a
tighter bound, but its computational complexity is high, especially for large
scale problems. In this work, we present a new SDP formulation for BQPs, with
two desirable properties. First, it has a similar relaxation bound to
conventional SDP formulations. Second, compared with conventional SDP methods,
the new SDP formulation leads to a significantly more efficient and scalable
dual optimization approach, which has the same degree of complexity as spectral
methods. We then propose two solvers, namely, quasi-Newton and smoothing Newton
methods, for the dual problem. Both of them are significantly more efficiently
than standard interior-point methods. In practice, the smoothing Newton solver
is faster than the quasi-Newton solver for dense or medium-sized problems,
while the quasi-Newton solver is preferable for large sparse/structured
problems. Our experiments on a few computer vision applications including
clustering, image segmentation, co-segmentation and registration show the
potential of our SDP formulation for solving large-scale BQPs.Comment: Fixed some typos. 18 pages. Accepted to IEEE Transactions on Pattern
Analysis and Machine Intelligenc
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