113 research outputs found
A Parametric Non-Convex Decomposition Algorithm for Real-Time and Distributed NMPC
A novel decomposition scheme to solve parametric non-convex programs as they
arise in Nonlinear Model Predictive Control (NMPC) is presented. It consists of
a fixed number of alternating proximal gradient steps and a dual update per
time step. Hence, the proposed approach is attractive in a real-time
distributed context. Assuming that the Nonlinear Program (NLP) is
semi-algebraic and that its critical points are strongly regular, contraction
of the sequence of primal-dual iterates is proven, implying stability of the
sub-optimality error, under some mild assumptions. Moreover, it is shown that
the performance of the optimality-tracking scheme can be enhanced via a
continuation technique. The efficacy of the proposed decomposition method is
demonstrated by solving a centralised NMPC problem to control a DC motor and a
distributed NMPC program for collaborative tracking of unicycles, both within a
real-time framework. Furthermore, an analysis of the sub-optimality error as a
function of the sampling period is proposed given a fixed computational power.Comment: 16 pages, 9 figure
Low Complexity Regularization of Linear Inverse Problems
Inverse problems and regularization theory is a central theme in contemporary
signal processing, where the goal is to reconstruct an unknown signal from
partial indirect, and possibly noisy, measurements of it. A now standard method
for recovering the unknown signal is to solve a convex optimization problem
that enforces some prior knowledge about its structure. This has proved
efficient in many problems routinely encountered in imaging sciences,
statistics and machine learning. This chapter delivers a review of recent
advances in the field where the regularization prior promotes solutions
conforming to some notion of simplicity/low-complexity. These priors encompass
as popular examples sparsity and group sparsity (to capture the compressibility
of natural signals and images), total variation and analysis sparsity (to
promote piecewise regularity), and low-rank (as natural extension of sparsity
to matrix-valued data). Our aim is to provide a unified treatment of all these
regularizations under a single umbrella, namely the theory of partial
smoothness. This framework is very general and accommodates all low-complexity
regularizers just mentioned, as well as many others. Partial smoothness turns
out to be the canonical way to encode low-dimensional models that can be linear
spaces or more general smooth manifolds. This review is intended to serve as a
one stop shop toward the understanding of the theoretical properties of the
so-regularized solutions. It covers a large spectrum including: (i) recovery
guarantees and stability to noise, both in terms of -stability and
model (manifold) identification; (ii) sensitivity analysis to perturbations of
the parameters involved (in particular the observations), with applications to
unbiased risk estimation ; (iii) convergence properties of the forward-backward
proximal splitting scheme, that is particularly well suited to solve the
corresponding large-scale regularized optimization problem
A Gradually Reinforced Sample-Average-Approximation Differentiable Homotopy Method for a System of Stochastic Equations
This paper intends to apply the sample-average-approximation (SAA) scheme to
solve a system of stochastic equations (SSE), which has many applications in a
variety of fields. The SAA is an effective paradigm to address risks and
uncertainty in stochastic models from the perspective of Monte Carlo principle.
Nonetheless, a numerical conflict arises from the sample size of SAA when one
has to make a tradeoff between the accuracy of solutions and the computational
cost. To alleviate this issue, we incorporate a gradually reinforced SAA scheme
into a differentiable homotopy method and develop a gradually reinforced
sample-average-approximation (GRSAA) differentiable homotopy method in this
paper. By introducing a series of continuously differentiable functions of the
homotopy parameter ranging between zero and one, we establish a
differentiable homotopy system, which is able to gradually increase the sample
size of SAA as descends from one to zero. The set of solutions to the
homotopy system contains an everywhere smooth path, which starts from an
arbitrary point and ends at a solution to the SAA with any desired accuracy.
The GRSAA differentiable homotopy method serves as a bridge to link the
gradually reinforced SAA scheme and a differentiable homotopy method and
retains the nice property of global convergence the homotopy method possesses
while greatly reducing the computational cost for attaining a desired solution
to the original SSE. Several numerical experiments further confirm the
effectiveness and efficiency of the proposed method
Optimization with Sparsity-Inducing Penalties
Sparse estimation methods are aimed at using or obtaining parsimonious
representations of data or models. They were first dedicated to linear variable
selection but numerous extensions have now emerged such as structured sparsity
or kernel selection. It turns out that many of the related estimation problems
can be cast as convex optimization problems by regularizing the empirical risk
with appropriate non-smooth norms. The goal of this paper is to present from a
general perspective optimization tools and techniques dedicated to such
sparsity-inducing penalties. We cover proximal methods, block-coordinate
descent, reweighted -penalized techniques, working-set and homotopy
methods, as well as non-convex formulations and extensions, and provide an
extensive set of experiments to compare various algorithms from a computational
point of view
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