62,964 research outputs found
Constrained Global Optimization by Smoothing
This paper proposes a novel technique called "successive stochastic
smoothing" that optimizes nonsmooth and discontinuous functions while
considering various constraints. Our methodology enables local and global
optimization, making it a powerful tool for many applications. First, a
constrained problem is reduced to an unconstrained one by the exact nonsmooth
penalty function method, which does not assume the existence of the objective
function outside the feasible area and does not require the selection of the
penalty coefficient. This reduction is exact in the case of minimization of a
lower semicontinuous function under convex constraints. Then the resulting
objective function is sequentially smoothed by the kernel method starting from
relatively strong smoothing and with a gradually vanishing degree of smoothing.
The finite difference stochastic gradient descent with trajectory averaging
minimizes each smoothed function locally. Finite differences over stochastic
directions sampled from the kernel estimate the stochastic gradients of the
smoothed functions. We investigate the convergence rate of such stochastic
finite-difference method on convex optimization problems. The "successive
smoothing" algorithm uses the results of previous optimization runs to select
the starting point for optimizing a consecutive, less smoothed function.
Smoothing provides the "successive smoothing" method with some global
properties. We illustrate the performance of the "successive stochastic
smoothing" method on test-constrained optimization problems from the
literature.Comment: 17 pages, 1 tabl
Optimization viewpoint on Kalman smoothing, with applications to robust and sparse estimation
In this paper, we present the optimization formulation of the Kalman
filtering and smoothing problems, and use this perspective to develop a variety
of extensions and applications. We first formulate classic Kalman smoothing as
a least squares problem, highlight special structure, and show that the classic
filtering and smoothing algorithms are equivalent to a particular algorithm for
solving this problem. Once this equivalence is established, we present
extensions of Kalman smoothing to systems with nonlinear process and
measurement models, systems with linear and nonlinear inequality constraints,
systems with outliers in the measurements or sudden changes in the state, and
systems where the sparsity of the state sequence must be accounted for. All
extensions preserve the computational efficiency of the classic algorithms, and
most of the extensions are illustrated with numerical examples, which are part
of an open source Kalman smoothing Matlab/Octave package.Comment: 46 pages, 11 figure
Smoothing with Curvature Constraints based on Boosting Techniques
In many applications it is known that the underlying smooth function is constrained to have a specific form. In the present paper, we propose an estimation method based on the regression spline approach, which allows to include concavity or convexity constraints in an appealing way. Instead of using linear or quadratic programming routines, we handle the required inequality constraints on basis coefficients by boosting techniques. Therefore, recently developed componentwise boosting methods for regression purposes are applied, which allow to control the restrictions in each iteration. The proposed approach is compared to several competitors in a simulation study. We also consider a real world data set
A Smooth Primal-Dual Optimization Framework for Nonsmooth Composite Convex Minimization
We propose a new first-order primal-dual optimization framework for a convex
optimization template with broad applications. Our optimization algorithms
feature optimal convergence guarantees under a variety of common structure
assumptions on the problem template. Our analysis relies on a novel combination
of three classic ideas applied to the primal-dual gap function: smoothing,
acceleration, and homotopy. The algorithms due to the new approach achieve the
best known convergence rate results, in particular when the template consists
of only non-smooth functions. We also outline a restart strategy for the
acceleration to significantly enhance the practical performance. We demonstrate
relations with the augmented Lagrangian method and show how to exploit the
strongly convex objectives with rigorous convergence rate guarantees. We
provide numerical evidence with two examples and illustrate that the new
methods can outperform the state-of-the-art, including Chambolle-Pock, and the
alternating direction method-of-multipliers algorithms.Comment: 35 pages, accepted for publication on SIAM J. Optimization. Tech.
Report, Oct. 2015 (last update Sept. 2016
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