arXiv subjects

Mattias Fält

Publications and source records attributed to Mattias Fält.

8 recordsLinked to original sources

arXiv2023

Generalized Alternating Projections on Manifolds and Convex Sets

In this paper, we extend the previous convergence results for the generalized alternating projection method applied to subspaces in [arXiv:1703.10547] to hold also for smooth manifolds. We show that the algorithm locally behaves similarly in the subspace and manifold settings and that the same rates are obtained. We also present convergence rate results for when the algorithm is applied to non-empty, closed, and convex sets. The results are based on a finite identification property that implies that the algorithm after an initial identification phase solves a smooth manifold feasibility problem. Therefore, the rates in this paper hold asymptotically for problems in which this identification property is satisfied. We present a few examples where this is the case and also a counter example for when this is not.

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arXiv2019

QPDAS: Dual Active Set Solver for Mixed Constraint Quadratic Programming

We present a method for solving the general mixed constrained convex quadratic programming problem using an active set method on the dual problem. The approach is similar to existing active set methods, but we present a new way of solving the linear systems arising in the algorithm. There are two main contributions; we present a new way of factorizing the linear systems, and show how iterative refinement can be used to achieve good accuracy and to solve both types of sub-problems that arise from semi-definite problems.

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arXiv2019

System Identification for Hybrid Systems using Neural Networks

With new advances in machine learning and in particular powerful learning libraries, we illustrate some of the new possibilities they enable in terms of nonlinear system identification. For a large class of hybrid systems, we explain how these tools allow for identification of complex dynamics using neural networks. We illustrate the method by examining the performance on a quad-rotor example.

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arXiv2018

A Refined Algorithm for Curve Fitting by Segmented Straight Lines

We consider least squares approximation of a function of one variable by a continuous, piecewise-linear approximand that has a small number of breakpoints. This problem was notably considered by Bellman who proposed an approximate algorithm based on dynamic programming. Many suboptimal approaches have been suggested, but so far, the only exact methods resort to mixed integer programming with superpolynomial complexity growth. In this paper, we present an exact and efficient algorithm based on dynamic programming with a hybrid value function. The achieved time-complexity seems to be polynomial.

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arXiv2017

Envelope Functions: Unifications and Further Properties

Recently, the forward-backward and Douglas-Rachford envelope functions were proposed in the literature. The stationary points of these envelope functions have a close relationship with the solutions of the possibly nonsmooth optimization problem to be solved. The envelopes were shown to be smooth and convex under some additional assumptions. Therefore, these envelope functions create powerful bridges between nonsmooth and smooth optimization. In this paper, we present a general envelope function that unifies and generalizes these envelope functions. We provide properties of the general envelope function that sharpen corresponding known results for the special cases. We also present an envelope function for the generalized alternating projections method (GAP), named the GAP envelope. It enables for convex feasibility problems with two sets, of which one is affine, to be solved by finding any stationary point of the smooth and under some assumptions convex GAP envelope.

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arXiv2017

Optimal Convergence Rates for Generalized Alternating Projections

Generalized alternating projections is an algorithm that alternates relaxed projections onto a finite number of sets to find a point in their intersection. We consider the special case of two linear subspaces, for which the algorithm reduces to a matrix teration. For convergent matrix iterations, the asymptotic rate is linear and decided by the magnitude of the subdominant eigenvalue. In this paper, we show how to select the three algorithm parameters to optimize this magnitude, and hence the asymptotic convergence rate. The obtained rate depends on the Friedrichs angle between the subspaces and is considerably better than known rates for other methods such as alternating projections and Douglas-Rachford splitting. We also present an adaptive scheme that, online, estimates the Friedrichs angle and updates the algorithm parameters based on this estimate. A numerical example is provided that supports our theoretical claims and shows very good performance for the adaptive method.

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arXiv2016

Line Search For Generalized Alternating Projections

This paper is about line search for the generalized alternating projections (GAP) method. This method is a generalization of the von Neumann alternating projections method, where instead of performing alternating projections, relaxed projections are alternated. The method can be interpreted as an averaged iteration of a nonexpansive mapping. Therefore, a recently proposed line search method for such algorithms is applicable to GAP. We evaluate this line search and show situations when the line search can be performed with little additional cost. We also present a variation of the basic line search for GAP - the projected line search. We prove its convergence and show that the line search condition is convex in the step length parameter. We show that almost all convex optimization problems can be solved using this approach and numerical results show superior performance with both the standard and the projected line search, sometimes by several orders of magnitude, compared to the nominal method.

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arXiv2016

Line Search for Averaged Operator Iteration

Many popular first order algorithms for convex optimization, such as forward-backward splitting, Douglas-Rachford splitting, and the alternating direction method of multipliers (ADMM), can be formulated as averaged iteration of a nonexpansive mapping. In this paper we propose a line search for averaged iteration that preserves the theoretical convergence guarantee, while often accelerating practical convergence. We discuss several general cases in which the additional computational cost of the line search is modest compared to the savings obtained.

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