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A Nonsmooth, Nonconvex Optimization Approach to Robust Stabilization by Static Output Feedback and Low-Order Controllers

James V. Burke, Adrian S. Lewis, Michael L. Overton
2003 IFAC Proceedings Volumes  
Finding global optimizers of these functions is hard, so we use a recently developed gradient sampling method that approximates local optimizers.  ...  We formulate two related nonsmooth, nonconvex optimization problems over K, respectively with the following objectives: minimization of the -pseudospectral abscissa of A + BKC, for a fixed ≥ 0, and maximization  ...  We thank Bill Helton for bringing the importance of low-order controllers to our attention.  ... 
doi:10.1016/s1474-6670(17)35659-8 fatcat:yjzwi3wfl5eefda62x2fkxfzqm

Modified Three-Term Conjugate Gradient Method and Its Applications

Jiankun Liu, Shouqiang Du
2019 Mathematical Problems in Engineering  
We propose a modified three-term conjugate gradient method with the Armijo line search for solving unconstrained optimization problems. The proposed method possesses the sufficient descent property.  ...  Due to simplicity, low storage, and nice convergence properties, the proposed method is used to solve M-tensor systems and a kind of nonsmooth optimization problems with l1-norm.  ...  The conjugate gradient method is suitable for solving unconstrained optimization problems.  ... 
doi:10.1155/2019/5976595 fatcat:2fwq5tqwffcrfagstcfqy3xfwu

A new smoothing modified three-term conjugate gradient method for l1$l_{1}$-norm minimization problem

Shouqiang Du, Miao Chen
2018 Journal of Inequalities and Applications  
three-term conjugate gradient method.  ...  The smoothing modified three-term conjugate gradient method is based on Polak-Ribière-Polyak conjugate gradient method.  ...  modified three-term conjugate gradient method for solving l 1 -norm nonsmooth problems.  ... 
doi:10.1186/s13660-018-1696-9 pmid:29755245 pmcid:PMC5934501 fatcat:fle2xx3eljhone5qsg3w7ogtki

Stabilization via Nonsmooth, Nonconvex Optimization

James V. Burke, Didier Henrion, Adrian S. Lewis, Michael L. Overton
2006 IEEE Transactions on Automatic Control  
Nonsmooth variational analysis and computational methods are powerful tools that can be effectively applied to find local minimizers of nonconvex optimization problems arising in fixed-order controller  ...  We support this claim by applying nonsmooth analysis and methods to a challenging "Belgian chocolate" stabilization problem posed in 1994: find a stable, minimum phase, rational controller that stabilizes  ...  These stabilizing controllers were obtained by application of a new numerical method for nonsmooth, nonconvex optimization called gradient sampling.  ... 
doi:10.1109/tac.2006.884944 fatcat:akxutzgwmbcv5edb2o5y3paaau

Smoothing Nonlinear Conjugate Gradient Method for Image Restoration Using Nonsmooth Nonconvex Minimization

Xiaojun Chen, Weijun Zhou
2010 SIAM Journal of Imaging Sciences  
of a sequence generated by this method is a Clarke stationary point of the nonsmooth and nonconvex optimization problem.  ...  Image restoration problems are often converted into large-scale, nonsmooth and nonconvex optimization problems. Most existing minimization methods are not efficient for solving such problems.  ...  Ng for his helpful comments and providing us the code of [24] for numerical test.  ... 
doi:10.1137/080740167 fatcat:xgtearfmyvbt3i22ajw3ahcp3i

Page 1809 of Mathematical Reviews Vol. , Issue 89C [page]

1989 Mathematical Reviews  
Finally, a formula for second-order directional derivatives is derived for a special class of nonsmooth functions.  ...  We consider ba- sic procedures that reduce by 7-10 times the number of sample permutations in the search for the optimal values of an objective function.  ... 

A nonsmooth optimisation approach for the stabilisation of time-delay systems

Joris Vanbiervliet, Koen Verheyden, Wim Michiels, Stefan Vandewalle
2007 E S A I M: Control, Optimisation and Calculus of Variations  
In general, the spectral abscissa is a nonsmooth and nonconvex function, precluding the use of standard optimisation methods.  ...  Instead, we use a recently developed bundle gradient optimisation algorithm which has already been successfully applied to fixed-order controller design problems for systems of ordinary differential equations  ...  Overton for making the matlab implementation of the gradient sampling algorithm described in [6]  ... 
doi:10.1051/cocv:2007060 fatcat:vmwxf5qgsje5lcmojkocn2lo2y

Randomized Smoothing SVRG for Large-scale Nonsmooth Convex Optimization [article]

Wenjie Huang
2018 arXiv   pre-print
We develop and analyze a new algorithm that achieves robust linear convergence rate, and both its time complexity and gradient complexity are superior than state-of-art nonsmooth algorithms and subgradient-based  ...  In this paper, we consider the problem of minimizing the average of a large number of nonsmooth and convex functions.  ...  In ( [Mäkelä, 2002] ), Bundle method has been investigated in nonsmooth optimization.  ... 
arXiv:1805.05189v1 fatcat:7mjcgk7acrg2zgw3nv2tqc5cr4

Accelerated Stochastic Gradient Method for Composite Regularization

Wenliang Zhong, James Tin-Yau Kwok
2014 International Conference on Artificial Intelligence and Statistics  
In this paper, we propose a novel extension with accelerated gradient method for stochastic optimization.  ...  Regularized risk minimization often involves nonsmooth optimization. This can be particularly challenging when the regularizer is a sum of simpler regularizers, as in the overlapping group lasso.  ...  Nesterov pioneered the accelerated gradient descent (AGD) method for smooth optimization, which achieves the optimal convergence rate for a black-box model [12] .  ... 
dblp:conf/aistats/ZhongK14 fatcat:7a5i7mxsjjap3j74xnvc7cutua

Towards an augmented domain decomposition method for nonsmooth contact dynamics models

Vincent Visseq, Pierre Alart, David Dureisseix
2014 Computational Particle Mechanics  
This paper explores the numerical performances of algorithms enriched by an augmented interface problem in a domain decomposition method dedicated to nonsmooth dynamic systems.  ...  Dureisseix, Towards an augmented domain decomposition method for nonsmooth contact dynamics models, Computational Particle Mechanics 1(1):15-26, 2014, The final publication is available at Springer via  ...  Annex: Parallel conjugate gradient The domain decomposition distributed database is herein used in a parallel conjugate gradient algorithm.  ... 
doi:10.1007/s40571-014-0005-8 fatcat:j7rwl4ckabfybadtcubdva6mbu

Better Approximation and Faster Algorithm Using the Proximal Average

Yaoliang Yu
2013 Neural Information Processing Systems  
The new approximation is justified using a recent convex analysis toolproximal average, and yields a novel proximal gradient algorithm that is strictly better than the one based on smoothing, without incurring  ...  We re-examine this powerful methodology and point out a nonsmooth approximation which simply pretends the linearity of the proximal map.  ...  for drawing his attention to the proximal average; and the reviewers for their valuable comments.  ... 
dblp:conf/nips/Yu13a fatcat:6bvsfb2ohrbypho37djwiv3mxa

Proximal Gradient Algorithms: Applications in Signal Processing [article]

Niccolò Antonello, Lorenzo Stella, Panagiotis Patrinos, Toon van Waterschoot
2020 arXiv   pre-print
This paper focuses on the recent enhanced variants of the proximal gradient numerical optimization algorithm, which combine quasi-Newton methods with forward-adjoint oracles to tackle large-scale problems  ...  These proximal gradient algorithms are here described in an easy-to-understand way, illustrating how they are able to address a wide variety of problems arising in signal processing.  ...  nonlinear conjugate gradient or (quasi-)Newton methods [43] .  ... 
arXiv:1803.01621v4 fatcat:aybrk3icwrcxpnvle3eaxmwgw4

Online multi-label learning with accelerated nonsmooth stochastic gradient descent

Sunho Park, Seungjin Choi
2013 2013 IEEE International Conference on Acoustics, Speech and Signal Processing  
In this paper we present a method for online multi-label learning where we minimize the primal form using the accelerated nonsmooth stochastic gradient descent which has been recently developed to extend  ...  The primal form is appealing for the development of online learning but involves a nonsmooth convex loss function.  ...  nonsmooth stochastic gradient descent optimization [17] .  ... 
doi:10.1109/icassp.2013.6638273 dblp:conf/icassp/ParkC13 fatcat:zfc2gebh3neiljnpr3vranq77a

Preface

Efim A. Galperin
2006 Computers and Mathematics with Applications  
It will be interesting for the reader to compare gradient methods developed in this paper with nongradient approaches presented elsewhere in this issue.  ...  The next paper presents a new notion of distance uniformity and a fast algorithm for generating uniformly distributed small samples of points in R n by using the combination of a popular 0898-1221/06/$  ...  Global optimization is performed by combining evolutionary global search with "genetic operators" and a local descent (conjugate gradient local search, or Powell's local search).  ... 
doi:10.1016/j.camwa.2006.08.001 fatcat:rinpkoy655c6rkshbbjj7p5vty

A BFGS-SQP method for nonsmooth, nonconvex, constrained optimization and its evaluation using relative minimization profiles

Frank E. Curtis, Tim Mitchell, Michael L. Overton
2016 Optimization Methods and Software  
gradient sampling (GS) technique of Burke et al. [9] for unconstrained nonsmooth optimization.  ...  For general constrained nonsmooth, nonconvex optimization problems-i.e. where p ≥ 1 in (1)-the aforementioned sequential quadratic optimization approach employing gradient sampling (SQP-GS) was presented  ...  Acknowledgements The authors are grateful to the two anonymous referees for carefully reading the paper and for their helpful comments.  ... 
doi:10.1080/10556788.2016.1208749 fatcat:jyxhhnstnrgrpchu5hm4bw2p2q
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