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An active set algorithm for nonlinear optimization with polyhedral constraints

Science China Mathematics 2016年第8期59卷 1525-1542页

作者： HAGER William W. ZHANG Hongchao Department of Mathematics University of FloridaGainesvilleFL 32611-8105USA Department of Mathematics Louisiana State UniversityBaton RougeLA 70803-4918USA

A polyhedral active set algorithm PASA is developed for solving a nonlinear optimization problem whose feasible set is a polyhedron. Phase one of the algorithm is the gradient projection method, while phase two is any algorithm for solving a linearly constrained optimization problem. Rules are provided for branching between the two phases. Global convergence to a stationary point is established, while asymptotically PASA performs only phase two when either a nondegeneracy assumption holds, or the active constraints are linearly independent and a strong second-order sufficient optimality condition holds.

关键词： polyhedral constrained optimization active set algorithm PASA gradient projection algorithm local and global convergence

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An active set algorithm to estimate parameters in generalized linear models with ordered predictors

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COMPUTATIONAL STATISTICS & DATA ANALYSIS 2010年第6期54卷 1442-1456页

作者： Rufibach, Kaspar Univ Zurich Biostat Unit Inst Social & Prevent Med CH-8001 Zurich Switzerland

In biomedical studies, researchers are often interested in assessing the association between one or more ordinal explanatory variables and an outcome variable, at the same time adjusting for covariates of any type. The outcome variable may be continuous, binary, or represent censored survival times. In the absence of precise knowledge of the response function, using monotonicity constraints on the ordinal variables improves efficiency in estimating parameters, especially when sample sizes are small. An active set algorithm that can efficiently compute such estimators is proposed, and a characterization of the solution is provided. Flaying an efficient algorithm at hand is especially relevant when applying likelihood ratio tests in restricted generalized linear models, where one needs the value of the likelihood at the restricted maximizer. The algorithm is illustrated on a real life data set from oncology. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Ordered explanatory variable Constrained estimation Least-squares Logistic regression Cox regression active set algorithm Likelihood ratio test under linear constraints

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A new active set algorithm for box constrained optimization

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SIAM JOURNAL ON OPTIMIZATION 2006年第2期17卷 526-557页

作者： Hager, William W. Zhang, Hongchao Univ Florida Dept Math Gainesville FL 32611 USA

An active set algorithm (ASA) for box constrained optimization is developed. The algorithm consists of a nonmonotone gradient projection step, an unconstrained optimization step, and a set of rules for branching between the two steps. Global convergence to a stationary point is established. For a nondegenerate stationary point, the algorithm eventually reduces to unconstrained optimization without restarts. Similarly, for a degenerate stationary point, where the strong second-order sufficient optimality condition holds, the algorithm eventually reduces to unconstrained optimization without restarts. A specific implementation of the ASA is given which exploits the recently developed cyclic Barzilai - Borwein (CBB) algorithm for the gradient projection step and the recently developed conjugate gradient algorithm CG_DESCENT for unconstrained optimization. Numerical experiments are presented using box constrained problems in the CUTEr and MINPACK-2 test problem libraries.

关键词： nonmonotone gradient projection box constrained optimization active set algorithm ASA cyclic BB method CBB conjugate gradient method CG_DESCENT degenerate optimization

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PMSM Drive system Control Based on Improved active set algorithm

PMSM Drive system Control Based on Improved Active Set Algor...

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20th Chinese Control and Decision Conference

作者： Chen, Shujin Wu, Mingfang Lai, Zhongmin Jiangsu Univ Sci & Technol Prov Key Lab Adv Welding Technol Zhenjiang 212003 Peoples R China

ISBN: (纸本)9781424417339

In order to obtain excellent varied speed performance of welding robot, the predictive model of motor drive system was established. On the basis of the analysis of the permanent magnet synchronous motor drive system, generalized predictive control method was adopted to control the rotor speed and position of the rotor. Because of the input and output constraint of the system, convex quadratic problem was proposed, and then an active set algorithm was adopted to deal with these constrained input and output. At last, the performance of this method is demonstrated by a real-time implementation using a digital signal processor (DSP) chip on a permanent-magnet synchronous motor with sinusoidal flux distribution. Theoretic analysis and experiments verify the validity and feasibility of this approach.

关键词： Generalized Predictive Control active set algorithm Motor Drive System Welding Robot

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Group Control Strategy of Welding Machine Based on Improved active set algorithm

Group Control Strategy of Welding Machine Based on Improved ...

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The 2011 International Conference on Electric and Electronics(EEIC 2011)

作者： Chen Shujin Zhang Junjun Provincial key Lab of Advanced Welding Technology Jiangsu University of Science and Technology

In order to solve the group optimal control problem on welding machine of shipyard, this paper established the optimization model of operating parameters. Optimization object includes minimum the average waiting time of welding, the largest number of online welding machine and minimum transformer energy consumption. Group control system has nonlinear, multiconstraint characteristics, but the inequality constraints in this study are linear. When quadratic programming problem was solved by improved active set algorithm, the overall control strategy was obtained, which can achieve objective optimization by priority scheduling, multi-step optimization strategy, guaranteeing the production quality at the same time. This research provides a theoretical basis and foundation technology for the stable operation of welding machine group.

关键词： Welding machine Group control active set algorithm

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Archetypal Analysis for Binary Data

Archetypal Analysis for Binary Data

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2025 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2025

作者： Wedenborg, A. Emilie J. Mørup, Morten The Technical University of Denmark DTU Compute - Cognitive Systems Section Denmark

ISBN: (纸本)9798350368741

Archetypal analysis (AA) is a matrix decomposition method that identifies distinct patterns using convex combinations of the data points denoted archetypes with each data point in turn reconstructed as convex combinations of the archetypes. AA thereby forms a polytope representing trade-offs of the distinct aspects in the data. Most existing methods for AA are designed for continuous data and do not exploit the structure of the data distribution. In this paper, we propose two new optimization frameworks for archetypal analysis for binary data. i) A second order approximation of the AA likelihood based on the Bernoulli distribution with efficient closed-form updates using an active set procedure for learning the convex combinations defining the archetypes, and a sequential minimal optimization strategy for learning the observation specific reconstructions. ii) A Bernoulli likelihood based version of the principal convex hull analysis (PCHA) algorithm originally developed for least squares optimization. We compare these approaches with the only existing binary AA procedure relying on multiplicative updates and demonstrate their superiority on both synthetic and real binary data. Notably, the proposed optimization frameworks for AA can easily be extended to other data distributions providing generic efficient optimization frameworks for AA based on tailored likelihood functions reflecting the underlying data distribution. © 2025 IEEE.

关键词： active set algorithm Archetypal Analysis principal convex hull quadratic programming

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The primal-dual active set method for a crack problem with non-penetration

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IMA JOURNAL OF APPLIED MATHEMATICS 2004年第1期69卷 1-26页

作者： Hintermüller, M Kovtunenko, VA Kunisch, K Graz Univ Inst Math A-8010 Graz Austria MA Lavrentyev Hydrodynam Inst Novosibirsk 630090 Russia

The Lame problem in a 2D domain with a crack under a non-penetration condition is considered as a variational inequality. A primal-dual active set method is proposed as an efficient numerical solution technique and compared to a previously employed iterative method for a penalized formulation. Sufficient conditions for monotonic convergence of a discretized version of the proposed algorithm are given and numerical experiments are presented.

关键词： crack non-penetration variational inequality active set algorithm numerical simulation

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Error estimation in nonlinear optimization

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JOURNAL OF GLOBAL OPTIMIZATION 2014年第2-3期59卷 327-341页

作者： Hager, William W. Mico-Umutesi, Delphine Univ Florida Dept Math Gainesville FL 32611 USA

Methods are developed and analyzed for estimating the distance to a local minimizer of a nonlinear programming problem. One estimate, based on the solution of a constrained convex quadratic program, can be used when strict complementary slackness and the second-order sufficient optimality conditions hold. A second estimate, based on the solution of an unconstrained nonconvex, nonsmooth optimization problem, is valid even when strict complementary slackness is violated. Both estimates are valid in a neighborhood of a local minimizer. An active set algorithm is developed for computing a stationary point of the nonsmooth error estimator. Each iteration of the algorithm requires the solution of a symmetric, positive semidefinite linear system, followed by a line search. Convergence is achieved in a finite number of iterations. The error bounds are based on stability properties for nonlinear programs. The theory is illustrated by some numerical examples.

关键词： Error bounds KKT conditions active set algorithm Nonconvex quadratic programming Nonlinear programming

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Structured Variable Selection with Sparsity-Inducing Norms

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JOURNAL OF MACHINE LEARNING RESEARCH 2011年第10期12卷 2777-2824页

作者： Jenatton, Rodolphe Audibert, Jean-Yves Bach, Francis Ecole Normale Super INRIA SIERRA Project Team Lab Informat INRIA ENS CNRS UMR 8548 F-75214 Paris France Univ Paris Est Imagine ENPC CSTB Ecole Normale Super Lab InformatINRIA ENS CNRS UMR 8548 F-77455 Marne La Vallee France

We consider the empirical risk minimization problem for linear supervised learning, with regularization by structured sparsity-inducing norms. These are defined as sums of Euclidean norms on certain subsets of variables, extending the usual l(1)-norm and the group l(1)-norm by allowing the subsets to overlap. This leads to a specific set of allowed nonzero patterns for the solutions of such problems. We first explore the relationship between the groups defining the norm and the resulting nonzero patterns, providing both forward and backward algorithms to go back and forth from groups to patterns. This allows the design of norms adapted to specific prior knowledge expressed in terms of nonzero patterns. We also present an efficient active set algorithm, and analyze the consistency of variable selection for least-squares linear regression in low and high-dimensional settings.

关键词： sparsity consistency variable selection convex optimization active set algorithm

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Numerical Simulations of Vaccination and Wolbachia on Dengue Transmission Dynamics in the Nonlinear Model

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IEEE ACCESS 2022年 10卷 31116-31144页

作者： Junsawang, Prem Zuhra, Samina Sabir, Zulqurnain Raja, Muhammad Asif Zahoor Shoaib, Muhammad Botmart, Thongchai Weera, Wajaree Khon Kaen Univ Fac Sci Dept Stat Khon Kaen 40002 Thailand Abasyn Univ Dept Comp Peshawar 25000 Pakistan Hazara Univ Dept Math & Stat Mansehra 21120 Pakistan Natl Yunlin Univ Sci & Technol Future Technol Res Ctr Touliu 64002 Yunlin Taiwan COMSATS Univ Islamabad Dept Math Attock Campus Attock 43600 Pakistan Khon Kaen Univ Fac Sci Dept Math Khon Kaen 40002 Thailand

In this study, it is indicated that the world can get rid of the dengue virus by using vaccines and Wolbachia. In many findings, it is observed that Wolbachia therapy is efficacious in those regions that display the minimal to moderate the transmission level. On the contrary, vaccination is highly successful when used in serologically persons and places with large transmission levels. The resilience of stochastic methodology based on the numerical computing schemes will be used to exploit the artificial neural networks (ANNs) modelling legacy, as well as the metaheuristic intelligence using the hybrid of global and local search schemes thru genetic algorithms (GAs) and active-set method (ASA). The combination of both strategies is used to manage the numerical therapies of the mathematical form of the dengue model. The optimal control results through GA-ASA can be retrieved by offering an error-based fitness function generated for dengue model represented via nonlinear systems of equations. The acquired findings are compared to the Adams numerical results to ensure that the suggested stochastic system is accurate. For determining convergence, the training contours are based on various contact rate values. Furthermore, the statistical achievements of the suggested stochastic scheme to solve the novel developed dengue model, which demonstrate the stability and dependability of the dynamical system scheme.

关键词： Viruses (medical) Mathematical models Vaccines Diseases Numerical models Statistics Sociology Dengue model vaccination Wolbachia treatment genetic algorithm active set algorithm

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