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An active set strategy for solving optimization problems with up to 200,000,000 nonlinear constraints

APPLIED NUMERICAL MATHEMATICS 2009年第12期59卷 2999-3007页

作者： Schittkowski, Klaus Univ Bayreuth Dept Comp Sci D-95440 Bayreuth Germany

Numerical test results are presented for solving smooth nonlinear programming problems with a large number of constraints, but a moderate number of variables. The active set method proceeds from a given bound for the maximum number of expected active constraints at an optimal solution, which must be less than the total number of constraints. A quadratic programming subproblem is generated with a reduced number of linear constraints from the so-called working set, which is internally changed from one iterate to the next. Only for active constraints, i.e., a certain subset of the working set, flew gradient values must be computed. The line search is adapted to avoid too many active constraints which do not fit into the working set. The active set strategy is an extension of an algorithm described earlier by the author together with a rigorous convergence proof. Numerical results for some simple academic test problems show that nonlinear programs with up to 200,000,000 nonlinear constraints are efficiently solved on a standard PC. (C) 2009 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： SQP sequential quadratic programming Nonlinear programming Many constraints Active set strategy

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Design of Mexican Hat Wavelet neural networks for solving Bratu type nonlinear systems

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NEUROCOMPUTING 2017年 221卷 1-14页

作者： Masood, Zaheer Majeed, Khalid Samar, Raza Raja, Muhammad Asif Zahoor Muhammad Ali Jinnah Univ Dept Elect Engn Islamabad Pakistan COMSATS Inst Informat Technol Dept Elect Engn Attock Pakistan

In this paper a Mexican Hat Wavelet based neural network is designed and applied for solving the nonlinear Bratu type equation. This equation is widely used in fuel ignition models, electrically conducting solids and heat transfer studies. The Mexican Hat Wavelet Differential equation artificial neural networks (MHW-DEANN) are used for the first time to construct an energy function of the system in an unsupervised manner. The tunable parameters of MHW-DEANN are trained with a hybrid evolutionary computing approach: we exploit the strength of Genetic Algorithms (GA) and sequential quadratic programming (SQP) to find the best weights. Monte-Carlo simulations are performed for the proposed scheme with statistical analysis to validate the effectiveness and convergence of the proposed method for Bratu-type equations. It is observed that the proposed method converges in all cases and can solve the equation with high accuracy and reliability.

关键词： Wavelet-based neural networks, fuel ignition model Genetic algorithm sequential quadratic programming Electrical conducting solids Boundary value problems

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Exactly satisfying initial conditions neural network models for numerical treatment of first Painleve equation

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APPLIED SOFT COMPUTING 2015年 26卷 244-256页

作者： Raja, Muhammad Asif Zahoor Khan, Junaid Ali Siddiqui, A. M. Behloul, D. Haroon, T. Samar, Raza COMSATS Inst Informat Technol Dept Elect Engn Attock Pakistan Hamdard Univ Fac Engn Sci & Technol Karachi Pakistan Penn State Univ Dept Math York PA 17403 USA USTHB Dept Comp Sci Algiers 16111 Algeria COMSATS Inst Informat Technol Dept Math Islamabad Pakistan Mohammad Ali Jinnah Univ Dept Elect Engn Islamabad Pakistan

In this paper, novel computing approach using three different models of feed-forward artificial neuralnetworks (ANNs) are presented for the solution of initial value problem (IVP) based on first Painleve equation. These mathematical models of ANNs are developed in an unsupervised manner with capability to satisfy the initial conditions exactly using log-sigmoid, radial basis and tan-sigmoid transfer functions in hidden layers to approximate the solution of the problem. The training of design parameters in each model is performed with sequential quadratic programming technique. The accuracy, convergence and effectiveness of the proposed schemes are evaluated on the basis of the results of statistical analyses through sufficient large number of independent runs with different number of neurons in each model as well. The comparisons of these results of proposed schemes with standard numerical and analytical solutions validate the correctness of the design models. (C) 2014 Elsevier B.V. All rights reserved.

关键词： Painleve transcendents Neural network sequential quadratic programming Nonlinear differential equations Activation functions

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Momentum Model-Based Minimal Parameter Identification of a Space Robot

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JOURNAL OF GUIDANCE CONTROL AND DYNAMICS 2019年第3期42卷 508-523页

作者： Naveen, B. Shah, Suril V. Misra, Arun K. Indian Inst Technol Dept Mech Engn NH 65 Nagaur Rd Jodhpur 342037 Rajasthan India McGill Univ Dept Mech Engn 817 Sherbrooke St West Montreal PQ H3A 0C3 Canada

Accurate information of inertial parameters is critical to motion planning and control of space robots. Before the launch, only a rudimentary estimate of the inertial parameters is available from experiments and CAD models. After the launch, on-orbit operations substantially alter the value of inertial parameters. In this work, a new momentum model-based method is proposed for identifying the minimal parameters of a space robot while on orbit. Minimal parameters are combinations of the inertial parameters of the links and uniquely define the momentum and dynamic models. Consequently, they are sufficient for motion planning and control of both the satellite and robotic arms mounted on it. The key to the proposed framework is the unique formulation of the momentum model in the linear form of minimal parameters. Further, to estimate the minimal parameters, a novel joint trajectory planning and optimization technique based on direction combinations of joints' velocity are proposed. The efficacy of the identification framework is demonstrated on a 12-degree-of-freedom, spatial, dual-arm space robot. The methodology is developed for tree-type space robots, requires just the pose and twist data, and is scalable with increasing number of joints.

关键词： Multi Arm Space Robots Motion Planning Satellites Robots Reaction Wheels Computing Trajectory Optimization sequential quadratic programming Particle Swarm Optimization Rigid Body Kinematics

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Maximum Volume Clustering: A New Discriminative Clustering Approach

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JOURNAL OF MACHINE LEARNING RESEARCH 2013年 14卷 2641-2687页

作者： Niu, Gang Dai, Bo Shang, Lin Sugiyama, Masashi Tokyo Inst Technol Dept Comp Sci Meguro Ku Tokyo 1528552 Japan Georgia Inst Technol Coll Comp Atlanta GA 30332 USA Nanjing Univ State Key Lab Novel Software Technol Nanjing 210023 Jiangsu Peoples R China

The large volume principle proposed by Vladimir Vapnik, which advocates that hypotheses lying in an equivalence class with a larger volume are more preferable, is a useful alternative to the large margin principle. In this paper, we introduce a new discriminative clustering model based on the large volume principle called maximum volume clustering (MVC), and then propose two approximation schemes to solve this MVC model: A soft-label MVC method using sequential quadratic programming and a hard-label MVC method using semi-definite programming, respectively. The proposed MVC is theoretically advantageous for three reasons. The optimization involved in hard-label MVC is convex, and under mild conditions, the optimization involved in soft-label MVC is akin to a convex one in terms of the resulting clusters. Secondly, the soft-label MVC method possesses a clustering error bound. Thirdly, MVC includes the optimization problems of a spectral clustering, two relaxed k-means clustering and an information-maximization clustering as special limit cases when its regularization parameter goes to infinity. Experiments on several artificial and benchmark data sets demonstrate that the proposed MVC compares favorably with state-of-the-art clustering methods.

关键词： discriminative clustering large volume principle sequential quadratic programming semi-definite programming finite sample stability clustering error bound

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Optimization of a piezoelectric ceramic actuator

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SENSORS AND ACTUATORS A-PHYSICAL 2000年第1-2期84卷 81-94页

作者： Jiang, TY Ng, TY Lam, KY Natl Univ Singapore Inst High Performance Comp Singapore 118261 Singapore

In this paper, the dynamic model of the prototyped piezoelectric ceramic (PZT) linear actuator is established and optimized to obtain optimal performance. The PZT linear actuator construction is a cylindrical block with two cylindrical brass blocks attached to each side via PZT plates in between. Previous experimental and analytical studies on the prototyped PZT actuator concluded that velocity of the actuator had to be increased. Also, for practical applications, low voltages (applied across the PZT plates) were included as a criterion in the design scheme. The sequential quadratic programming (SQP) method was employed in the optimization process. Optimization of the velocity was performed in detail, with important design parameters being the driving voltages and system parameters such as the masses of the metal blocks constructing the system, linear parameters of PZT, and coefficient of friction. Various waveforms of driving voltage were explored in an attempt to make the actuator move faster. (C) 2000 Elsevier Science S.A. All rights reserved.

关键词： optimization piezoelectric ceramic actuator sequential quadratic programming

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A generalized Burr mixture autoregressive models for modeling non linear time series

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COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 2024年第19期53卷 6832-6851页

作者： Low, Victor Jian Ming Khoo, Wooi Chen Khoo, Hooi Ling Univ Tunku Abdul Rahman Lee Kong Chian Fac Engn & Sci Jalan Sungai Long Kajang Selangor Malaysia Sunway Univ Dept Appl Stat Darul Ehsan Selangor Malaysia Sunway Univ Sch Math Sci Dept Appl Stat 5 Jalan Univ Bandar Sunway 47500 Selangor Malaysia

A more flexible type of mixture autoregressive model, namely the Burr mixture autoregressive, BMAR model is studied in this article for modeling non linear time series. The model consists of a mixture of K autoregressive components with each conditional distribution of the component following a Burr distribution. The BMAR model enjoys some nice statistical properties which allow it to capture time series with: (1) unimodal or multimodal;(2) asymmetry or symmetry conditional distribution;(3) conditional heteroscedasticity;(4) cyclical or seasonal;and (5) conditional leptokurtic distribution. Sufficient and less restrictive conditions for the ergodicity of the BMAR model are derived and discussed. A more robust constrained optimization algorithm (EM - sequential quadratic programming method) is proposed for the non linear optimization problem. From the simulation studies carried out, the parameters estimation method showed satisfying results. The variance of the estimated parameters is also addressed with the missing information principle. Real datasets from two different fields of study are used to assess the performance of the BMAR model compared to other competing models. The comparison done in the empirical examples reveals the supremacy of the BMAR model in capturing the data behavior.

关键词： Burr mixture autoregressive model EM algorithm sequential quadratic programming stationary conditions

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Linear Feedback Guidance for Low-Thrust Many-Revolution Earth-Orbit Transfers

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JOURNAL OF SPACECRAFT AND ROCKETS 2009年第6期46卷 1320-1325页

作者： Gao, Yang Chinese Acad Sci Beijing 100190 Peoples R China

The article focuses on a study on the development of a linear feedback guidance scheme for-low thrust Earth-orbit transfers that involve many orbital revolutions. The key concept of the proposed guidance scheme is cited. It notes that the study used the parameterized control law in both optimization and guidance. According to the author, the scheme possesses near-optimal performance because the optimal trajectory and control can tracked by means of space vehicles.

关键词： Thrust Earth Linear quadratic Regulator Geostationary Transfer Orbit Algebraic Riccati Equation Propellant sequential quadratic programming Numerical Simulation Nonlinear programming Iterative Algorithm

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An active-set algorithm for norm constrained quadratic problems

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MATHEMATICAL programming 2022年第1期193卷 447-483页

作者： Rontsis, Nikitas Goulart, Paul J. Nakatsukasa, Yuji Univ Oxford Dept Engn Sci Oxford OX1 3PN England Univ Oxford Math Inst Oxford OX2 6GG England Natl Inst Informat Tokyo 1018430 Japan

We present an algorithm for the minimization of a nonconvex quadratic function subject to linear inequality constraints and a two-sided bound on the 2-norm of its solution. The algorithm minimizes the objective using an active-set method by solving a series of trust-region subproblems (TRS). Underpinning the efficiency of this approach is that the global solution of the TRS has been widely studied in the literature, resulting in remarkably efficient algorithms and software. We extend these results by proving that nonglobal minimizers of the TRS, or a certificate of their absence, can also be calculated efficiently by computing the two rightmost eigenpairs of an eigenproblem. We demonstrate the usefulness and scalability of the algorithm in a series of experiments that often outperform state-of-the-art approaches;these include calculation of high-quality search directions arising in sequential quadratic programming on problems of the CUTEst collection, and Sparse Principal Component Analysis on a large text corpus problem (70 million nonzeros) that can help organize documents in a user interpretable way.

关键词： Nonconvex optimization Trust-region Active-set sequential quadratic programming Dimensionality reduction

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Loss of superlinear convergence for an SQP-type method with conic constraints

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SIAM JOURNAL ON OPTIMIZATION 2006年第4期16卷 1201-1210页

作者： Diehl, M Jarre, F Vogelbusch, CH Univ Heidelberg Interdisciplinary Ctr Sci Comp D-69120 Heidelberg Germany Univ Dusseldorf Math Inst D-40225 Dusseldorf Germany

In this short note we consider a sequential quadratic programming (SQP) - type method with conic subproblems and compare this method with a standard SQP method in which the conic constraint is linearized at each step. For both approaches we restrict our attention to convex subproblems since these are easy to solve and guarantee a certain global descent property. Using the example of a simple nonlinear program (NLP) and its conic reformulation we show that the SQP method with conic subproblems displays a slower rate of convergence than standard SQP methods. We then explain why an SQP subproblem that is based on a better approximation of the feasible set of the NLP results in a much slower algorithm.

关键词： semidefinite programming nonlinear programming sequential quadratic programming second order cone

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