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A Lagrange-Newton algorithm for sparse nonlinear programming

MATHEMATICAL programming 2022年第1-2期195卷 903-928页

作者： Zhao, Chen Xiu, Naihua Qi, Houduo Luo, Ziyan Beijing Jiaotong Univ Dept Math Beijing 100044 Peoples R China Univ Southampton Sch Math Southampton SO17 1BJ Hants England

The sparse nonlinear programming (SNP) problem has wide applications in signal and image processing, machine learning and finance, etc. However, the computational challenge posed by SNP has not yet been well resolved due to the nonconvex and discontinuous to-norm involved. In this paper, we resolve this numerical challenge by developing a fast Newton-type algorithm. As a theoretical cornerstone, we establish a first-order optimality condition for SNP based on the concept of strong beta-Lagrangian stationarity via the Lagrangian function, and reformulate it as a system of nonlinear equations called the Lagrangian equations. The nonsingularity of the corresponding Jacobian is discussed, based on which the Lagrange-Newton algorithm (LNA) is then proposed. Under mild conditions, we establish the locally quadratic convergence and its iterative complexity estimation. To further demonstrate the efficiency and superiority of our proposed algorithm, we apply LNA to two specific problems arising from compressed sensing and sparse high-order portfolio selection, in which significant benefits accrue from the restricted Newton step.

关键词： sparse nonlinear programming Lagrangian equation The Newton method Locally quadratic convergence Application

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Optimality conditions for sparse nonlinear programming

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Science China Mathematics 2017年第5期60卷 759-776页

作者： PAN LiLi XIU NaiHua FAN Jun Department of Mathematics Beijing Jiaotong University Department of Mathematics Shandong University of Technology

The sparse nonlinear programming （SNP） is to minimize a general continuously differentiable func- tion subject to sparsity, nonlinear equality and inequality constraints. We first define two restricted constraint qualifications and show how these constraint qualifications can be applied to obtain the decomposition properties of the Frechet, Mordukhovich and Clarke normal cones to the sparsity constrained feasible set. Based on the decomposition properties of the normal cones, we then present and analyze three classes of Karush-Kuhn- Tucker （KKT） conditions for the SNP. At last, we establish the second-order necessary optimality condition and sufficient optimality condition for the SNP.

关键词： sparse nonlinear programming constraint qualification normal cone first-order optimality con-dition second-order optimality condition

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Large scale parameter estimation using sparse nonlinear programming methods

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SIAM JOURNAL ON OPTIMIZATION 2003年第1期14卷 223-244页

作者： Betts, JT Huffman, WP Boeing Co Math & Engn Anal Seattle WA 98124 USA

Traditional methods for solving "in verse problems" usually combine a standard initial-value numerical integration technique with a Gauss-Newton method for optimization. This paper presents an approach that uses neither of the traditional processes. Initial-value integration is replaced by a discretization of the relevant differential-algebraic equations. We then exploit sparse finite difference approximations to the Hessian matrix which permits us to construct a quadratically convergent algorithm for solving parameter estimation problems. Computational results with the approach are presented in a series of examples.

关键词： parameter estimation sparse nonlinear programming inverse problems

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Very low-thrust trajectory optimization using a direct SQP method

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2000年第1-2期120卷 27-40页

作者： Betts, JT Boeing Informat & Support Serv Math & Engn Anal Seattle WA 98124 USA

The direct transcription or collocation method has demonstrated notable success in the solution of trajectory optimization and optimal control problems. This approach combines a sparse nonlinear programming algorithm with a discretization of the trajectory dynamics. A challenging class of optimization problems occurs when the spacecraft trajectories are characterized by thrust levels that are very low relative to the vehicle weight. Low-thrust trajectories are demanding because realistic forces due to oblateness, aerodynamic drag, and third-body perturbions often dominate the thrust. Furthermore because the thrust is so low, significant changes to the orbits require very long duration trajectories. When a collocation method is applied to a problem of this type, the resulting nonlinear program is very large because the trajectories are long, and very nonlinear because of the perturbing forces. This paper describes the application of the transcription method to the solution of very low-thrust orbit transfers. The vehicle dynamics are defined using a modified set of equinoctial coordinates, and the trajectory modeling is described using these dynamics. A solution is presented for a representative transfer using a spacecraft with a thrust acceleration of approximately 1.25 x 10(-7) km/s(2). This transfer requires over 578 revolutions, and leads to a sparse optimization problem with 416 123 variables and 249 674 constraints. Issues related to the numerical conditioning and problem formulation are discussed. (C) 2000 Elsevier Science B.V. All rights reserved.

关键词： optimal control sparse nonlinear programming trajectory optimization

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A sparse nonlinear OPTIMIZATION ALGORITHM

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1994年第3期82卷 519-541页

作者： BETTS, JT FRANK, PD 1. Mathematics and Engineering Analysis Department Research and Technology Division Boeing Computer Services Seattle Washington

One of the most effective numerical techniques for solving nonlinear programming problems is the sequential quadratic programming approach. Many large nonlinear programming problems arise naturally in data fitting and when discretization techniques are applied to systems described by ordinary or partial differential equations. Problems of this type are characterized by matrices which are large and sparse. This paper describes a nonlinear programming algorithm which exploits the matrix sparsity produced by these applications. Numerical experience is reported for a collection of trajectory optimization problems with nonlinear equality and inequality constraints.

关键词： sparse nonlinear programming SEQUENTIAL QUADRATIC programming TRAJECTORY OPTIMIZATION

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Exploiting sparsity in the direct transcription method for optimal control

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 1999年第2期14卷 179-201页

作者： Betts, JT Huffman, WP Boeing Co Informat & Support Serv Math & Engn Anal Seattle WA 98124 USA

In the direct transcription method an approximation to an optimal control problem is constructed by discretization of the state and control variables. The control problem is thus transcribed into a large scale constrained optimization problem with a finite number of variables. It is necessary to solve the nonlinear programming (NLP) problem produced by the discretization as efficiently as possible, and research has focused on methods for solving the underlying NLP when the relevant Jacobian and Hessian matrices are sparse. However little attention has been given to computing the Jacobian and Hessian, particularly for applications that require finite difference techniques. Typically it is assumed that these matrices have a block structure, and when standard finite difference estimates are used this operation is a major computational expense. This paper describes how to actually construct the Jacobian and Hessian matrices efficiently, and exploit sparsity inherent in the problem. This new technique can significantly reduce the computational costs when the original optimal control problem has many state and control variables.

关键词： optimal control sparse nonlinear programming

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CORRECTED SEQUENTIAL LINEAR-programming FOR sparse MINIMAX OPTIMIZATION

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BIT NUMERICAL MATHEMATICS 1994年第3期34卷 372-387页

作者： JONASSON, K MADSEN, K TECH UNIV DENMARK INST MATH MODELLING DK-2800 LYNGBY DENMARK

We present a new algorithm for nonlinear minimax optimization which is well suited for large and sparse problems. The method is based on trust regions and sequential linear programming. On each iteration a linear minimax problem is solved for a basic step. If necessary, this is followed by the determination of a minimum norm corrective step based on a first-order Taylor approximation. No Hessian information needs to be stored. Global convergence is proved. This new method has been extensively tested and compared with other methods, including two well known codes for nonlinear programming. The numerical tests indicate that in many cases the new method can find the solution in just as few iterations as methods based on approximate second-order information. The tests also show that for some problems the corrective steps give much faster convergence than for similar methods which do not employ such steps.

关键词： nonlinear MINIMAX OPTIMIZATION SEQUENTIAL LINEAR programming sparse nonlinear programming

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Optimal low thrust trajectories to the moon

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SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 2003年第2期2卷 144-170页

作者： Betts, JT Erb, SO Boeing Co Math & Comp Technol Seattle WA 98124 USA Univ Stuttgart Inst Flugmech & Flugregelung D-70550 Stuttgart Germany

The direct transcription or collocation method has demonstrated notable success in the solution of trajectory optimization and optimal control problems. This approach combines a sparse nonlinear programming algorithm with a discretization of the trajectory dynamics. A challenging class of optimization problems occurs when the spacecraft trajectories are characterized by thrust levels that are very low relative to the vehicle weight. Low thrust trajectories are demanding because realistic forces, due to oblateness, and third-body perturbations often dominate the thrust. Furthermore, because the thrust is so low, significant changes to the orbits require very long duration trajectories. When a collocation method is applied to a problem of this type, the resulting nonlinear program is very large, because the trajectories are long, and very nonlinear because of the perturbing forces. This paper describes the application of the transcription method to compute an optimal low thrust transfer from an Earth orbit to a specified lunar mission orbit. It is representative of the SMART-1 or "Small Missions for Advanced Research in Technology" of the ESA scientific program [J. Schoenmaekers, J. Pulido, and R. Jehn, Tech. report S1-ESC-RP-5001, European Space Agency, 1998]. The spacecraft is deployed from an Ariane-5 into an elliptic Earth centered park orbit. The goal is to insert the spacecraft into a lunar orbit that is polar and elliptic and has its pericenter above the south pole. The spacecraft utilizes a solar electric propulsion system. The goal is to compute the optimal steering during the orbit transfer, which takes over 200 days, so that the fuel consumption is minimized. The vehicle dynamics are defined using a modified set of equinoctial coordinates, and the trajectory modeling is described using these dynamics. A solution is presented that requires the solution of a sparse optimization problem with 211031 variables and 146285 constraints. The trajectory we present re

关键词： optimal control sparse nonlinear programming orbit transfer

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