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A Primal-dual Interior Point Method for nonlinear programming

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Northeastern Mathematical Journal 2008年第3期24卷 275-282页

作者：张珊姜志侠 School of Mathematics Jilin University Changchun130012 Department of Mathematics Changchun University of Science and Technology Changchun 130022

In this paper, we propose a primal-dual interior point method for solving general constrained nonlinear programming problems. To avoid the situation that the algorithm we use may converge to a saddle point or a local maximum, we utilize a merit function to guide the iterates toward a local minimum. Especially, we add the parameter ε to the Newton system when calculating the decrease directions. The global convergence is achieved by the decrease of a merit function. Furthermore, the numerical results confirm that the algorithm can solve this kind of problems in an efficient way.

关键词： primal-dual interior point algorithm merit function global convergence nonlinear programming

An inexact projected gradient method with rounding and lifting by nonlinear programming for solving rank-one semidefinite relaxation of polynomial optimization

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arXiv 2021年

作者： Yang, Heng Liang, Ling Carlone, Luca Toh, Kim-Chuan Laboratory for Information and Decision Systems Massachusetts Institute of Technology Department of Mathematics National University of Singapore Department of Mathematics Institute of Operations Research and Analytics National University of Singapore

We consider solving high-order semidefinite programming (SDP) relaxations of nonconvex polynomial optimization problems (POPs) that often admit degenerate rank-one optimal solutions. Instead of solving the SDP alone, we propose a new algorithmic framework that blends local search using the nonconvex POP into global descent using the convex SDP. In particular, we first design a globally convergent inexact projected gradient method (iPGM) for solving the SDP that serves as the backbone of our framework. We then accelerate iPGM by taking long, but safeguarded, rank-one steps generated by fast nonlinear programming algorithms. We prove that the new framework is still globally convergent for solving the SDP. To solve the iPGM subproblem of projecting a given point onto the feasible set of the SDP, we design a two-phase algorithm with phase one using a symmetric Gauss-Seidel based accelerated proximal gradient method (sGS-APG) to generate a good initial point, and phase two using a modified limited-memory BFGS (L-BFGS) method to obtain an accurate solution. We analyze the convergence for both phases and establish a novel global convergence result for the modified L-BFGS that does not require the objective function to be twice continuously differentiable. We conduct numerical experiments for solving second-order SDP relaxations arising from a diverse set of POPs. Our framework demonstrates state-of-the-art efficiency, scalability, and robustness in solving degenerate rank-one SDPs to high accuracy, even in the presence of millions of equality *** Codes 90C06, 90C22, 90C23, 90C55 © 2021, CC BY.

关键词： nonlinear programming

Optimal Design of a Photovoltaic Hydrogen Production System via nonlinear programming

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Optimal Design of a Photovoltaic Hydrogen Production System ...

第35届中国过程控制会议

作者： Yuan Liu Lei Xie Hongye Su Yiqin Wang Yangyang Luo Zhejiang University

This paper presents a mathematical model for a photovoltaic hydrogen production system, along with an optimal design study applied to this system. Precise modeling of hydrogen production has proven to be a challenging task. Based on previous work, a general framework is proposed for battery pack-assisted photovoltaic hydrogen production. Specifically, explicit mathematical expressions are given for the hydrogen production efficiency of the electrolyzer, and a current source model is established for the photovoltaic panels. Regarding the optimal design, a systematic framework is outlined to obtain the optimal strategy by converting the control problem into a boundary value problem, subsequently solved via nonlinear programming. Simulation results show that under the optimal strategy, the battery energy consumption by the photovoltaic hydrogen production system is significantly reduced, while maintaining minimal power consumption operation when the irradiation is insufficient, without the need for frequent start-stop cycles. In addition, the calculation time of the IPOPT solver to find the optimal strategy is discussed, proving its ability to address nonlinear planning problems at a millisecond-level calculation time.

关键词： Optimal design Photovoltaic hydrogen production nonlinear programming

A simple canonical form for nonlinear programming problems and its use

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arXiv 2018年

作者： Mascarenhas, Walter F. Instituto de Matemática e Estatística Universidade de São Paulo Cidade Universitária Rua do Matão 1010 São Paulo SPCEP 05508-090 Brazil

We argue that reducing nonlinear programming problems to a simple canonical form is an effective way to analyze them, specially when the problem is degenerate and the usual linear independence hypothesis does not hold. To illustrate this fact we solve an open problem about constraint qualifications using this simple canonical form. Copyright © 2018, The Authors. All rights reserved.

关键词： nonlinear programming

NPASA: An algorithm for nonlinear programming - Motivation and global convergence

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arXiv 2021年

作者： Diffenderfer, James Hager, William W. Lawrence Livermore National Laboratory LivermoreCA United States The University of Florida GainesvilleFL United States

In this paper, we present a two phase method for solving nonlinear programming problems called nonlinear Polyhedral Active Set Algorithm (NPASA) that has global and local convergence guarantees under reasonable assumptions. The first phase consists of an augmented Lagrangian method to ensure global convergence while the second phase is designed to promote fast local convergence by performing a balanced reduction of two error estimators for nonlinear programs. After presenting error estimators for nonlinear programs and our algorithm NPASA, we establish global convergence properties for NPASA. Local quadratic convergence of NPASA is established in a companion paper [11]. © 2021, CC BY.

关键词： nonlinear programming

AN AGGREGATE CONSTRAINT METHOD FOR nonlinear-programming

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JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY 1991年第11期42卷 1003-1010页

作者： LI, XS DALIAN UNIV TECHNOL ENGN MECH RES INSTDALIAN 116023PEOPLES R CHINA

This paper presents a new and efficient method, referred to as an aggregate constraint method, for the solution of general non-linear programming problems in which a multi-constrained problem is converted to one with a single parametric constraint. This is derived by using the two concepts of constraint surrogation and maximum entropy, and represents a uniform approximation to the original constraint set. An augmented Lagrangean algorithm is then used to solve the resulting problem. Numerical examples are given to show the high efficiency of the present method.

关键词： nonlinear programming OPTIMIZATION INFORMATION THEORY

Position location scheme using nonlinear programming based on RSSI and DV-Hop

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International Journal of Hybrid Information Technology 2015年第5期8卷 1-10页

作者： Shang, Fengjun Lan, Lan Dong, Maojian College of Computer Science and Technology Chongqing University of Posts and Telecommunications Chongqing400065 China

Localization is used in location-aware applications such as navigation, autonomous robotic movement, and asset tracking to position a moving object on a coordinate system. In this paper, a nonlinear programming algorithm is proposed based on RSSI and improved DV-Hop algorithm, called NPRDV-Hop. The algorithm makes four major contributions to the localization problem in the wireless sensor networks (WSNs). Firstly, a hop distance is improved, called Hop Distance. This scheme could assure that the most nodes receive the Hop Distance from beacon node who has the least hops between them. This practical localization scheme is relatively high accuracy and low cost for WSNs. Secondly, Heron's formula is introduced as objective function. Thirdly, Gauss distribution is introduced to select RSSI so that the error of distance is little. Lastly, the general problem is considered by the nonlinear programming to solve for the locations of the sensors. Simulation results show that the proposed method can improve location accuracy and coverage without increasing hardware cost of sensor node. The performance of this algorithm is superior to the original DV-Hop algorithm. © 2015 SERSC.

关键词： nonlinear programming

Analytical Delta-V Approximation for nonlinear programming of Multi-target Rendezvous and Flyby Trajectories

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arXiv 2024年

作者： Huang, An-Yi Li, Heng-Nian Luo, Ya-Zhong State Key Laboratory of Astronautic Dynamics Xi’an710043 China National University of Defense Technology Changsha410073 China College of Aerospace Science and Engineering Hunan Key Laboratory of Intelligent Planning and Simulation for Aerospace Missions China

This study proposes an analytical Delta-V approximation of short-timetransfers based on the linear relative motion and a gradient-based nonlinear programming model of multi-target rendezvous and fly by trajectories. In previous studies, the Lambert's solution is commonly used to evaluate Delta-Vof short-duration transfers. In this study, to avoid the iteration process forobtaining the Lambert's solution and its gradient, the linear relative motion equations are applied to form an analytical two-point boundary value model for the near-circular orbit rendezvous problems. Although the relative motion equations are usually applicable when the two orbits are close enough, and the position and velocity errors would become more significant as the orbital differences increase, the errors of the velocity increments were proved acceptable in our simulations. Moreover, the analytical formula facilitates the calculation of the gradients to the start epoch and flight time, which are used to establish a nonlinear programming model for sequence optimization that gradient-based algorithms can easily solve. Simulation results demonstrated that the analytical Delta-V approximation requires much less calculation than the Lambert's solution, and the proposed gradient-based nonlinear programming algorithms can obtain similar results in less time than previous methods. Copyright © 2024, The Authors. All rights reserved.

关键词： nonlinear programming

Exact Matrix Factorization Updates for nonlinear programming

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arXiv 2022年

作者： Escobedo, Adolfo R. School of Computing and Augmented Intelligence TempeAZ85281 United States

LU and Cholesky matrix factorization algorithms are core subroutines used to solve systems of linear equations (SLEs) encountered while solving an optimization problem. Standard factorization algorithms are highly efficient but remain susceptible to the accumulation of roundoff errors, which can lead solvers to return feasibility and optimality claims that are actually invalid. This paper introduces a novel approach for solving sequences of closely related SLEs encountered in nonlinear programming efficiently and without roundoff errors. Specifically, it introduces rank-one update algorithms for the roundoff-error-free (REF) factorization framework, a toolset built on integer-preserving arithmetic that has led to the development and implementation of fail-proof SLE solution subroutines for linear programming. The formal guarantees of the proposed algorithms are established through the derivation of theoretical insights. Their advantages are supported with computational experiments, which demonstrate upwards of 75x-improvements over exact factorization run-times on fully dense matrices with over one million entries. A significant advantage of the methodology is that the length of any coefficient calculated via the proposed algorithms is bounded polynomially in the size of the inputs without having to resort to greatest common divisor operations, which are required by and thereby hinder an efficient implementation of exact rational arithmetic approaches. Copyright © 2022, The Authors. All rights reserved.

关键词： nonlinear programming