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A projected derivative-free algorithm for nonlinear equations with convex constraints

OPTIMIZATION METHODS & SOFTWARE 2014年第1期29卷 24-41页

作者： La Cruz, William Cent Univ Venezuela Fac Ingn Dept Elect Comp & Control Caracas 1051DF Venezuela

This paper presents a derivative-free algorithm for solving nonlinear equations with convex constraints. The new method uses in a systematic way the projection and the residual to generate search directions and feasible iterates. A convergence analysis is described. Extensive numerical experiences are included to highlight the efficacy of the proposed algorithm for the solution of nonlinear equations with convex constraints.

关键词： nonlinear equations convex constraints derivative-free algorithm

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A derivative-free algorithm for bound constrained optimization

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2002年第2期21卷 119-142页

作者： Lucidi, S Sciandrone, M Univ Roma La Sapienza Dipartimento Informat & Sistemist Rome Italy CNR Ist Anal Sistemi & Informat I-00185 Rome Italy

In this work, we propose a new globally convergent derivative-free algorithm for the minimization of a continuously differentiable function in the case that some of (or all) the variables are bounded. This algorithm investigates the local behaviour of the objective function on the feasible set by sampling it along the coordinate directions. Whenever a "suitable" descent feasible coordinate direction is detected a new point is produced by performing a linesearch along this direction. The information progressively obtained during the iterates of the algorithm can be used to build an approximation model of the objective function. The minimum of such a model is accepted if it produces an improvement of the objective function value. We also derive a bound for the limit accuracy of the algorithm in the minimization of noisy functions. Finally, we report the results of a preliminary numerical experience.

关键词： derivative-free algorithm bound constraints linesearch technique

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A fourth-order derivative-free algorithm for nonlinear equations

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2011年第8期235卷 2551-2559页

作者： Peng, Yehui Feng, Heying Li, Qiyong Zhang, Xiaoqing Hunan Univ Sci & Technol Sch Math & Comp Sci Xiangtan 411201 Hunan Peoples R China Huazhong Univ Sci & Technol Sch Energy & Power Engn Wuhan 430074 Hubei Peoples R China Huaihua Coll Dept Math Huaihua 418008 Hunan Peoples R China

A two-step derivative-free iterative algorithm is presented for solving nonlinear equations. Error analysis shows that the algorithm is fourth-order with efficiency index equal to 1.5874. A lot of numerical results show that the algorithm is effective and is preferable to some existing derivative-free methods in terms of computation cost. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Nonlinear equation derivative-free algorithm Root finding Error analysis

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A penalty derivative-free algorithm for nonlinear constrained optimization

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OPTIMIZATION LETTERS 2015年第6期9卷 1213-1229页

作者： Lv, Wei Sun, Qiang Lin, He Sui, Ruirui Shanghai Univ Dept Math Shanghai 200444 Peoples R China Chinese Acad Sci Dept Mat & Energy Res Shanghai Inst Appl Phys Shanghai 201204 Peoples R China

In this paper, we modify a derivative-free line search algorithm (DFL) proposed in the Ref. (Liuzzi et al. SIAM J Optimiz 20(5):2614-2635, 2010) to minimize a continuously differentiable function of box constrained variables or unconstrained variables with nonlinear constraints. The first-order derivatives of the objective function and of the constraints are assumed to be neither calculated nor explicitly approximated. Different line-searches are used for box-constrained variables and unconstrained variables. Accordingly the convergence to stationary points is proved. The computational behavior of the method has been evaluated on a set of test problems. The performance and data profiles are used to compare with DFL.

关键词： derivative-free algorithm Nonlinear constrained optimization Box constrained variables Unconstrained variables

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An efficient derivative-free algorithm for bound constrained mixed-integer optimization

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EVOLUTIONARY INTELLIGENCE 2024年第1期17卷 11-20页

作者： Yang, Shanxue Liu, Hongwei Pan, Chen Xidian Univ Sch Math & Stat Xian 710071 Shaanxi Peoples R China Xian Univ Finance & Econ Sch Stat Xian 710100 Shaanxi Peoples R China Huanggang Normal Univ Coll Math & Stat Huanggang 438000 Hubei Peoples R China

Mixed integer optimization is very important and complicated task in the optimization field, which widely exists in the engineering problems. In order to improve the efficiency of derivative-free algorithm when solving the mixed integer optimization problems, we propose an efficient derivative-free algorithm, which is based on the modified minimal positive base and the technique of search directions rotation. The method using the modified minimal positive base only needs at most function evaluations at every iteration, compared with the derivative-free algorithms based on the maximal 2n positive base, where n is the number of variables. Meantime, the technique of search directions rotation we proposed can overcome the disadvantage of the method based on the minimal positive base which can cause undesirable large angles between some positive base directions and large unexplored feasible domain. Accordingly the convergence to stationary points is proved. To evaluate the performance of our method, we compare it with two classical algorithms on 50 benchmark problems. The results of numerical experiments show that the method can reduce the number of function evaluations and improve the efficiency of the algorithm.

关键词： derivative-free algorithm Bound constrained optimization Mixed-integer programming Number of function evaluations Alternating minimal positive base

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A derivative-free algorithm for nonlinear equations and its applications in multibody dynamics

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JOURNAL OF algorithmS & COMPUTATIONAL TECHNOLOGY 2018年第1期12卷 30-42页

作者： Tang, Di Bao, Shiyi Lv, Binbin Guo, Hongtao Luo, Lijia Mao, Jianfeng Zhejiang Univ Technol Coll Mech Engn Hangzhou Zhejiang Peoples R China China Aerodynam Res & Dev Ctr High Speed Aerodynam Inst Mianyang Peoples R China

Local floating coordinate system is used to represent the deployment motion of each rigid and flexible body of multibody system dynamics. Normal substructure modes are employed to describe the flexibility of a flexible body. Constraint equations establish the linkage between different bodies, part of them to specify positions and the others to specify orientations. System's governing equations are then derived using generalized coordinates by Lagrange methods. The resulting differential-algebraic equations are transformed to algebraic equations using backward differential formula corrector method, thus highly coupled nonlinear equations are obtained. However, Jacobian matrix of the nonlinear equations is hard to calculate, and then a quasi-Newton method based on Broyden-Fletcher-Goldfarb-Shanno update approach for the solution of the nonlinear equations is proposed. And a suitable line search approach is combined with the Broyden-Fletcher-Goldfarb-Shanno method to improve its efficiency. Some numerical results are reported to show efficiency of the proposed method. Afterwards, the Broyden-Fletcher-Goldfarb-Shanno method is integrated into multibody dynamics method. A rigid multibody case and a rigid-flex multibody case are further studied to show the efficiency of the proposed multibody solver.

关键词： derivative-free algorithm nonlinear equations multibody dynamics differential-algebraic equations

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Reducing the number of function evaluations in derivative-free algorithm for bound constrained optimization

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EVOLUTIONARY INTELLIGENCE 2023年第6期16卷 1779-1788页

作者： Yang, Shanxue Yang, Zuqiao Fu, Yiping Liu, Hongwei Xidian Univ Sch Math & Stat Xian 710071 Shaanxi Peoples R China Xian Univ Finance & Econ Sch Stat Xian 710100 Shaanxi Peoples R China Huanggang Normal Univ Coll Math & Stat Huanggang 438000 Hubei Peoples R China Tianjin Univ Coll Management & Econ Tianjin 300072 Peoples R China

In this paper, we present a derivative-free algorithm based on modified minimal positive base for bound constrained optimization problems. Compared with the derivative-free algorithms based on the maximal 2n positive base, the algorithms based on the minimal n + 1 positive base only need at most n + 1 function evaluations at every iteration, where n is the number of variables. Therefore, we can reduce the number of function evaluations from 2n to n + 1 at each iteration. But the minimal positive base can cause undesirable large angles between some positive base directions and large unexplored feasible domain. In order to overcome this defect, we propose a modified set of feasible directions based on the minimal positive base and the technique of search direction rotation to investigate the unexplored domain at the next iteration. Accordingly, convergence to stationary points is proved. Moreover, the numerical experiments show that the method based on modified minimal positive base can reduce the number of function evaluations and is beneficial in a derivative-free context.

关键词： derivative-free algorithm Bound constrained optimization Number of function evaluations Modified minimal positive base Technique of search direction rotation

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Two derivative-free algorithms for nonlinear equations

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OPTIMIZATION METHODS & SOFTWARE 2008年第3期23卷 395-410页

作者： Deng, Youjun Liu, Zhenhai Cent S Univ Sch Math Sci & Comp Technol Changsha Hunan Peoples R China

In this paper, we propose two new derivative-free algorithms for nonlinear equations. The first is based on quasi-Newton method and is globally and superlinearly convergent under some mild assumptions. The second combines the ideas of the first with the filter strategy, which helps to reduce the backtracking steps in calculating the stepsizes, for evaluating candidate points. We show its convergence under the same assumptions. The resulting algorithms show some attractive features. Some encouraging preliminary computational results for both algorithms are reported.

关键词： derivative-free algorithm backtracking line search filter algorithm superlinear convergence nonlinear equations

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New sequential and parallel derivative-free algorithms for unconstrained minimization

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SIAM JOURNAL ON OPTIMIZATION 2002年第1期13卷 79-96页

作者： García-Palomares, UM Rodríguez, JF Univ Simon Bolivar Dept Proc & Sistemas Caracas 1080A Venezuela Dept Mecan Caracas 1080A Venezuela

This paper presents sequential and parallel derivative-free algorithms for finding a local minimum of smooth and nonsmooth functions of practical interest. It is proved that, under mild assumptions, a sufficient decrease condition holds for a nonsmooth function. Based on this property, the algorithms explore a set of search directions and move to a point with a sufficiently lower functional value. If the function is strictly differentiable at its limit points, a ( sub) sequence of points generated by the algorithm converges to a first-order stationary point (delf(x) = 0). If the function is convex around its limit points, convergence ( of a subsequence) to a point with nonnegative directional derivatives on a set of search directions is ensured. Preliminary numerical results on sequential algorithms show that they compare favorably with the recently introduced pattern search methods.

关键词： nonsmooth function unconstrained minimization derivative-free algorithm parallel algorithms necessary and sufficient conditions

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A derivative-free TRUST-REGION METHOD FOR BIOBJECTIVE OPTIMIZATION

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SIAM JOURNAL ON OPTIMIZATION 2014年第1期24卷 334-362页

作者： Ryu, Jong-Hyun Kim, Sujin Hongik Univ Coll Business Management Yeongi Gun Chungcheong Nam South Korea Natl Univ Singapore Ind & Syst Engn Dept Singapore 117576 Singapore

We consider unconstrained black-box biobjective optimization problems in which analytic forms of the objective functions are not available and function values can be obtained only through computationally expensive simulations. We propose a new algorithm to approximate the Pareto optimal solutions of such problems based on a trust-region approach. At every iteration, we identify a trust region, then sample and evaluate points from it. To determine nondominated solutions in the trust region, we employ a scalarization method to convert the two objective functions into one. We construct and optimize quadratic regression models for the two original objectives and the converted single objective. We then remove dominated points from the current Pareto approximation and construct a new trust region around the most isolated point in order to explore areas that have not been visited. We prove convergence of the method under general regularity conditions and present numerical results suggesting that the method efficiently generates well-distributed Pareto optimal solutions.

关键词： biobjective optimization multiobjective optimization derivative-free algorithm trust-region method Pareto dominance

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