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Duality in the fuzzy-parametric space for fuzzy-parametric nonlinear programming problem

OPSEARCH 2018年第3-4期55卷 662-676页

作者： Osman, M. S. Abd Elazeem, A. M. Elsisy, M. A. Rashwan, M. M. Al Asher Univ El Asher City Egypt October High Inst Engn & Technol Giza Egypt Benha Univ Fac Engn Benha Banha Egypt Cairo Univ Fac Econ & Polit Sci Cairo Egypt

The concepts of fuzziness and parametric analysis are of importance to treat uncertainty in mathematical model and may offer certain more viewpoints. The basic notions of the parametric study in nonlinear programming problem are presented by Osman (Aplikace matematiky 22(5):318-332, 1977;Aplikace matematiky 22(5):333-348, 1977). In general, a parametric programming problem is not easy to be solved. In addition, sometime, solving a parametric programming problem with parameters in the objective is easier than solving a parametric problem with parameters in the constraints and vice versa. Therefore, a parametric study in duality space is important to facilitate solving a parametric programming problem. The fuzzy nonlinear problem is interested area for research as one of the tools for treating uncertainty. The fuzzy nonlinear problem when parameters in the objective function or constrains or both is called the fuzzy parametric nonlinear problem. Therefore, dealing with fuzziness and duality parametric space is concerned. In this paper, a novelty introduction to the fuzzy basic notions of parametric programming problem are clarified, the relations between the concepts concerning duality in parametric spaces which introduced by Osman et al. (Int J Math Arch 6(12):161-165, 2016) and the fuzzy concepts are presented. We present and define the fuzzy parametric notions of the set of feasible parameters, the solvability set, and the stability sets of the first and second kind. These notions are not defined before. The theoretical relations and an illustration example are introduced.

关键词： nonlinear programming problem Parametric Duality Fuzzy set Fuzzy nonlinear programming problem

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Energy optimal path planning for wheeled mobile robots with circular obstacle

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OPTIMAL CONTROL APPLICATIONS & METHODS 2025年第2期46卷 583-601页

作者： Kim, Youngjin Singh, Tarunraj Univ Buffalo Dept Mech Engn & Aerosp Engn Buffalo NY 14214 USA

The energy optimal motion planning algorithm for a differential drive robot with a circular obstacle in the maneuver space is introduced in this article. Rather than the traditional unicycle model, a non-canonical state space model of the kinematics of a wheeled mobile robot is used to formulate the optimal control problem. The necessary conditions for optimality are formally derived using calculus of variations, resulting in constraints for the costates when the inequality constraint become active, revealing that the controls are required to be continuous. The optimality conditions permit decomposing the motion planning problem in two independent segments: prior to and after activation of the inequality constraints. Analytical expressions for the evolving states and control are found to be a function of Jacobi elliptic functions, permitting reducing the optimal control problem to a root-finding problem. A comprehensive parametric study is included to demonstrate the impact on the optimal trajectories by varying the radius of the obstacle. The study shows that there are three distinct maneuver strategies with respect to the size of the obstacle. Furthermore, by including entering and exit time as optimization variables, a methodology for generalizing the development to any given motion scenario is developed and numerically illustrated. This article presents an analytical solution for energy-optimal control of a wheeled mobile robot navigating around a circular obstacle. The optimality conditions are derived using calculus of variations, leading to a closed-form solution for the pure-state inequality constrained problem. The resulting closed-form expressions reduce the optimal control problem to a root-finding problem. A rigorous parametric study reveals that there are three distinct maneuver strategies with respect to the size of the obstacle. image

关键词： calculus of variations mobile robot motion planning nonlinear programming problem obstacle avoidance optimal control

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A dual-dimensionality reduction strategy for optimization-based parallel parking path planner

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EXPERT SYSTEMS WITH APPLICATIONS 2025年 263卷

作者： Hu, Qiuxia Ma, Jie Zhan, Guanglun Gao, Feng Chongqing Univ Coll Mech & Vehicle Engn Chongqing 400044 Peoples R China

Optimization-based parking path planner has been attracting much more attention due to its capability to generate the optimal path rather than a feasible one. But when solving this optimal problem, it is difficult to make a good balance between the computational accuracy and efficiency. Because dense discretization points are necessary for accuracy, which conversely leads to a high dimensionality of optimization space. To deal with this challenge, this paper proposes a dual-dimensionality reduction strategy to accelerate the numerical optimization process of parallel parking path. One is to reduce the number of optimized parameters by approximating the optimal path with multiple circular arcs connected end to end, according to the numerical statistical analysis of the geometric features of the optimal path. The other is to reduce the number of nonlinear and non-convex constraints for obstacle avoidance by analyzing the geometry relationship between the parameterized arcs and the boundary lines of obstacles. This point-over-arc obstacle avoidance constraint can ensure that the planned path is collision-free by only checking the limitations at the endpoints of the circular arcs. The effectiveness of the proposed method on the improvement in computational efficiency is validated by several comparative simulation and experimental tests. Compared with the traditional numerical optimization methods, the presented algorithm improves the computational efficiency by nearly 40 times on average.

关键词： Automated parking Path planning Optimal control nonlinear programming problem Obstacle avoidance

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Improved genetic algorithm for nonlinear programming problems

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Journal of Systems Engineering and Electronics 2011年第3期22卷 540-546页

作者： Kezong Tang Jingyu Yang Haiyan Chen Shang Gao School of Computer Science and Technology Nanjing University of Science and TechnologyNanjing 210094P.R.China School of Computer Science and Engineering Jiangsu University of Science and TechnologyZhenjiang 212003P.R.China

An improved genetic algorithm（IGA） based on a novel selection strategy to handle nonlinear programming problems is *** individual in selection process is represented as a three-dimensional feature vector which is composed of objective function value,the degree of constraints violations and the number of constraints *** is easy to distinguish excellent individuals from general individuals by using an individuals＇ feature ***,a local search（LS） process is incorporated into selection operation so as to find feasible solutions located in the neighboring areas of some infeasible *** combination of IGA and LS should offer the advantage of both the quality of solutions and diversity of *** results over a set of benchmark problems demonstrate that IGA has better performance than other algorithms.

关键词： genetic algorithm(GA) nonlinear programming problem constraint handling non-dominated solution optimization problem.

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A high performance neural network for solving nonlinear programming problems with hybrid constraints

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PHYSICS LETTERS A 2001年第2期288卷 88-94页

作者： Tao, Q Cao, JD Xue, MS Qiao, H Univ Sci & Technol China Dept Automat Hefei 230027 Peoples R China Southeast Univ Dept Appl Math Nanjing 210096 Peoples R China City Univ Hong Kong Dept Manufacture Engn & Engn Management Hong Kong Hong Kong Peoples R China

A continuous neural network is proposed in this Letter for solving optimization problems. It not only can solve nonlinear programming problems with the constraints of equality and inequality, but also has a higher performance. The main advantage of the network is that it is an extension of Newton's gradient method for constrained problems, the dynamic behavior of the network under special constraints and the convergence rate can be investigated. Furthermore, the proposed network is simpler than the existing networks even for solving positive definite quadratic programming problems. The network considered is constrained by a projection operator on a convex set. The advanced performance of the proposed network is demonstrated by means of simulation of several numerical examples. (C) 2001 Elsevier Science B.V. All rights reserved.

关键词： nonlinear programming problem neural network energy function gradient method global asymptotic stability

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A Low-Dimensional Optimization Method for Parallel Parking Path Planning

A Low-Dimensional Optimization Method for Parallel Parking P...

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2025 International Conference on Electrical Automation and Artificial Intelligence, ICEAAI 2025

作者： Hu, Qiuxia Zhan, Guanglun Gao, Feng Liu, Hongwei College of Mechanical and Vehicle Engineering Chongqing University Chongqing China Western Science City ICV Innovation Center (Chongqing) Co. Ltd. Chongqing China

ISBN: (纸本)9798331506797

Optimization-based parking path planner can make full use of parking space, but its extensive computational consumption limits its practical application. This paper proposes a low-dimensional optimization path planning method, which accelerates the numerical computation process by reducing the number of the optimization variables and obstacle avoidance constraints. Since differential equations in the optimization-based planner require many discretize points to ensure numerical accuracy, the geometric characteristic of the optimal path is analyzed from a large amount of optimization results and further used to describe the parallel parking mission as a low-dimensional nonlinear programming problem that only consists of algebraic equations. As adjacent points on the parking path are connected by analytical curves, a simplified obstacle avoidance constraint is designed based on relative geometric position relationship between these curves and obstacle boundary lines. Based on this, a collision-free parking path can be optimized by verifying obstacle avoidance constraints only at the endpoints. Simulation results show that the optimized path is collision-free and the optimization process is accelerated by these techniques. © 2025 IEEE.

关键词： nonlinear programming problem Obstacle avoidance Optimization Parallel parking Path planning

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Existence and Uniqueness of a Solution for Some nonlinear programming problems

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MEDITERRANEAN JOURNAL OF MATHEMATICS 2015年第1期12卷 133-146页

作者： Gabeleh, Moosa Shahzad, Naseer Ayatollah Boroujerdi Univ Dept Math Boroujerd Iran King Abdulaziz Univ Dept Math Jeddah 21589 Saudi Arabia

Let us consider the mappings and , where A and B are two nonempty subsets of a metric space (X, d). The purpose of this paper is to provide sufficient conditions to ensure the existence and uniqueness of an element such that . In particular, the existence and uniqueness of a best proximity point for non-self-mappings are obtained. Examples are given to show usability of our results.

关键词： Best proximity point fixed point nonlinear programming problem WP-property

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Particle swarm optimization based heuristics for nonlinear programming problems

Particle swarm optimization based heuristics for nonlinear p...

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International Multiconference of Engineers and Computer Scientists

作者： Matsui, Takeshi Kato, Kosuke Sakawa, Masatoshi Uno, Takeshi Morihara, Kenji Hiroshima Univ Grad Sch Engn Hiroshima Japan

ISBN: (纸本)9789889867140

In recent years, a high speed and accurate optimization method has been expected for large scale nonlinear programming problems. As a new solution method for nonlinear programming problems, a particle swarm optimization (PSO) method was proposed by Kennedy et al. and has attracted considerable attention. However, there exist few reports with respect to successful results for constrained nonlinear programming problems by PSO-based methods. Furthermore, PSO-based methods have a shortcoming that the search is liable to stopping at a certain local solution. In this paper, incorporating the bisection method and a homomorphous mapping to carry out the search considering constraints as well as the multiple stretching technique and modified move schemes of particles to restraining the stopping around local solutions. Furthermore, we show the efficiency of the proposed revised PSO method (rPSO) by comparing it with an existing method through the application of them into the numerical examples.

关键词： nonlinear programming problem particle swarm optimization bisection method homomorphous mapping multiple stretching

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An improved Genetic Algorithm for a Type of nonlinear programming problems

An improved Genetic Algorithm for a Type of Nonlinear Progra...

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IEEE International Conference on Automation and Logistics

作者： He Dakuo Wang Fuli Jia Mingxing Minist Educ Key Lab Proc Ind Automat Shenyang 110004 Peoples R China

ISBN: (纸本)9781424425020

Based on the study on how to apply penalty strategy for solving a type of nonlinear programming problems by genetic algorithm, such conclusion can be drawn that only applying penalty strategy is inadequate to deal with nonlinear programming problems well. It is important to lead infeasible individuals into the feasible set during the evolution process. Penalty and repair strategy are associated to improve the performance of the algorithm. Based on such thought that the constraint which has the highest degree of violation can be satisfied first by enlarging the penalty on the individuals and repair, repair operator is proposed to perform repair operation of infeasible individuals. At the same time, based on optimization design theory, a method has been proposed to establish initial population by using uniform design. Thus, repair genetic algorithm (RGA) is proposed.

关键词： nonlinear programming problem genetic algorithm uniform design penalty strategy repair strategy repair operator

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A quadratically convergent scaling Newton's method for nonlinear programming problems

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Optimization 1998年第3期44卷 219-233页

作者： Lin, Zhenghua Song, Daicai Department of Mathematics Jilin University Changchun 130023 China Fushun Petroleum Institute Fushun 113001 China

Similar to [6], we present a W-quadratically convergent scaling Newton's method for nonlinear programming problems. At each iteration, the multiplier variables and slack variables generated by this algorithm all keep in positive orthant of R^m, we only compute a Newton step and determine the step length with at most 2 m division operations, where m is the number of constraints. In Section 4, three examples also support the Q-quadratic convergence. © 1998 OPA (Overseas Publishers Association) N.V. Published by license under the Gordon and Breach Science Publishers imprint.

关键词： Interior point algorithm nonlinear programming problem Q-quadratic convergence Scaling Newton's method

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