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separable programming problem with bipolar max-T fuzzy relation equation constraints

FUZZY SETS AND SYSTEMS 2024年 484卷

作者： Molai, Ali Abbasi Damghan Univ Sch Math & Comp Sci POB 36715-364 Damghan Iran

One of the most important classes of nonlinear programming problems is separable programming problem due to its applications in theory and practice. This paper studies this class of the problems subject to a system of bipolar fuzzy relation equations using the max -T composition operator, where T is a continuous and Archimedean t -norm. Its feasible solution set structure is determined by two vectors called lower and upper bound vector of its feasible domain. It is shown that there exists an optimal solution for the problem with max -continuous and Archimedean t -norm such that its components are the corresponding components of either the lower or upper bound vector. Based on the interesting property, some sufficient conditions are proposed to detect some of its optimal components or one of its optimal solutions. These conditions can reduce the dimensions of the original problem when some its optimal components are determined. The objective function of the reduced problem is equivalently rewritten in a specific form. This form guides us to design a value matrix based on the characteristic matrix and ascending or descending of univariate functions constituting the objective function. An approach is extended on the matrix to find the optimal solution of the (reduced) problem. It is important to note that a number of problems can completely be solved using the sufficient conditions.

关键词： Fuzzy relations Fuzzy mathematical programming Bipolar fuzzy relation equation separable programming Max-continuous t-norm composition Matrix-based-algorithm

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Converting general nonlinear programming problems into separable programming problems with feedforward neural networks

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NEURAL NETWORKS 2003年第7期16卷 1059-1074页

作者： Lu, BL Ito, K Shanghai Jiao Tong Univ Dept Comp Sci & Engn Shanghai 200030 Peoples R China Tokyo Inst Technol Dept Computat Intelligence & Syst Sci Midori Ku Yokohama Kanagawa 2268502 Japan

In this paper we present a method for converting general nonlinear programming (NLP) problems into separable programming (SP) problems by using feedforward neural networks (FNNs). The basic idea behind the method is to use two useful features of FNNs: their ability to approximate arbitrary continuous nonlinear functions with a desired degree of accuracy and their ability to express nonlinear functions in terms of parameterized compositions of functions of single variables. According to these two features, any nonseparable objective functions and/or constraints in NLP problems can be approximately expressed as separable functions with FNNs. Therefore, any NLP problems can be converted into SP problems. The proposed method has three prominent features. (a) It is more general than existing transformation techniques: (b) it can be used to formulate optimization problems as SP problems even when their precise analytic objective function and/or constraints are unknown;(c) the SP problems obtained by the proposed method may highly facilitate the selection of grid points for piecewise linear approximation of nonlinear functions. We analyze the computational complexity of the proposed method and compare it with an existing transformation approach. We also present several examples to demonstrate the method and the performance of the simplex method with the restricted basis entry rule for solving SP problems. (C) 2003 Elsevier Science Ltd. All rights reserved.

关键词： nonlinear programming separable programming feedforward neural network piecewise linear approximation separable function simplex method

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GLOBAL MINIMIZATION OF LARGE-SCALE CONSTRAINED CONCAVE QUADRATIC PROBLEMS BY separable programming

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MATHEMATICAL programming 1986年第2期34卷 163-174页

作者： ROSEN, JB PARDALOS, PM 1.Computer Science Department University of Minnesota 55455 Minneapolis MN USA

The global minimization of a large-scale linearly constrained concave quadratic problem is considered. The concave quadratic part of the objective function is given in terms of the nonlinear variablesx ∈R ⁿ, while the linear part is in terms ofy ∈R ^k. For large-scale problems we may havek much larger thann. The original problem is reduced to an equivalent separable problem by solving a multiple-cost-row linear program with 2n cost rows. The solution of one additional linear program gives an incumbent vertex which is a candidate for the global minimum, and also gives a bound on the relative error in the function value of this incumbent. Ana priori bound on this relative error is obtained, which is shown to be ≤ 0.25, in important cases. If the incumbent is not a satisfactory approximation to the global minimum, a guaranteedε-approximate solution is obtained by solving a single liner zero–one mixed integer programming problem. This integer problem is formulated by a simple piecewise-linear underestimation of the separable problem.

关键词： Global Minimization separable programming Quadratic programming Large-Scale

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COMPUTATIONAL ASPECTS OF 2-SEGMENT separable programming

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MATHEMATICAL programming 1983年第1期26卷 21-39页

作者： MEYER, RR Computer Sciences Department University of Wisconsin 53706 Madison WI USA

Recursive separable programming algorithms based on local, two-segment approximations are described for the solution of separable convex programs. Details are also given for the computation of lower bounds on the optimal value by both a primal and a dual approach, and these approaches are compared. Computational comparisons of the methods are provided for a variety of test problems, including a water supply application (with more than 600 constraints and more than 900 variables) and an econometric modelling problem (with more than 200 variables).

关键词： separable programming Network Optimization Piecewise-Linear Approximation Error Bounds

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A global optimization method for nonconvex separable programming problems

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1999年第2期117卷 275-292页

作者： Li, HL Yu, CS Natl Chiao Tung Univ Inst Informat Management Hsinchu 30050 Taiwan Shih Chien Univ Dept Informat Management Taipei 10020 Taiwan

Conventional methods of solving nonconvex separable programming (NSP) problems by mixed integer programming methods requires adding numerous 0-1 variables. In this work, we present a new method of deriving the global optimum of a NSP program using less number of 0-1 variables. A separable function is initially expressed by a piecewise linear function with summation of absolute terms. Linearizing these absolute terms allows us to convert a NSP problem into a linearly mixed 0-1 program solvable for reaching a solution which is extremely close to the global optimum. (C) 1999 Elsevier Science B.V. All rights reserved.

关键词： goal programming piecewise linear function separable programming

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A modified separable programming approach to weapon system allocation problems.

A modified separable programming approach to weapon system a...

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作者： McLaughlin, Thomas Robert Jr. United States Naval Postgraduate School

学位级别：硕士

This thesis considers mathematical techniques for com- puting the optimal allocation of weapons from m different systems against n undefended targets. A standard nonlinear programming problem is considered. A discussion is given on John Danskin's Algorithm for the determination of the optimal values of the lagrange multipliers for this problem Using a transformation of variables, the nonlinear problem is reformulated as a separable problem and solved by sepa- rable programming. A new method, the hybrid algorithm, for the determination of the optimal lagrange multipliers is developed.

关键词： Weapon System Allocation Problems separable programming Danskin's Algorithm Nonlinear Allocation Models Computational Algorithm

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LINEAR, INTEGER, separable AND FUZZY-programming PROBLEMS - A UNIFIED APPROACH TOWARDS REFORMULATION

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JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY 1988年第2期39卷 161-171页

作者： DARBYDOWMAN, K LUCAS, C MITRA, G YADEGAR, J BRUNEL UNIV UXBRIDGE UB8 3PHMIDDXENGLAND UNIV LONDON QUEEN MARY COLL LONDON E1 4NSENGLAND

For mathematical programming (MP) to have greater impact as a decision tool, MP software systems must offer suitable support in terms of model communication and modelling techniques. In this paper, modelling techniques that allow logical restrictions to be modelled in integer programming terms are described, and their implications discussed. In addition, it is illustrated that many classes of non-linearities which are not variable separable may be, after suitable algebraic manipulation, put in a variable separable form. The methods of reformulating the fuzzy linear programming problem as a max-min problem is also introduced. It is shown that analysis of bounds plays a key role in the following four important contexts: model reduction, reformulation of logical restrictions as 0-1 mixed integer programmes, reformulation of non-linear programmes as variable separable programmes and reformulation of fuzzy linear programmes. It is observed that, as well as incorporating an interface between the modeller and the optimizer, there is a need to make available to the modeller software facilities which support the model reformulation techniques described here.

关键词： fuzzy sets integer programming linear programming modelling non-linear programming separable programming

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A BRANCH AND BOUND ALGORITHM FOR separable CONCAVE programming

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Journal of Computational Mathematics 2004年第6期22卷 895-904页

作者： Hong-gangXue Cheng-xianXu Feng-minXu DepartmentofMathematics FacultyofScienceXi'anJiaotongUniversityxi'an710049China DepartmentofMathematics FacultyofScienceXi'anJiaotongUniversityXi'an710049China

In this paper, we propose a new branch and bound algorithm for the solution of large scale separable concave programming problems. The largest distance bisection (LDB) technique is proposed to divide rectangle into sub-rectangles when one problem is branched into two subproblems. It is proved that the LDB method is a normal rectangle subdivision(NRS). Numerical tests on problems with dimensions from 100 to 10000 show that the proposed branch and bound algorithm is efficient for solving large scale separable concave programming problems, and convergence rate is faster than ω-subdivision method.

关键词： ω-subdivision Branch and bound algorithm Largest distance bisection Normal rectangle subdivision separable programming

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Branch and bound algorithm for separable concave programming

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Journal of Computational Mathematics 2004年第6期22卷 895-904页

作者： Xue, Hong-Gang Xu, Cheng-Xian Xu, Feng-Min Dept. of Math. Faculty of Sci. Xi'an Jiaotong Univ. Xi'an 710049 China Dept. of Math. Faculty of Sci. Xi'an Univ. of Technol. Xi'an 710049 China

We propose a new branch and bound algorithm for the solution of large scale separable concave programming problems. The largest distance bisection (LDB) technique is proposed to divide rectangle into sub-rectangles when one problem is branched into two subproblems. It is proved that the LDB method is a normal rectangle subdivision (NRS). Numerical tests on problems with dimensions from 100 to 10000 show that the proposed branch and bound algorithm is efficient for solving large scale separable concave programming problems, and convergence rate is faster than ω-subdivision method.

关键词： ω-subdivision Branch and bound algorithm Largest distance bisection Normal rectangle subdivision separable programming

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OPTIMAL OBJECTIVE FUNCTION APPROXIMATION FOR separable CONVEX QUADRATIC-programming

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MATHEMATICAL programming 1994年第1期67卷 133-142页

作者： GUDER, F MORRIS, JG UNIV WISCONSIN SCH BUSINESS MADISON WI 53706 USA

We present an optimal piecewise-linear approximation method for the objective function of separable convex quadratic programs. The method provides guidelines on how many grid points to use and how to position them for... 详细信息

关键词： separable programming QUADRATIC programming

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