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引用; SIAM REVIEW 1976年第1期18卷 1-51页; 作者： PETERSON, EL NORTHWESTERN UNIV DEPT IND ENGN MANAGEMENT SCIEVANSTONIL 60201 NORTHWESTERN UNIV DEPT MATHEVANSTONIL 60201; Contrary to popular belief, geometric programming is not just a special technique for studying the very important class of posynomial (optimization) problems. It is really a very general mathematical theory that is es... 详细信息; Contrary to popular belief, geometric programming is not just a special technique for studying the very important class of posynomial (optimization) problems. It is really a very general mathematical theory that is especially useful for studying a large class of separable problems. Its practical efficacy is due mainly to the fact that many important (seemingly inseparable) problems can actually be formulated as separable geometric programming problems, by fully exploiting their linear algebraic structure. Some examples are: nonlinear network flow problems (both single-commodity and multicommodity), discrete optimal control problems with linear dynamics, optimal location problems of the generalized Fermat type, ( $l_{P}$; 来源：评论; 学校读者我要写书评

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An efficient uncertain chance constrained geometric programming model based on value-at-risk for truss structure optimization problems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2025年 458卷

作者： Chen, Jie Li, Haoxuan Yang, Xiangfeng Univ Int Business & Econ Sch Informat Technol & Management Beijing 100029 Peoples R China

Uncertain geometric programming is a type of geometric programming involving uncertain variables. As described in the literature, the uncertain geometric programming model based on expected value cannot reflect the risk preference of decision-makers. It motivates us to establish an uncertain geometric programming model based on value-at-risk to describe the risk level that managers can tolerate. Firstly, we propose the uncertain geometric programming model based on value-at-risk. Then, according to the operational law in uncertainty theory, this model is transformed into a crisp and equivalent form. Three numerical examples are used to verify the model's efficacy, and the paper emphasizes the influence of confidence level in the objective function and the constraints. In addition, the paper discusses the expected value model under an uncertain environment and presents the difference between expected value and value-at-risk. Finally, we apply the model to the problem of a two-bar truss, and the optimal solution can be obtained within the risk level that the structural designer can accept.

关键词： Uncertainty theory Chance constraints geometric programming Value-at-risk Two-bar truss

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geometric programming problem with single-term exponents subject to max-product fuzzy relational equations

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MATHEMATICAL AND COMPUTER MODELLING 2011年第1-2期53卷 55-62页

作者： Zhou, XueGang Ahat, Rashida GuangDong Univ Finance Dept Appl Math Guangzhou 510521 Guangdong Peoples R China Xinjiang Univ Sch Math & Syst Sci Urumqi 830046 Xinjiang Peoples R China

In this paper, an optimization model for minimizing an objective function with single-term exponents subject to fuzzy relational equations specified in max-product composition is presented. The solution set of such a fuzzy relational equation is a non-convex set. First, we present some properties for the optimization problem under the assumptions of both negative and nonnegative exponents in the objective function. Second, an efficient procedure is developed to find an optimal solution without looking for all the potential minimal solutions and without using the value matrix. An example is provided to illustrate the procedure. (C) 2010 Elsevier Ltd. All rights reserved.

关键词： Fuzzy optimization Max-product composition Fuzzy relational equations geometric programming

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geometric programming Applied to Multipoint-to-Multipoint MIMO Relay Networks

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JOURNAL OF COMMUNICATIONS AND NETWORKS 2015年第3期17卷 241-246页

作者： Kim, Jaesin Kim, Suil Pak, Ui-Young Agcy Def Dev Taejon 300600 South Korea

In this paper, we consider a relaying system which employs a single relay in a wireless network with distributed sources and destinations. Here, all source, destination, and relay nodes are equipped with multiple antennas. For amplify-and-forward relay systems, we confirm the achievable sum rate through a joint multiple source precoders and a single relay filter design. To this end, we propose a new linear processing scheme in terms of maximizing the sum rate performance by applying a blockwise relaying method combined with geometric programming techniques. By allowing the global channel knowledge at the source nodes, we show that this joint design problem is formulated as a standard geometric program, which can guarantees a global optimal value under the modified sum rate criterion. Simulation results show that the proposed blockwise relaying scheme with the joint power allocation method provides substantial sum rate gain compared to the conventional schemes.

关键词： Amplify-and-forward block diagonalization geometric programming multiple-input multiple-output (MIMO) relay

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geometric programming with a single-term exponent subject to bipolar max-product fuzzy relation equation constraints

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FUZZY SETS AND SYSTEMS 2020年 397卷 61-83页

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

We study geometric programming with a single-term exponent subject to bipolar max-product fuzzy relation equation constraints in the area of economics and the covering problem. The structure of its feasible domain is characterized and the lower and upper bound vectors on its solution set are determined. It is shown that each component of one of its optimal solutions is the corresponding component of either the lower bound or the upper bound vector. This interesting property helps us to create a value matrix and present some necessary and sufficient conditions for its consistency checking. Moreover, some sufficient conditions are proposed to detect one of its optimal solutions without solving the problem. A modified branch-and-bound method is extended to solve the problem, in a general case, with use of the value matrix. An efficient algorithm is then designed to solve the problem using the sufficient conditions and the modified branch-and-bound method. Its computational complexity is also analyzed. Finally, some examples are provided to illustrate its importance and the steps of the algorithm. (C) 2019 Elsevier B.V. All rights reserved.

关键词： Bipolar fuzzy relation equation geometric programming Max-product composition Modified branch-and-bound method

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geometric programming with fuzzy parameters in engineering optimization

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INTERNATIONAL JOURNAL OF APPROXIMATE REASONING 2007年第3期46卷 484-498页

作者： Liu, Shiang-Tai Vanung Univ Grad Sch Business & Management Chungli 320 Taiwan

geometric programming provides a powerful tool for solving a variety of engineering optimization problems. Many applications of geometric programming are engineering design problems in which some of the problem parameters are estimates of actual values. When the parameters in the problem are imprecise, the calculated objective value should be imprecise as well. This paper develops a procedure to derive the fuzzy objective value of the fuzzy posynomial geometric programming problem when the exponents of decision variables in the objective function, the cost and the constraint coefficients, and the right-hand sides are fuzzy numbers. The idea is based on Zadeh's extension principle to transform the fuzzy geometric programming problem into a pair of two-level of mathematical programs. Based on duality algorithm and a simple algorithm, the pair of two-level mathematical programs is transformed into a pair of conventional geometric programs. The upper bound and lower bound of the objective value are obtained by solving the pair of geometric programs. From different values of a, the membership function of the objective value is constructed. Two examples are used to illustrate that the whole idea proposed in this paper. (C) 2007 Elsevier Inc. All rights reserved.

关键词： geometric programming fuzzy set optimization

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geometric programming Problems with Triangular and Trapezoidal Twofold Uncertainty Distributions

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2024年第3期200卷 978-1016页

作者： Mondal, Tapas Ojha, Akshay Kumar Pani, Sabyasachi Indian Inst Technol Bhubaneswar Sch Basic Sci Bhubaneswar India

geometric programming is a well-known optimization tool for dealing with a wide range of nonlinear optimization and engineering problems. In general, it is assumed that the parameters of a geometric programming problem are deterministic and accurate. However, in the real-world geometric programming problem, the parameters are frequently inaccurate and ambiguous. To tackle the ambiguity, this paper investigates the geometric programming problem in an uncertain environment, with the coefficients as triangular and trapezoidal twofold uncertain variables. In this paper, we introduce uncertain measures in a generalized version and focus on more complicated twofold uncertainties to propose triangular and trapezoidal twofold uncertain variables within the context of uncertainty theory. We develop three reduction methods to convert triangular and trapezoidal twofold uncertain variables into singlefold uncertain variables using optimistic, pessimistic, and expected value criteria. Reduction methods are used to convert the geometric programming problem with twofold uncertainty into the geometric programming problem with singlefold uncertainty. Furthermore, the chance-constrained uncertain-based framework is used to solve the reduced singlefold uncertain geometric programming problem. Finally, a numerical example is provided to demonstrate the effectiveness of the procedures.

关键词： geometric programming Triangular twofold uncertain variable Trapezoidal twofold uncertain variable Reduction method

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geometric programming with several discrete variables: Algorithms employing generalized benders' decomposition

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ENGINEERING OPTIMIZATION 1995年第3期25卷 201-212页

作者： Choi, JC Bricker, DL Department of Industrial Engineering The University of Iowa Iowa City Iowa 52242 United States

geometric programming problems in which several of the variables are restricted to assume either integer values or one of a set of standard sizes are known as Semi-Discrete geometric programming problems. In this paper several variations of Generalized Benders' Decomposition are described for these problems and some computational experience is presented.

关键词： geometric programming discrete optimization generalized Benders' decomposition

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geometric programming solution of second degree difficulty for carbon ejection controlled reliable smart production system

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RAIRO-OPERATIONS RESEARCH 2022年第2期56卷 1013-1029页

作者： Kugele, Andreas Se Ho Ahmed, Waqas Sarkar, Biswajit Yonsei Univ Dept Ind Engn 50 Yonsei Ro Seoul 03722 South Korea Natl Univ Sci & Technol NUST Business Sch Dept Operat & Supply Chain Islamabad 44000 Pakistan Saveetha Univ Saveetha Dent Coll Ctr Transdisciplinary Res CFTR Saveetha Inst Med & Tech Sci 162 Poonamallee High Rd Chennai 600077 Tamil Nadu India

Smart manufacturing systems should always aim to be fully sustainable while simultaneously being as reliable as possible which is difficult to reach. Furthermore, climate change especially by carbon emission in the industry is a significant topic and carbon emission should be controlled and reduced to save the environment. Contributing towards a greener environment in a positive manner is done by reducing the number of insufficient items that are produced in a smart production system which also can be reached with higher reliability in the system. Therefore, this study models a smart reliable production system with controlled carbon ejection. To solve the proposed smart production system in this study, a geometric programming approach with a degree of difficulty level two is used which results in optimum results that are quasi-closed. Furthermore, numerical experiments are conducted to validate the proposed model and prove that by using a higher degree geometric programming approach, an optimal solution is found. The numerical results do not only show optimal solutions but also that the smart production system with controlled carbon ejection is reliable.

关键词： Smart production system reliability geometric programming setup reduction controlled carbon ejection

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geometric programming approach to doping profile design optimization of metal-oxide-semiconductor devices

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MATHEMATICAL AND COMPUTER MODELLING 2013年第1-2期58卷 344-354页

作者： Li, Yiming Chen, Ying-Chieh Natl Chiao Tung Univ Dept Elect & Comp Engn Parallel & Sci Comp Lab Hsinchu 300 Taiwan

We study one-dimensional doping profile design optimization problem of metal-oxide-semiconductor (MOS) devices using a geometric programming (GP) technique. To model the explored optimal doping profile into a GP problem, the subthreshold swing is formulated as an objective function and the on- and off-state currents are considered as constraints for solving the corresponding optimal doping profile. The GP problem is a special type of convex optimization and is solved globally and efficiently using the existing numerical solvers in GGPLAB. The accuracy of optimized results is validated by comparing with numerical semiconductor device simulation. This approach provides a way to optimize doping problem which may benefit manufacturing of MOS devices. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Doping profile Design optimization geometric programming MOS devices

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