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Solving a class of geometric programming problems by an efficient dynamic model

COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2013年第3期18卷 692-709页

作者： Nazemi, Alireza Sharifi, Elahe Shahrood Univ Technol Sch Math Sci Dept Math Shahrood Iran

In this paper, a neural network model is constructed on the basis of the duality theory, optimization theory, convex analysis theory, Lyapunov stability theory and LaSalle invariance principle to solve geometric programming (GP) problems. The main idea is to convert the GP problem into an equivalent convex optimization problem. A neural network model is then constructed for solving the obtained convex programming problem. By employing Lyapunov function approach, it is also shown that the proposed neural network model is stable in the sense of Lyapunov and it is globally convergent to an exact optimal solution of the original problem. The simulation results also show that the proposed neural network is feasible and efficient. (C) 2012 Elsevier B.V. All rights reserved.

关键词： Neural network geometric programming Convex programming Convergent Stability

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Optimal allocation of local feedback in multistage amplifiers via geometric programming

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-REGULAR PAPERS 2001年第1期48卷 1-11页

作者： Dawson, JL Boyd, SP Hershenson, MD Lee, TH Stanford Univ Dept Elect Engn Stanford CA 94305 USA

We consider the problem of optimally allocating local feedback to the stages of a multistage amplifier. The local feed-back gains affect many performance indexes for the overall amplifier, such as bandwidth, gain, rise time, delay, output signal swing, linearity, and noise performance, in a complicated and nonlinear fashion, making optimization of the feedback gains a challenging problem. In this paper, we show that this problem, though complicated and nonlinear, can be formulated as a special type of optimization problem called geometric programming. geometric programs can be solved globally and efficiently using recently developed interior-point methods. Our method, therefore, gives a complete solution to the problem of optimally allocating local feedback gains, taking into account a wide variety of constraints.

关键词： amplifiers analog circuits circuit optimization design automation geometric programming sensitivity

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Neutrosophic Goal geometric programming Problem and Its Application to Multi-objective Reliability Optimization Model

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INTERNATIONAL JOURNAL OF FUZZY SYSTEMS 2018年第6期20卷 1986-1994页

作者： Kundu, Tanmay Islam, Sahidul Univ Kalyani Dept Math Kalyani 741235 W Bengal India

This paper presents the goal geometric programming method in neutrosophic environment. Neutrosophic set is one of the most useful tools to express uncertainty, impreciseness in a more general way compare to fuzzy set and intuitionistic fuzzy set. Thus, the proposed approach is described here as an extension of fuzzy goal geometric programming and intuitionistic fuzzy goal geometric programming. To demonstrate the methodology and applicability of the proposed approach, a multi-objective nonlinear reliability optimization model is taken here and it is evaluated comparing the result obtained by the proposed method with the solution obtained in intuitionistic fuzzy goal geometric programming technique at the end of this paper.

关键词： Goal programming geometric programming Neutrosophic set Reliability optimization

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Multi-objective geometric programming problem with ε-constraint method

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APPLIED MATHEMATICAL MODELLING 2014年第2期38卷 747-758页

作者： Ojha, A. K. Biswal, K. K. Indian Inst Technol Bhubaneswar Sch Basic Sci Bhubaneswar 751013 Orissa India NIIS Inst Business Adm Dept Math Bhubaneswar 752054 Orissa India

In multi-objective geometric programming problem there are more than one objective functions: There is no single optimal solution which simultaneously optimizes all the objective functions. Under these conditions the decision makers always search for the most "preferred" solution, in contrast to the optimal solution. A few mathematical programming methods namely fuzzy programming, goal programming and weighting methods have been applied in the recent past to find the compromise solution. In this paper epsilon-constraint method has been applied to find the non-inferior solution. A brief solution procedure of epsilon-constraint method has been presented to find the non-inferior solution of the multi-objective programming problems. Further, the multi-objective programming problems is solved by the fuzzy programming technique to find the optimal compromise solution. Finally, two numerical examples are solved by both the methods and compared with their obtained solutions. (C) 2013 Elsevier Inc. All rights reserved.

关键词： geometric programming Vector minimization epsilon-Constraint method Fuzzy programming Duality theory Pareto optimal solution

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Variation-Aware geometric programming Models for the Clock Network Buffer Sizing Problem

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IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS 2014年第4期33卷 532-545页

作者： Rakai, Logan Farshidi, Amin Westwick, David Behjat, Laleh Univ Calgary Dept Elect & Comp Engn Calgary AB T2N 1N4 Canada

In this paper, we present and analyze four efficient models that produce significantly improved results by optimizing conflicting power and skew objectives in the clock network buffer sizing problem. Each model is in geometric programming format and has certain advantages, such as maximum reduction in power, robustness to process variation, and striking a balance between skew and power optimization. The buffer sizing problem is formulated as a geometric programming problem to provide globally optimal solutions to the four models. We also show that a geometric programming multiobjective model can be used to optimize both power and skew without requiring any tuning from a designer. The presented self-tuning multiobjective formulation not only provides optimal solutions for buffer sizes, but also finds the tuning parameters that result in overall combined reduction in power and skew without loss of convexity. The effectiveness of the models are illustrated on several publicly available benchmarks. The models provide on average 40% to 60% improvement in power while reducing skew in several cases. We have also proposed a smart heuristic for discretization of the continuous geometric programming solution that preserves skew and power. Finally, we provide a guideline for designers to decide which one of the proposed models is the most appropriate for their needs.

关键词： Clock networks geometric programming robust optimization

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Global optimization of signomial geometric programming problems

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2014年第3期233卷 500-510页

作者： Xu, Gongxian Bohai Univ Dept Math Jinzhou 121013 Peoples R China

This paper presents a global optimization approach for solving signomial geometric programming problems. In most cases nonconvex optimization problems with signomial parts are difficult, NP-hard problems to solve for global optimality. But some transformation and convexification strategies can be used to convert the original signomial geometric programming problem into a series of standard geometric programming problems that can be solved to reach a global solution. The tractability and effectiveness of the proposed successive convexification framework is demonstrated by seven numerical experiments. Some considerations are also presented to investigate the convergence properties of the algorithm and to give a performance comparison of our proposed approach and the current methods in terms of both computational efficiency and solution quality. (C) 2013 Elsevier B.V. All rights reserved.

关键词： geometric programming Signomial geometric programming Global optimization Convexification

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Temperature-aware floorplanning via geometric programming

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MATHEMATICAL AND COMPUTER MODELLING 2010年第7-8期51卷 927-934页

作者： Chen, Ying-Chieh Li, Yiming Natl Chiao Tung Univ Inst Commun Engn Hsinchu 300 Taiwan Natl Chiao Tung Univ Dept Elect Engn Hsinchu 300 Taiwan Natl Nano Device Labs Hsinchu 300 Taiwan

With microprocessor power densities escalating rapidly when technology scales below nanometer regime, there is an exigent need for developing innovative cooling systems for electronic product design. The high temperature of chips greatly affects its reliability, raises the leakage power consumed to unprecedented levels, and makes cooling systems significantly more expensive. The maximum temperature of a block in a chip depends not only on its own power density, but also on the chip area in each blocks. In this paper, we employ geometric programming (GP) for the optimization problem of temperature reduction and chip area floorplanning. We notice that the formulated model is a nonlinear convex problem;consequently, its solution can be solved GP method. Based upon an incremental floorplanning problem together with the GP model, the temperature-aware floorplanning scheme significantly reduces peak module temperature with minimal chip area impact. For Microelectronics Center of North Carolina (MCNC) ami33 under a testing environment temperature of 0 degrees C, compared with the maximum temperature of the original module, the maximum temperature of the optimized one could be reduced from 90 degrees C to 10 degrees C, where the minimized chip area is about 700 mm(2). For the case of MCNC ami49, the maximum temperature reduction is 60 degrees C (i.e., its reduction is from 65 degrees C to 5 degrees C) with a minimal chip area of 2500 mm(2). We have numerically found a floorplan which can reduce the maximum temperature of the chip and minimize the chip area while maintaining comparable performance simultaneously. (C) 2009 Elsevier Ltd. All rights reserved.

关键词： Optimal designs Floorplanning geometric programming Nonlinear programming Temperature aware programming involving graphs or networks Numerical optimization Incremental floorplanning VLSI circuit physical design

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A flexible programming approach based on intuitionistic fuzzy optimization and geometric programming for solving multi-objective nonlinear programming problems

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EXPERT SYSTEMS WITH APPLICATIONS 2018年 93卷 245-256页

作者： Jafarian, E. Razmi, J. Baki, M. F. Univ Tehran Coll Engn Sch Ind Engn Tehran Iran Univ Windsor Odette Sch Business Windsor ON Canada

In this paper, a novel method is proposed to support the process of solving multi-objective nonlinear programming problems subject to strict or flexible constraints. This method assumes that the practical problems are expressed in the form of geometric programming problems. Integrating the concept of intuitionistic fuzzy sets into the solving procedure, a rich structure is provided which can include the inevitable uncertainties into the model regarding different objectives and constraints. Another important feature of the proposed method is that it continuously interacts with the decision maker. Thus, the decision maker could learn about the problem, thereby a compromise solution satisfying his/hers preferences could be obtained. Further, a new two-step geometric programming approach is introduced to determine Pareto-optimal compromise solutions for the problems defined during different iterative steps. Employing the compensatory operator of "weighted geometric mean", the first step concentrates on finding an intuitionistic fuzzy efficient compromise solution. In the cases where one or more intuitionistic fuzzy objectives are fully achieved, a second geometric programming model is developed to improve the resulting compromise solution. Otherwise, it is concluded that the resulting solution vectors simultaneously satisfy both of the conditions of intuitionistic fuzzy efficiency and Pareto-optimality. The models forming the proposed solving method are developed in a way such that, the posynomiality of the defined problem is not affected. This property is of great importance when solving nonlinear programming problems. A numerical example of multi-objective nonlinear programming problem is also used to provide a better understanding of the proposed solving method. (c) 2017 Published by Elsevier Ltd.

关键词： Multi-objective nonlinear programming geometric programming Intuitionistic fuzzy optimization Interactive decision making

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Posynomial geometric programming with interval exponents and coefficients

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2008年第1期186卷 17-27页

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

geometric programming provides a powerful tool for solving nonlinear problems where nonlinear relations can be well presented by an exponential or power function. In the real world, many applications of geometric programming are engineering design problems in which some of the problem parameters are estimates of actual values. This paper develops a solution method when the exponents in the objective function, the cost and the constraint coefficients, and the right-hand sides are imprecise and represented as interval data. Since the parameters of the problem are imprecise, the objective value should be imprecise as well. A pair of two-level mathematical programs is formulated to obtain the upper bound and lower bound of the objective values. Based on the duality theorem and by applying a variable separation technique, the pair of two-level mathematical programs is transformed into a pair of ordinary one-level geometric programs. Solving the pair of geometric programs produces the interval of the objective value. The ability of calculating the bounds of the objective value developed in this paper might help lead to more realistic modeling efforts in engineering optimization areas. (c) 2007 Elsevier B.V. All rights reserved.

关键词： geometric programming interval parameter duality theorem optimization

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OPTIMIZING THE NUMBER OF TOOLS AND CUTTING PARAMETERS IN MULTITOOL TURNING FOR MULTIPLE OBJECTIVES THROUGH geometric-programming

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APPLIED MATHEMATICAL MODELLING 1986年第3期10卷 162-169页

作者： JHA, NK Mechanical Engineering Department Manhattan College Bronx New York 10471USA

Multiple tool turning is often considered an economic alternative to single tool turning but for its rational application, however, accurate process planning is essential. This paper investigates the automatic (computerized) process planning for a multiple tool turning operation. A unified or multiple objective function based on cost of production and rate of production has been developed. The unified objective function is optimized, subject to constraints, such as component characteristics, machine power available, cutting speed, feed, and depth of cut rate. In all, 14 constraints are applied to the multiple objective function. During analysis, however, it was discovered that the integrality requirement on the number of tools, poses a great problem. Due to this constraint, the optimization problem became one of the ‘nonlinear mixed integer programming’ class. Nonlinear programming, coupled with integer programming, is always a complex problem and was tackled by suitably modifying the geometric programming technique. The mathematical model developed was further modified to include discrete steps of speed and feed on the machine. It was found that in discrete cases, the number of passes required for completing the job was more than in the continuous case. The approach has been demonstrated through an example.

关键词： multi-tool turning geometric programming cutting parameters

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