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Efficient Convexification Strategy for generalized geometric programming Problems

INFORMS JOURNAL ON COMPUTING 2019年第2期31卷 226-234页

作者： Lu, Hao-Chun Yao, Liming Fu Jen Catholic Univ Dept Informat Management New Taipei 24205 Taiwan Sichuan Univ Business Sch Chengdu 610065 Sichuan Peoples R China

generalized geometric programming (GGP) problems consist of a signomial being minimized in the objective function subject to signomial constraints, and such problems have been utilized in various fields. After modeling numerous applications as GGP problems, solving them has become a significant requirement. A convex underestimator is considered an important concept to solve GGP problems for obtaining the global minimum. Among convex underestimators, variable transformation is one of the most popular techniques. This study utilizes an estimator to solve the difficulty of selecting an appropriate transformation between the exponential transformation and power convex transformation techniques and considers all popular types of transformation techniques to develop a novel and efficient convexification strategy for solving GGP problems. This proposed convexification strategy offers a guide for selecting the most appropriate transformation techniques on any condition of a signomial term to obtain the tightest convex underestimator. Several numerical examples in the online supplement are presented to investigate the effects of different convexification strategies on GGP problems and demonstrate the effectiveness of the proposed convexification strategy with regard to both solution quality and computation efficiency.

关键词： convex programming generalized geometric programming convexification strategy variables transformation technique global optimization

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A MILP formulation for generalized geometric programming using piecewise-linear approximations

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2015年第2期245卷 360-370页

作者： Tseng, Chung-Li Zhan, Yiduo Zheng, Qipeng P. Kumar, Manish Univ New S Wales UNSW Australia Business Sch Sydney NSW 2052 Australia Univ Cent Florida Dept Ind Engn & Management Syst Orlando FL 32816 USA Nexant Inc San Francisco CA 94105 USA

generalized geometric programming (GGP) problems are converted to mixed-integer linear programming (MILP) problems using piecewise-linear approximations. Our approach is to approximate a multiple-term log-sum function of the form log(x(1) + x(2) + ... +x(n)) in terms of a set of linear equalities or inequalities of logx(1), logx(2), ... , and logx(n), where x(1,) ... , x(n), are strictly positive. The advantage of this approach is its simplicity and readiness to implement and solve using commercial MILP solvers. While MILP problems in general are no easier than GGP problems, this approach is justified by the phenomenal progress of computing power of both personal computers and commercial MILP solvers. The limitation of this approach is discussed along with numerical tests. (C) 2015 Elsevier B.V. All rights reserved.

关键词： generalized geometric programming Piecewise-linear approximation Mixed-integer linear programming Gglobal optimization

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Global optimization for generalized geometric programming problems with discrete variables

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OPTIMIZATION 2013年第7期62卷 895-917页

作者： Shen, Pei-Ping Bai, Xiao-Di Henan Normal Univ Coll Math & Informat Sci Xinxiang 453007 Peoples R China

generalized geometric programming (GGP) problems occur frequently in engineering design and management, but most existing methods for solving GGP actually only consider continuous variables. This article presents a new branch-and-bound algorithm for globally solving GGP problems with discrete variables. For minimizing the problem, an equivalent monotonic optimization problem (P) with discrete variables is presented by exploiting the special structure of GGP. In the algorithm, the lower bounds are computed by solving ordinary linear programming problems that are derived via a linearization technique. In contrast to pure branch-and-bound methods, the algorithm can perform a domain reduction cut per iteration by using the monotonicity of problem (P), which can suppress the rapid growth of branching tree in the branch-and-bound search so that the performance of the algorithm is further improved. Computational results for several sample examples and small randomly generated problems are reported to vindicate our conclusions.

关键词： global optimization generalized geometric programming discrete variable reduction cut linearization technique

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Branch-reduction-bound algorithm for generalized geometric programming

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JOURNAL OF GLOBAL OPTIMIZATION 2013年第3期56卷 1123-1142页

作者： Shen, Peiping Li, Xiaoai Henan Normal Univ Coll Math & Informat Sci Xinxiang 453007 Peoples R China

This article presents a branch-reduction-bound algorithm for globally solving the generalized geometric programming problem. To solve the problem, an equivalent monotonic optimization problem whose objective function is just a simple univariate is proposed by exploiting the particularity of this problem. In contrast to usual branch-and-bound methods, in the algorithm the upper bound of the subproblem in each node is calculated easily by arithmetic expressions. Also, a reduction operation is introduced to reduce the growth of the branching tree during the algorithm search. The proposed algorithm is proven to be convergent and guarantees to find an approximative solution that is close to the actual optimal solution. Finally, numerical examples are given to illustrate the feasibility and efficiency of the present algorithm.

关键词： generalized geometric programming Global optimization Monotonic function Reduction operation

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Branch-reduction-bound algorithm for generalized geometric programming

Branch-reduction-bound algorithm for generalized geometric p...

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Global Optimization Workshops (GOWs)

作者： Shen, Peiping Li, Xiaoai Henan Normal Univ Coll Math & Informat Sci Xinxiang 453007 Peoples R China

关键词： generalized geometric programming Global optimization Monotonic function Reduction operation

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Improved logarithmic linearizing method for optimization problems with free-sign pure discrete signomial terms

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JOURNAL OF GLOBAL OPTIMIZATION 2017年第1期68卷 95-123页

作者： Lu, Hao-Chun Fu Jen Catholic Univ Dept Informat Management Coll Management 510 Jhongjheng Rd Sinjhuang City 242 Taipei County Taiwan

Free-sign pure discrete signomial (FPDS) terms are vital to and are frequently observed in many nonlinear programming problems, such as geometric programming, generalized geometric programming, and mixed-integer non-linear programming problems. In this study, all variables in the FPDS term are discrete variables. Any improvement to techniques for linearizing FPDS term contributes significantly to the solving of nonlinear programming problems;therefore, relative techniques have continually been developed. This study develops an improved exact method to linearize a FPDS term into a set of linear programs with minimal logarithmic numbers of zero-one variables and constraints. This method is tighter than current methods. Various numerical experiments demonstrate that the proposed method is significantly more efficient than current methods, especially when the problem scale is large.

关键词： Pure discrete signomial Free-sign variables Linearization generalized geometric programming Global optimization

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Nonmonotonic Trust Region Algorithm via the Conjugate Gradient Path for Unconstrained generalized geometric programming

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Chinese Quarterly Journal of Mathematics 2011年第2期26卷 285-289页

作者： DANG Ya-zheng JING Shu-jie LI Yu School of Management University of Shanghai for Science and Technology Shanghai 200093 China College of Mathematics and Information Science Henan Polytechnic University Jiaozuo 454000 China

In this paper,on the basis of making full use of the characteristics of unconstrained generalized geometric programming（GGP）,we establish a nonmonotonic trust region algorithm via the conjugate path for solving unconstrained GGP problem.A new type of condensation problem is presented,then a particular conjugate path is constructed for the problem,along which we get the approximate solution of the problem by nonmonotonic trust region algorithm,and further prove that the algorithm has global convergence and quadratic convergence properties.

关键词： generalized geometric programming condensation conjugate path trust region

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Global Optimization Method for Model Predictive Control Based on Wiener Model 12

Global Optimization Method for Model Predictive Control Base...

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International Multi-Conference on Systems, Signals and Devices (SSD)

作者： Degachi, Hajer Chagra, Wassila Ksouri, Moufida Tunis El Manar Univ Natl Engn Sch Tunis Anal Concept & Control Syst Lab LR11ES20 Tunis Tunisia Tunis El Manar Univ El Manar Preparatory Inst Engn Studies Anal Concept & Control Syst Lab LR11ES20 Tunis Tunisia

ISBN: (纸本)9781479917587

This paper deals with model predictive control based on Wiener model. The nonlinear block of the considered model is represented by a polynomial relation and the model's parameters are determined using the neural networks. A global optimization method, i.e. the generalized geometric programming method, is used to solve the nonconvex optimization control problem. The efficiency of the proposed controller is illustrated through a simulation example.

关键词： Model predictive control Wiener model neural network optimization generalized geometric programming

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A general algorithm for solving generalized geometric programming with nonpositive degree of difficulty

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2009年第1期44卷 139-158页

作者： Wang Yanjun Li Tao Liang Zhian Shanghai Univ Finance & Econ Shanghai 200433 Peoples R China

In this paper, a general algorithm for solving generalized geometric programming with nonpositive degree of difficulty is proposed. It shows that under certain assumptions the primal problem can be transformed and decomposed into several subproblems which are easy to solve, and furthermore we verify that through solving these subproblems we can obtain the optimal value and solutions of the primal problem which are global solutions. At last, some examples are given to vindicate our conclusions.

关键词： generalized geometric programming Degree of difficulty Decomposition method

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Global Optimization Method for Model Predictive Control Based on Wiener Model

Global Optimization Method for Model Predictive Control Base...

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International Multi-Conference on Systems, Signals and Devices

作者： Hajer Degachi Wassila Chagra Moufida Ksouri Tunis El Manar University National Engineering School of Tunis LR11ES20 Analysis Conception and Control of Systems Laboratory Tunis El Manar University El Manar Preparatory Institute for Engineering Studies LR11ES20 Analysis Conception and Control of Systems Laboratory

ISBN: (纸本)9781479917594

关键词： Model predictive control Wiener model neural network optimization generalized geometric programming

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