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Linearization method of global optimization for generalized geometric programming

APPLIED MATHEMATICS AND COMPUTATION 2005年第1期162卷 353-370页

作者： Shen, P Henan Normal Univ Sch Math & Informat Sci Xinxiang 453002 Peoples R China

Many methods for solving generalized geometric programming (GGP) problem can only find locally optimal solutions. But up to now, less work has been devoted to solving global optimization of GGP due to the inherent difficulty. This paper gives a method for finding the globally optimal solutions of GGP. Utilizing an exponentially variable transformation and some other techniques the initial nonlinear and nonconvex GGP problem is reduced to a sequence of linear programming problems. The proposed algorithm is proven that it is convergent to the global minimum through the solutions of a series of linear programming problems. Several GGP examples in the literatures are tested to demonstrate that the proposed method can systematically solve these examples to find the global optimum within a prespecified error. (C) 2004 Elsevier Inc. All rights reserved.

关键词： generalized geometric programming global optimization linearization branch-and-bound

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A tutorial on geometric programming

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OPTIMIZATION AND ENGINEERING 2007年第1期8卷 67-127页

作者： Boyd, Stephen Kim, Seung-Jean Vandenberghe, Lieven Hassibi, Arash Stanford Univ Dept Elect Engn Informat Syst Lab Stanford CA 94305 USA Univ Calif Los Angeles Dept Elect Engn Los Angeles CA 90095 USA Clear Shape Technol Inc Sunnyvale CA 94086 USA

A geometric program (GP) is a type of mathematical optimization problem characterized by objective and constraint functions that have a special form. Recently developed solution methods can solve even large-scale GPs extremely efficiently and reliably;at the same time a number of practical problems, particularly in circuit design, have been found to be equivalent to (or well approximated by) GPs. Putting these two together, we get effective solutions for the practical problems. The basic approach in GP modeling is to attempt to express a practical problem, such as an engineering analysis or design problem, in GP format. In the best case, this formulation is exact;when this is not possible, we settle for an approximate formulation. This tutorial paper collects together in one place the basic background material needed to do GP modeling. We start with the basic definitions and facts, and some methods used to transform problems into GP format. We show how to recognize functions and problems compatible with GP, and how to approximate functions or data in a form compatible with GP (when this is possible). We give some simple and representative examples, and also describe some common extensions of GP, along with methods for solving (or approximately solving) them.

关键词： convex optimization geometric programming generalized geometric programming interior-point methods

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Solving non convex Min-Max predictive controller

Solving non convex Min-Max predictive controller

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Conference on Information Decision and Control

作者： Badreddine, Bouzouita Faouzi, Bouani Mekki, Ksouri Laboratory of Analyse and Control of Systems (ACS) ENIT Tunisia National Institute of Applied Sciences and Technology Tunis Tunisia

ISBN: (纸本)9781424409013

This paper focuses on robust predictive control (RPC) using uncertain Controlled Auto Regressive Integrated Moving Average (CARIMA) model. To take into account the uncertain behavior of physical process, the parametric uncertainties are considered. The proposed controller is based on worst case strategy. Consequently, the control law is obtained by resolution of a non convex min-max optimization problem. To overcome the drawbacks of classical optimization methods, a global deterministic optimization, generalized geometric programming (GGP), is proposed. This technique is addressed to non convex polynomial problem which the case of most robust and nonlinear control system analysis and design problem. The efficiency of this technique is tested on benchmark functions and compared with LMI and genetic algorithms optimisation methods. Simulation results obtained with an uncertain process are also presented to illustrate the performance of the proposed controller.

关键词： robust predictive control generalized geometric programming min-max optimization problem LMI genetic algorithms parametric uncertainties

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MINIMUM DISCRIMINATION INFORMATION PROBLEMS VIA generalized geometric programming

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Applied Mathematics(A Journal of Chinese Universities) 2003年第1期18卷 103-114页

作者： Zhu DetongDept. of Math., Shanghai Normal Univ., Shanghai 200234,China. Dept. of Math. Shanghai Normal Univ. Shanghai China

In this paper,the quadratic program problem and minimum discrimination information (MDI) problem with a set of quadratic inequality constraints and entropy constraints of density are *** on the properties of the generalized geometric programming,the dual programs of these two problems are ***,the duality theorems and related Kuhn-Tucker conditions for two pairs of the prime-dual programs are also established by the duality theory.

关键词： generalized geometric programming Kuhn-Tucker condition entropy of the density.

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A new type of condensation curvilinear path algorithm for unconstrained generalized geometric programming

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MATHEMATICAL AND COMPUTER MODELLING 2002年第11-12期35卷 1209-1219页

作者： Wang, YJ Zhang, KC Shen, PP Xian Jiaotong Univ Sch Sci Xian 710049 Peoples R China

A new type of condensation curvilinear path algorithm is proposed for unconstrained generalized geometric programming (GGP). First, a new type of condensation problem is presented based on the special structure of GGP. Then a particular curvilinear path for the problem is constructed, along which we get the approximate solution of the problem within a trust region. It is proved that the method is globally convergent and that the convergence rate is quadratic. Numerical experiments are given to show the effectiveness and feasibility of the algorithm. (C) 2002 Elsevier Science Ltd. All rights reserved.

关键词： generalized geometric programming condensation curvilinear path trust region

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Treating free variables in generalized geometric global optimization programs

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JOURNAL OF GLOBAL OPTIMIZATION 2005年第1期33卷 1-13页

作者： Li, HL Tsai, JF Natl Chiao Tung Univ Inst Informat Management Hsinchu 300 Taiwan Natl Taipei Univ Technol Dept Business Management Taipei 10608 Taiwan

generalized geometric programming (GGP) problems occur frequently in engineering design and management. Recently, some exponential-based decomposition methods [Maranas and Floudas, 1997,Computers and Chemical Engineering 21(4), 351-370;Floudas et al., 1999 , Handbook of Test Problems in Local and Global Optimization, Kluwer Academic Publishers, Boston, pp. 5-105;Floudas, 2000 Deterministic Global Optimizaion: Theory, Methods and Application, Kluwer Academic Publishers, Boston, pp. 257-306] have been developed for GGP problems. These methods can only handle problems with positive variables, and are incapable of solving more general GGP problems. This study proposes a technique for treating free (i.e., positive, zero or negative) variables in GGP problems. Computationally effective convexification rules are also provided for signomial terms with three variables.

关键词： generalized geometric programming global optimization

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Artificial immune system for solving generalized geometric problems: a preliminary results 05

Artificial immune system for solving generalized geometric p...

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Proceedings of the 7th annual conference on Genetic and evolutionary computation

作者： Jui-Yu Wu Yun-Kung Chung Yuan Ze University Chung-Li Taiwan R.O.C.

ISBN: (纸本)9781595930101

generalized geometric programming (GGP) is an optimization method in which the objective function and constraints are nonconvex functions. Thus, a GGP problem includes multiple local optima in its solution space. When using conventional nonlinear programming methods to solve a GGP problem, local optimum may be found, or the procedure may be mathematically tedious. To find the global optimum of a GGP problem, a bio-immune-based approach is considered. This study presents an artificial immune system (AIS) including: an operator to control the number of antigen-specific antibodies based on an idiotypic network hypothesis; an editing operator of receptor with a Cauchy distributed random number, and a bone marrow operator used to generate diverse antibodies. The AIS method was tested with a set of published GGP problems, and their solutions were compared with the known global GGP solutions. The testing results show that the proposed approach potentially converges to the global solutions.

关键词： nonlinear programming generalized geometric programming artificial immune system

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Simultaneous gate sizing and placement

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IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS 2000年第2期19卷 206-214页

作者： Chen, W Hsieh, CT Pedram, M Univ So Calif Dept Elect Engn Syst Los Angeles CA 90089 USA

This paper presents an iterative optimization technique for improving delay in integrated circuits. The basic idea is to perform timing analysis to identify the set of k most-critical paths in the circuit followed by cell resizing and replacement along the critical path set and their neighboring cells. The process is repeated until no further reduction in circuit delay is possible. At the core of this technique lies a mathematical formulation for simultaneous cell sizing and placement subject to timing and position constraints, We show that the resulting problem formulation is a generalized geometric program, which can be solved by solving a sequence of geometric programs. Experimental results on a set of benchmark circuits demonstrate the effectiveness of our approach compared to the conventional approaches which separate gate sizing from gate placement.

关键词： cell placement critical path optimization gate sizing generalized geometric programming

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Global optimization in generalized geometric programming

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COMPUTERS & CHEMICAL ENGINEERING 1997年第4期21卷 351-369页

作者： Maranas, CD Floudas, CA Department of Chemical Engineering Princeton University Princeton NJ 08544-5263 U.S.A.

A deterministic global optimization algorithm is proposed for locating the global minimum of generalized geometric (signomial) problems (GGP). By utilizing an exponential variable transformation the initial nonconvex problem (GGP) is reduced to a (DC) programming problem where both the constraints and the objective are decomposed into the difference of two convex functions. A convex relaxation of problem (DC) is then obtained based on the linear lower bounding of the concave parts of the objective function and constraints inside some box region. The proposed branch and bound type algorithm attains finite E-convergence to the global minimum through the successive refinement of a convex relaxation of the feasible region and/or of the objective function and the subsequent solution of a series of nonlinear convex optimization problems. The efficiency of the proposed approach is enhanced by eliminating variables through monotonicity analysis, by maintaining tightly bound variables through rescaling, by further improving the supplied variable bounds through convex minimization, and finally by transforming each inequality constraint so as the concave part lower bounding is as tight as possible. The proposed approach is illustrated with a large number of test examples and robust stability analysis problems. Copyright (C) 1996 Elsevier Science Ltd

关键词： global optimization generalized geometric programming signomials robust stability analysis

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INTERACTIVE OPTIMIZATION OF PLATE GIRDER BRIDGES SUBJECTED TO MOVING LOADS

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COMPUTER-AIDED DESIGN 1990年第6期22卷 368-376页

作者： ADELI, H MAK, KY Department of Civil Engineering Ohio State University 470 Hitchcock Hall 2070 Neil Avenue Columbus OH 43210 USA

A robust and efficient optimization algorithm based on the generalized geometric programming technique has been developed for minimum-weight design of multispan steel plate girders used in highway bridges. Plate girders can be of homogeneous or hybrid construction, using high-strength flange plates and low-strength web plate. The basis of the design is the American Association of State Highway and Transportation Officials (AASHTO) specifications. The plate girders are subjected to live (moving) loads according to AASHTO specifications. The algorithm has been implemented in an interactive FORTRAN -77 program with graphic capabilities.

关键词： construction engineering, girder bridge design generalized geometric programming

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