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FPGA Architecture Optimization Using geometric programming

IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS 2010年第8期29卷 1163-1176页

作者： Smith, Alastair M. Constantinides, George A. Cheung, Peter Y. K. Univ London Imperial Coll Sci Technol & Med Circuits & Syst Res Grp Dept Elect & Elect Engn London SW7 2BT England

This paper is concerned with the application of geometric programming to the design of homogeneous field programmable gate array (FPGA) architectures. The paper builds on an increasing body of work concerned with modeling reconfigurable architectures, and presents a full area and delay model of an FPGA. We use a geometric programming framework to show how transistor sizing and high-level architecture parameter selection can now be solved as a concurrent optimization problem. We validate the model through the use of simulation program with integrated circuit emphasis (SPICE) models and the versatile place and route (VPR) FPGA architecture simulation tool. Not only does the optimization framework allow architectures to be optimized orders of magnitude faster than previous work, but the combined optimization can lead to different architectural conclusions compared to conventional methods by exploring the coupling between the two sets of optimization variables. Specifically, we show that as delay takes more significance in the objective of the optimization, there should be more lookup tables in a logic block, whereas conventional techniques suggest that there should be fewer lookup tables in an FPGA logic block.

关键词： Convex optimization field programmable gate array (FPGA) geometric programming reconfigurable architectures

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Tractable approximate robust geometric programming

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OPTIMIZATION AND ENGINEERING 2008年第2期9卷 95-118页

作者： Hsiung, Kan-Lin Kim, Seung-Jean Boyd, Stephen Stanford Univ Dept Elect Engn Informat Syst Lab Stanford CA 94305 USA

The optimal solution of a geometric program (GP) can be sensitive to variations in the problem data. Robust geometric programming can systematically alleviate the sensitivity problem by explicitly incorporating a model of data uncertainty in a GP and optimizing for the worst-case scenario under this model. However, it is not known whether a general robust GP can be reformulated as a tractable optimization problem that interior-point or other algorithms can efficiently solve. In this paper we propose an approximation method that seeks a compromise between solution accuracy and computational efficiency. The method is based on approximating the robust GP as a robust linear program (LP), by replacing each nonlinear constraint function with a piecewise-linear (PWL) convex approximation. With a polyhedral or ellipsoidal description of the uncertain data, the resulting robust LP can be formulated as a standard convex optimization problem that interior-point methods can solve. The drawback of this basic method is that the number of terms in the PWL approximations required to obtain an acceptable approximation error can be very large. To overcome the "curse of dimensionality" that arises in directly approximating the nonlinear constraint functions in the original robust GP, we form a conservative approximation of the original robust GP, which contains only bivariate constraint functions. We show how to find globally optimal PWL approximations of these bivariate constraint functions.

关键词： geometric programming linear programming piecewise-linear function robust geometric programming robust linear programming robust optimization

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Radio Resource Sharing and Edge Caching with Latency Constraint for Local 5G Operator: geometric programming Meets Stackelberg Game

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IEEE TRANSACTIONS ON MOBILE COMPUTING 2021年第2期20卷 707-721页

作者： Sanguanpuak, Tachaporn Niyato, Dusit Rajatheva, Nandana Latva-Aho, Matti Univ Oulu Ctr Wireless Commun CWC 6G Flagship Oulu 90014 Finland Nanyang Technol Univ NTU Sch Comp Sci & Engn Singapore 639798 Singapore

The rapidly increasing demand in indoor small cell networks has given rise to the concept of local 5G operator (OP) for local service delivery. In this regard, we develop a novel game-theoretic framework with geometric programming to model and analyze cache-enabled small cell base stations (SBSs) with infrastructure sharing for local 5G OP networks. In such a network, the local 5G OP provides wireless network in indoor area and rent out the infrastructure which are RAN and cache storage to multiple mobile network operators (MNOs) while guarantee the quality-of-experience (QoE) at the users (UEs) of MNOs. We formulate a Stackelberg game model where the local 5G OP is the leader and the MNOs are the followers. The local 5G OP aims to maximize its profit by optimizing its infrastructure rental fee, and the MNOs aim to minimize their renting cost of infrastructure by minimizing the cache intensity subject to latency constraint at each UE. Here, the cache intensity is defined as the product of the number of SBSs per unit area and the number of popular files stored in each SBS. The optimization problems of the local 5G OP and the MNOs are transformed into geometric programming. Accordingly, the subgame perfect equilibrium of Stackelberg game is obtained through the succesive geometric programming (SGP) method. Since the MNOs share their rented infrastructure, for cost sharing, we apply the concept of Shapley value to divide the cost among the MNOs. We show that the cost sharing problem can be mapped into a simplified "airport runway cost sharing problem", in which the Shapley value can be computed efficiently. Finally, we present an extensive performance evaluation that reveals interesting insights into designing resource sharing with edge caching in local 5G OP networks.

关键词： 5G beyond 5G (B5G) edge caching latency constraint local 5G operator micro-operator stochastic geometry geometric programming Stackelberg game Shapley value

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LOWER BOUNDS FOR POLYNOMIALS USING geometric programming

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SIAM JOURNAL ON OPTIMIZATION 2012年第2期22卷 460-473页

作者： Ghasemi, Mehdi Marshall, Murray Univ Saskatchewan Dept Math & Stat Saskatoon SK S7N 5E6 Canada

We make use of a result of Hurwitz [J. Reine Angew. Math., 108 (1891), pp. 266-268] and Reznick [Math. Ann., 283 (1989), pp. 431-464], and a consequence of this result due to Fidalgo and Kovacec [Math. Z., 269 (2011), pp. 629-645], to establish, in Theorem 2.3, a new sufficient condition for a polynomial f is an element of R[X-1, . . . , X-n] of even degree to be a sum of squares. Theorem 2.3 yields as special cases the results of Ghasemi and Marshall in [Arch. Math. (Basel), 95 (2010), pp. 343-353] and, consequently, also those of Fidalgo and Kovacec and Lasserre [Arch. Math. (Basel), 89 (2007), pp. 390-398]. We apply Theorem 2.3 to obtain a new lower bound f(gp) for f, and we explain how f(gp) can be computed using geometric programming. The lower bound f(gp) is generally not as good as the lower bound f(sos) introduced by Lasserre [SIAM J. Optim., 11 (2001), pp. 796-817] and Parrilo and Sturmfels [Algorithmic and Quantitative Real Algebraic Geometry, AMS, Providence, RI, 2003, pp. 88-99], which can be computed using semidefinite programming, but a run time comparison shows that, in practice, the computation of fgp is faster, and larger problems can be handled. The computation is simplest when the highest degree component of f has the form Sigma(n)(i=1) a(i)X(i)(2d), a(i) > 0, i = 1, . . . , n. The lower bounds for f established by Ghasemi and Marshall are obtained by evaluating the objective function of the geometric program at appropriate feasible points.

关键词： positive polynomials sums of squares optimization geometric programming

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NEW geometric-programming ALGORITHMS WITH NUMERICAL EXPERIMENTS

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1986年第2期24卷 194-205页

作者： POKATAYEV, AV ACAD SCI BESSR INST MATHSURGANOV STR 11MINSK 220604BELORUSSIAUSSR

geometric programming (GP) is one of the most important parts of mathematical programming. GP models are widely applied in different spheres of science, technology, national economy [1,2]. The paper presents six new algorithms of linear and polynomial approximation for the solution of GP programs. The results of extensive numerical experiments are also included in the paper. They indicate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

关键词： experiment geometric programming gradient methods

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Market exchange models and geometric programming

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CENTRAL EUROPEAN JOURNAL OF OPERATIONS RESEARCH 2019年第2期27卷 415-435页

作者： Eisenberg-Nagy, Marianna Illes, Tibor Lovics, Gabor Budapest Univ Technol & Econ Inst Math Budapest Hungary

Finding the equilibrium solution of the Market Exchange Models is an interesting topic. Here we discuss some relationship of the Fisher type Homogenous Market Exchange Model and the geometric programming. We also discuss that the equilibrium solution of the Linear and Cobb-Douglas Market Exchange Model as two special cases of the Homogenous Market Exchange Model can be found by geometric programming in polynomial time.

关键词： Market exchange models Homogeneous concave utility functions geometric programming Fisher type homogeneous market exchange models Linear and Cobb-Douglas utility functions

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Levelized Cost of Energy-Oriented Modular String Inverter Design Optimization for PV Generation System Using geometric programming

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IEEE ACCESS 2022年 10卷 27561-27578页

作者： Son, Yeongrack Mukherjee, Satyaki Mallik, Rahul Majmunovi'c, Branko Dutta, Soham Johnson, Brian Maksimovic, Dragan Seo, Gab-Su Natl Renewable Energy Lab Power Syst Engn Ctr Golden CO 80401 USA Univ Colorado Dept Elect Comp & Energy Engn Boulder CO 80309 USA Delta Elect Amer Milan M Jovanovid Power Elect Lab Durham NC 27709 USA Univ Washington Dept Elect & Comp Engn Seattle WA 98195 USA

Levelized cost of energy (LCOE) is a commonly used metric to assess the cost-to-benefit ratio over the lifetime of an energy resource, such as photovoltaics (PV);however, power electronics engineers tend to rely on metrics such as efficiency and power density, which do not guarantee lifetime cost optimality. Recent work has shown that an LCOE-focused optimization approach can yield improved system designs, leading to improved lifetime performance with balanced lifetime cost and energy generation. This paper outlines an LCOE optimization framework for PV power electronics that uses geometric programming. The large number of circuit parameters and nonlinear nature of the system equations pose significant barriers. Our approach allows for decoupling the design variables, which, in turn, enables superior computational efficiency and a near-optimal solution. By incorporating the power electronics design process and magnetic loss mechanism into the convex design framework, the optimization engine yields practically implementable parameters for a PV converter that minimizes LCOE. An optimization example for a cascaded modular PV inverter architecture is presented that suggests 3.35% LCOE improvement can be achieved by the new power electronics and the advanced optimization. The proposed optimization framework can be applied to other power generation systems to evaluate the effect of the power electronics design on system lifetime costs and efficiency.

关键词： Costs Optimization Inverters Power conversion Fitting Measurement Renewable energy sources Convex optimization geometric programming levelized cost of energy lifetime energy cost magnetic loss modeling solar inverters modular multilevel converter

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Optimizing the geometric programming problem with single-term exponents subject to max-min fuzzy relational equation constraints

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MATHEMATICAL AND COMPUTER MODELLING 2008年第3-4期47卷 352-362页

作者： Wu, Yan-Kuen Vanung Univ Dept Ind Management Tao Yuan 320 Taiwan

In this study we investigate the problem of minimizing an objective function with single-term exponents subject to fuzzy relational equations specified in max-min composition. Two folds are presented. First, we present some properties for this optimization problem under the assumption of both negative and nonnegative exponents in the objective function. Second, we provide an efficient procedure to solve this optimization problem without looking for all the potential minimal solutions. Two concrete examples are provided to illustrate the procedure. (c) 2007 Elsevier Ltd. All rights reserved.

关键词： fuzzy optimization fuzzy relational equations max-min composition geometric programming

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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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Design and optimization of buck and double buck converters by means of geometric programming

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MATHEMATICS AND COMPUTERS IN SIMULATION 2012年第8期82卷 1516-1530页

作者： Leyva, R. Ribes-Mallada, U. Garces, P. Reynaud, J. F. Univ Rovira & Virgili DEEEA Tarragona 43007 Spain CNRS Lab Anal & Architecture Syst Toulouse France

This paper describes a new method for determining the optimal components values and switching frequencies of buck DC-DC converters. First, we revisit some concepts of the optimization technique named geometric programming. Then, we observe that the problem of converter designing can be modeled by means of an objective function and certain constraints which can be written in a specific form known as the posynomial form. The constraints involve expressions that depend on magnitudes such as efficiency, bandwidth, and current and voltage ripples. Specifically, we apply the design method in a synchronous buck converter and a synchronous cascade buck converter. This technique can efficiently determine the optimal sizing of the converter or the infeasibility of the set of design constraints in a quickly manner and, therefore, it can eases the cumbersome task of manually designing buck DC-DC converters. As an additional result, we conclude that optimal design of the synchronous cascade buck converter performs more efficiently than the optimal design of the synchronous buck converter, given certain realistic set of specifications for wide-range voltage conversion. (C) 2012 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Synchronous buck converter Synchronous cascade buckconverter geometric programming Optimization Switchingconverter design

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