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Optimal design of a CMOS op-amp via geometric programming

IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS 2001年第1期20卷 1-21页

作者： Hershenson, MD Boyd, SP Lee, TH Barcelona Design Inc Sunnyvale CA 94086 USA Stanford Univ Dept Elect Engn Stanford CA 94305 USA

We describe a new method for determining component values and transistor dimensions for CMOS operational amplifiers (op-amps). We observe that a wide variety of design objectives and constraints have a special form, i.e., they are posynomial functions of the design variables. As a result, the amplifier design problem can be expressed as a special form of optimization problem called geometric programming, for which very efficient global optimization methods have been developed. As a consequence we can efficiently determine globally optimal amplifier designs or globally optimal tradeoffs among competing performance measures such as power, open-loop gain, and bandwidth. Our method, therefore, yields completely automated sizing of (globally) optimal CMOS amplifiers, directly from specifications. In this paper, we apply this method to a specific widely used operational amplifier architecture, showing in detail how to formulate the design problem as a geometric program. We compute globally optimal tradeoff curves relating performance measures such as power dissipation, unity-gain bandwidth, and open-loop gain. We show how the method can be used to size robust designs, i.e., designs guaranteed to meet the specifications for a variety of process conditions and parameters.

关键词： circuit optimization CMOS analog integrated circuits design automation geometric programming mixed analog-digital integrated circuits operational amplifiers

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Automatic CNN Model Partitioning for GPU/FPGA-based Embedded Heterogeneous Accelerators using geometric programming

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JOURNAL OF SIGNAL PROCESSING SYSTEMS FOR SIGNAL IMAGE AND VIDEO TECHNOLOGY 2023年第10期95卷 1203-1218页

作者： Carballo-Hernandez, Walther Pelcat, Maxime Berry, Francois Univ Clermont Auvergne Inst Pascal CNRS Clermont Auvergne INP F-63000 Clermont Ferrand France Univ Rennes IETR UMR CNRS 6164 INSA RennesUMR CNRS 6602 F-43017 Rennes France

Graphics Processing Unit (GPU), dedicated Application Specific Integrated Circuit (ASIC) and Field Programmable Gate Array (FPGA) accelerators are currently platforms of choice for porting Convolutional Neural Networks (CNNs). In this work, an automated Central Processing Unit (CPU)-GPU-FPGA partitioning selection is proposed for a given CNN layer. It is shown that using a Generalized geometric programming (GGP) optimization problem formulation, the CPU-GPU-FPGA partitioning problem can be modeled by considering a set of system performance metrics and constraints. Each metric is expressed in a posynomial form depending on CNN hyperparameters and architecture resource models. As for the partitioning method, the state-of-the-art techniques covered are: tiling, grouped convolution and fused-layer. The proposed analytical formalization is then employed to derive a set of objective functions and constraints as a GGP problem. It is demonstrated that it is possible to relax some problem constraints by including a penalization term, and reduce the problem to multiple simpler geometric programming (GP) sub-problems. Experimental results targeting an embedded FPGA-GPU platform with CNN layer configurations from state-of-the-art CNN models (AlexNet, VGG16 and ResNet18) show that the simplified problem is solvable in polynomial time with a speed-up gain and energy reduction of around 20% and 15%, respectively, when compared against an arbitrary balanced partitioning. If the models for objective and constraints functions preserve the posynomial form and log-log convexity, it is demonstrated that GGP is an efficient optimization solution to the Design Space Exploration (DSE) problem.

关键词： Heterogeneous platform Convolutional Neural Network geometric programming Mathematical optimization Embedded design

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Fuzzy multi-item economic production quantity model under space constraint: A geometric programming approach

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APPLIED MATHEMATICS AND COMPUTATION 2007年第2期184卷 326-335页

作者： Islam, Sahidul Roy, Tapan Kumar Bengal Engn & Sci Univ Dept Math Howrah 711103 W Bengal India

In this study we have formulated an economic production quantity (EPQ) model with flexibility and reliability consideration of production process and demand dependent unit production cost with fuzzy parameters. The model is restricted with available limited storage space constraint. The inventory related costs, storage spaces and others parameters are taken here as triangular fuzzy numbers. The problem is solved by modified geometric programming method. Finally, the model is illustrated by numerical example. (C) 2006 Elsevier Inc. All rights reserved.

关键词： inventory storage space triangular fuzzy number geometric programming

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Some heuristic approaches for solving extended geometric programming problems

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ENGINEERING OPTIMIZATION 2012年第12期44卷 1425-1446页

作者： Toscano, R. Amouri, S. B. Univ Lyon ENISE LTDS F-42023 St Etienne France Univ Montpellier 2 LMGC CC F-34095 Montpellier France

This article introduces an extension of standard geometric programming (GP) problems, called quasi geometric programming (QGP) problems. The idea behind QGP is very simple, it means that a problem becomes a GP problem when some variables are kept constant. The consideration of this particular kind of nonlinear and possibly non-smooth optimization problem is motivated by the fact that many engineering problems can be formulated, or well approximated, as a QGP problem. However, solving a QGP problem remains a difficult task due to its intrinsic nonconvex nature. This is why this article introduces some simple approaches for easily solving this kind of nonconvex problem. The interesting thing is that the proposed methods do not require the development of a customized solver and work well with any existing solver able to solve conventional GP problems. Some considerations on the robustness issue are also presented. Various optimization problems are considered to illustrate the ability of the proposed methods for solving a QGP problem. Comparison with previously published work is also given.

关键词： nonconvex optimization geometric programming quasi geometric programming GP-solver robust optimization

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Multitask learning and nonlinear optimal control of the COVID-19 outbreak: A geometric programming approach

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ANNUAL REVIEWS IN CONTROL 2021年 52卷 495-507页

作者： Hayhoe, Mikhail Barreras, Francisco Preciado, Victor M. Univ Penn Dept Elect & Syst Engn Philadelphia PA 19104 USA Univ Penn Dept Math Philadelphia PA 19104 USA

We propose a multitask learning approach to learn the parameters of a compartmental discrete-time epidemic model from various data sources and use it to design optimal control strategies of human-mobility restrictions that both curb the epidemic and minimize the economic costs associated with implementing non-pharmaceutical interventions. We develop an extension of the SEIR epidemic model that captures the effects of changes in human mobility on the spread of the disease. The parameters of the model are learned using a multitask learning approach that leverages both data on the number of deaths across a set of regions, and cellphone data on individuals' mobility patterns specific to each region. Using this model, we propose a nonlinear optimal control problem aiming to find the optimal mobility-based intervention strategy that curbs the spread of the epidemic while obeying a budget on the economic cost incurred. We also show that the solution to this nonlinear optimal control problem can be efficiently found, in polynomial time, using tools from geometric programming. Furthermore, in the absence of a straightforward mapping from human mobility data to economic costs, we propose a practical method by which a budget on economic losses incurred may be chosen to eliminate excess deaths due to over-utilization of hospital resources. Our results are demonstrated with numerical simulations using real data from the COVID-19 pandemic in the Philadelphia metropolitan area.

关键词： Epidemiology Mathematical modeling Multitask learning Optimal control geometric programming

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Disciplined geometric programming

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OPTIMIZATION LETTERS 2019年第5期13卷 961-976页

作者： Agrawal, Akshay Diamond, Steven Boyd, Stephen Stanford Univ Dept Elect Engn 450 Serra Mall Stanford CA 94305 USA Stanford Univ Dept Comp Sci 450 Serra Mall Stanford CA 94305 USA

We introduce log-log convex programs, which are optimization problems with positive variables that become convex when the variables, objective functions, and constraint functions are replaced with their logs, which we refer to as a log-log transformation. This class of problems generalizes traditional geometric programming and generalized geometric programming, and it includes interesting problems involving nonnegative matrices. We give examples of log-log convex functions, some well-known and some less so, and we develop an analog of disciplined convex programming, which we call disciplined geometric programming. Disciplined geometric programming is a subclass of log-log convex programming generated by a composition rule and a set of functions with known curvature under the log-log transformation. Finally, we describe an implementation of disciplined geometric programming as a reduction in CVXPY 1.0.

关键词： geometric programming Convex optimization Domain-specific languages

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Receding Horizon Control for Drinking Water Networks: The Case for geometric programming

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IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS 2020年第3期7卷 1151-1163页

作者： Wang, Shen Taha, Ahmad F. Gatsis, Nikolaos Giacomoni, Marcio H. Univ Texas San Antonio Dept Elect & Comp Engn San Antonio TX 78249 USA Univ Texas San Antonio Dept Civil & Environm Engn San Antonio TX 78249 USA

Optimal, network-driven control of water distribution networks (WDNs) is very difficult: valve and pump models form nontrivial, combinatorial logic;hydraulic models are nonconvex;water demand patterns are uncertain;and WDNs are naturally of large scale. Prior research on control of WDN addressed major research challenges, yet either i) adopted simplified hydraulic models, WDN topologies, and rudimentary valve/pump modeling or ii) used mixed-integer, nonconvex optimization to solve WDN control problems. The objective of this article is to develop tractable computational algorithms to manage WDN operation, while considering arbitrary topology, flow direction, an abundance of valve types, control objectives, hydraulic models, and operational constraints-all while only using convex, continuous optimization. Specifically, we propose new geometric programming (GP)-based model predictive control (MPC) algorithms, designed to solve the water flow equations and obtain WDN controls, i.e., pump/valve schedules alongside heads and flows. The proposed approach amounts to solving a series of convex optimization problems that graciously scale to large networks. The proposed approach is tested using a 126-node network with many valves and pumps and is shown to outperform traditional, rule-based control. The developed GP-based MPC algorithms, as well as the numerical test results, are all included on Github.

关键词： Valves Mathematical model Biological system modeling programming Topology Network topology geometric programming model predictive control pump and valve control water distribution networks

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Design Optimizaion of Gm-C Filters via geometric programming

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IEICE TRANSACTIONS ON ELECTRONICS 2017年第4期E100C卷 407-415页

作者： Yoon, Minyoung Kim, Byungjoon Kim, Jintae Nam, Sangwook Seoul Natl Univ Elect Engn Seoul South Korea Seoul Natl Univ Sch Elect Engn & Comp Sci Seoul South Korea Konkuk Univ Dept Elect Engn Seoul South Korea

This paper presents a design optimization method for a Gm-C active filter via geometric programming (GP). We first describe a GP-compatible model of a cascaded Gm-C filter that forms a biquadratic output transfer function. The bias, gain, bandwidth, and signal-to-noise ratio (SNR) of the Gm-C filter are described in a GP-compatible way. To further enhance the accuracy of the model, two modeling techniques are introduced. The first, a two-step selection method, chooses whether a saturation or subthreshold model should be used for each transistor in the filter to enhance the modeling accuracy. The second, a bisection method, is applied to include non-posynomial inequalities in the filter modeling. The presented filter model is optimized via a GP solver along with proposed modeling techniques. The numerical experiments over wide ranges of design specifications show good agreement between model and simulation results, with the average error for gain, bandwidth, and SNR being less than 9.9%, 4.4%, and 14.6%, respectively.

关键词： ultra-low power optimization geometric programming subthreshold mode operation active filter

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POSYNOMIAL geometric-programming AS A SPECIAL CASE OF SEMI-INFINITE LINEAR-programming

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1990年第3期66卷 455-475页

作者： RAJGOPAL, J BRICKER, DL UNIV IOWA DEPT IND & MANAGEMENT ENGNIOWA CITYIA 52242

This paper develops a wholly linear formulation of the posynomial geometric programming problem. It is shown that the primal geometric programming problem is equivalent to a semi-infinite linear program, and the dual problem is equivalent to a generalized linear program. Furthermore, the duality results that are available for the traditionally defined primal-dual pair are readily obtained from the duality theory for semi-infinite linear programs. It is also shown that two efficient algorithms (one primal based and the other dual based) for geometric programming actually operate on the semi-infinite linear program and its dual.

关键词： geometric programming semi-infinite linear programming duality generalized linear programming

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Fuzzy pricing, marketing and service planning in a fuzzy inventory model: A geometric programming approach

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APPLIED MATHEMATICAL MODELLING 2013年第10-11期37卷 6683-6694页

作者： Samadi, Farshid Mirzazadeh, Abolfazl Pedram, Mir Mohsen Kharazmi Univ Dept Ind Engn Tehran Iran Kharazmi Univ Dept Comp Engn Tehran Iran

In the real world markets, demand is influenced by different parameters. Recently, many researchers have been interested in integrated production and marketing planning strategies in inventory, models where demand depends on different parameters such as price and/or marketing expenditure. The quality of services that are offered to customers of a product is one of the most important parameters that affects demand in the real markets and has not been considered in development of inventory models. On the other hand, the cost parameters in real inventory systems and other parameters such as price, marketing and service elasticity to demand are imprecise and uncertain in nature. So, the notion of fuzziness can be applied to cope with this uncertainty. In this paper, a new fuzzy profit maximization inventory model with shortages is proposed. The demand is considered as a power function of price, marketing expenditure and service expenditure. Furthermore, unit cost is determined as a power function of order quantity. Since the proposed model is in a fuzzy environment, a fuzzy decision should be made to meet the decision criteria, and the results should be fuzzy. Therefore, the proposed model is formulated and solved using geometric programming and fuzzy optimization techniques to derive an approximation of the results' membership functions. The model is illustrated with a numerical example and finally a case study is provided for evaluation and validation of the results of model. (C) 2013 Elsevier Inc. All rights reserved.

关键词： Inventory geometric programming Profit maximization Service quality Marketing

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