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A tutorial on geometric programming

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Abstract

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.

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References

  • Abou-El-Ata M, Kotb K (1997) Multi-item EOQ inventory model with varying holding cost under two restrictions: a geometric programming approach. Prod Plan Control 8(6):608–611

    Google Scholar 

  • Abou-Seido A, Nowak B, Chu C (2004) Fitted Elmore delay: a simple and accurate interconnect delay model. IEEE Trans VLSI Syst 12(7):691–696

    Google Scholar 

  • Abuo-El-Ata M, Fergany H, El-Wakeel M (2003) Probabilistic multi-item inventory model with varying order cost under two restrictions: a geometric programming approach. Int J Prod Econ 83(3):223–231

    Google Scholar 

  • Adeli H, Kamal O (1986) Efficient optimization of space trusses. Comput Struct 24:501–511

    MATH  Google Scholar 

  • Aggarwal V, O’Reilly U-M (2006) Design of posynomial models for MOSFETs: symbolic regression using genetic algorithms. In: Riolo R, Soule T, Worzel B (eds) Genetic programming theory and practice. Genetic and evolutionary computation, vol 5. Springer, Ann Arbor, Chap 7

    Google Scholar 

  • Alejandre J, Allueva A, Gonzalez J (2004) A general alternative procedure for solving negative degree of difficulty problems in geometric programming. Comput Optim Appl 27(1):83–93

    MATH  MathSciNet  Google Scholar 

  • Andersen E, Ye Y (1998) A computational study of the homogeneous algorithm for large-scale convex optimization. Comput Optim Appl 10(3):243–269

    MATH  MathSciNet  Google Scholar 

  • Avriel M (1980) Advances in geometric programming. Mathematical concept and methods in science and engineering, vol 21. Plenum, New York

    MATH  Google Scholar 

  • Avriel M, Dembo R, Passy U (1975) Solution of generalized geometric programs. Int J Numer Methods Eng 9(1):149–168

    MATH  MathSciNet  Google Scholar 

  • Bazaraa M, Shetty C, Sherali H (1993) Non-linear programming: theory and algorithms. Wiley, New York

    Google Scholar 

  • Beightler C, Phillips D (1976) Applied geometric programming. Wiley, New York

    MATH  Google Scholar 

  • Ben-Tal A, Teboulle M (1986) Rate distortion theory with generalized information measures via convex programming duality. IEEE Trans Inf Theory 32(5):630–641

    MATH  MathSciNet  Google Scholar 

  • Ben-Tal A, Teboulle M, Charnes A (1988) The role of duality in optimization problems involving entropy functionals. J Optim Theory Appl 58(2):209–223

    MATH  MathSciNet  Google Scholar 

  • Bertsekas DP (1999) Nonlinear programming, 2nd edn. Athena Scientific

  • Bhardwaj S, Vrudhula S (2005) Leakage minimization of nano-scale circuits in the presence of systematic and random variations. In: Proceedings of the 42nd IEEE/ACM design automation conference (DAC), pp 535–540, 2005

  • Bhardwaj S, Cao Y, Vrudhula S (2006) Statistical leakage minimization through joint selection of gate sizes, gate lengths and threshold. In: Proceedings of the 12th conference on Asia and South Pacific design automation conference (ASP-DAC), pp 953–958, 2006

  • Boche H, Stańczak S (2004) Optimal QoS tradeoff and power control in CDMA systems. In: Proceedings of the 23rd IEEE INFOCOM, pp 477–486, 2004

  • Borah M, Owens R, Irwin M (1997) A fast algorithm for minimizing the Elmore delay to identified critical sinks. IEEE Trans Comput Aided Des Integr Circuits Syst 16(7):753–759

    Google Scholar 

  • Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Boyd S, Kim S-J, Patil D, Horowitz M (2005) Digital circuit optimization via geometric programming. Oper Res 53(6):899–932

    MathSciNet  MATH  Google Scholar 

  • Boyd S, Kim S-J, Patil D, Horowitz M (2006) A heuristic method for statistical digital circuit sizing. In: Proceedings of the 31st SPIE international symposium on microlithography, San Jose, 2006

  • Bricker D, Kortanek K, Xui L (1997) Maximum likelihood estimates with order restrictions on probabilities and odds ratios: a geometric programming approach. J Appl Math Decis Sci 1(1):53–65

    MATH  Google Scholar 

  • Chan A, Turlea E (1978) An approximate method for structural optimisation. Comput Struct 8(3–4):357–363

    MATH  Google Scholar 

  • Chandra D, Singh V, Kar H (2004) Improved Routh-Padé approximants: a computer-aided approach. IEEE Trans Autom Control 49(2):292–296

    MathSciNet  Google Scholar 

  • Chen C-P, Wong D (1999) Greedy wire-sizing is linear time. IEEE Trans Comput Aided Des Integr Circuits Syst 18(4):398–405

    MathSciNet  Google Scholar 

  • Chen C-P, Chu C, Wong D (1999) Fast and exact simultaneous gate and wire sizing by Lagrangian relaxation. IEEE Trans Comput Aided Des Integr Circuits Syst 18(7):1014–1025

    Google Scholar 

  • Chen T-Y (1992) Structural optimization using single-term posynomial geometric programming. Comput Struct 45(5–6):911–918

    MATH  Google Scholar 

  • Chen T-C, Pan S-R, Chang Y-W (2004) Timing modeling and optimization under the transmission line model. IEEE Trans Very Large Scale Integr Syst 12(1):28–41

    Google Scholar 

  • Chen W, Hseih C-T, Pedram M (2000) Simultaneous gate sizing and placement. IEEE Trans Comput Aided Des Integr Circuits Syst 19(2):206–214

    Google Scholar 

  • Cheng T (1991) An economic order quantity model with demand-dependent unit production cost and imperfect production processes. IIE Trans 23(1):23

    Google Scholar 

  • Cheng H, Fang S-C, Lavery J (2002) Univariate cubic L 1 splines—A geometric programming approach. Math Methods Oper Res 56(2):197–229

    MATH  MathSciNet  Google Scholar 

  • Cheng H, Fang S-C, Lavery J (2005a) A geometric programming framework for univariate cubic L 1 smoothing splines. Ann Oper Res 133(1–4):229–248

    MATH  MathSciNet  Google Scholar 

  • Cheng H, Fang S-C, Lavery J (2005b) A geometric programming framework for univariate cubic L 1 smoothing splines. J Comput Appl Math 174(2):361–382

    MATH  MathSciNet  Google Scholar 

  • Chiang M (2005a) Balancing transport and physical layers in wireless multihop networks: jointly optimal congestion control and power control. IEEE J Sel Areas Commun 23(1):104–116

    Google Scholar 

  • Chiang M (2005b) Geometric programming for communication systems. Found Trends Commun Inf Theory 2(1–2):1–154

    MATH  Google Scholar 

  • Chiang M, Boyd S (2004) Geometric programming duals of channel capacity and rate distortion. IEEE Trans Inf Theory 50(2):245–258

    MathSciNet  Google Scholar 

  • Chiang M, Chan B, Boyd S (2002a) Convex optimization of output link scheduling and active queue management in QoS constrained packet sitches. In: Proceedings of the 2002 IEEE international conference on communications (ICC), pp 2126–2130, 2002

  • Chiang M, Sutivong A, Boyd S (2002b) Efficient nonlinear optimization of queuing systems. In: Proceedings of the 2002 IEEE global telecommunications conference (GLOBECOM), pp 2425–2429, 2002

  • Choi J-C, Bricker D (1995) Geometric programming with several discrete variables: algorithms employing generalized benders. Eng Optim 3:201–212

    Google Scholar 

  • Choi J, Bricker D (1996a) Effectiveness of a geometric programming algorithm for optimization of machining economics models. Comp Oper Res 23(10):957–961

    MATH  Google Scholar 

  • Choi J-C, Bricker D (1996b) A heuristic procedure for rounding posynomial geometric programming solutions to discrete value. Comput Ind Eng 30(4):623–629

    Google Scholar 

  • Chu C, Wong D (1999) An efficient and optimal algorithm for simultaneous buffer and wire sizing. IEEE Trans Comput Aided Des Integr Circuits Syst 18(9):1297–1304

    Google Scholar 

  • Chu C, Wong D (2001a) Closed form solutions to simultaneous buffer insertion/sizing and wire sizing. ACM Trans Des Autom Electron Syst 6(3):343–371

    Google Scholar 

  • Chu C, Wong D (2001b) VLSI circuit performance optimization by geometric programming. Ann Oper Res 105(1–4):37–60

    MATH  MathSciNet  Google Scholar 

  • Clasen R (1963) The linear logarithmic programming problem. Rand Corp. Memo. RM-37-7-PR, June 1963

  • Clasen R (1984) The solution of the chemical equilibrium programming problem with generalized benders decomposition. Oper Res 32(1):70–79

    MATH  Google Scholar 

  • Colleran D, Portmann C, Hassibi A, Crusius C, Mohan S, Boyd S, Lee T, Hershenson M (2003) Optimization of phase-locked loop circuits via geometric programming. In: Proceedings of the 2003 IEEE custom integrated circuits conference (CICC), pp 326–328, 2003

  • Cong J, He L (1996) Optimal wire sizing for interconnects with multiple sources. ACM Trans Des Autom Electron Syst 1(4):478–511

    Google Scholar 

  • Cong J, He L (1998) Theory and algorithm of local-refinement-based optimization with application to device and interconnect sizing. IEEE Trans Comput Aided Des Integr Circuits Syst 18(4):406–420

    Google Scholar 

  • Cong J, Koh C-K (1994) Simultaneous driver and wire sizing for performance and power optimization. IEEE Trans Very Large Scale Integr Syst 2(4):408–423

    Google Scholar 

  • Cong J, Leung K-S (1995) Optimal wiresizing under Elmore delay model. IEEE Trans Comput Aided Des Integr Circuits Syst 14(3):321–336

    Google Scholar 

  • Cong J, Pan Z (2002) Wire width planning for interconnect performance optimization. IEEE Trans Comput Aided Des Integr Circuits Syst 21(3):319–329

    Google Scholar 

  • Corstjens M, Doyle P (1979) Channel optimization in complex marketing systems. Manag Sci 25(10):1014–1025

    MATH  Google Scholar 

  • Coudert O, Haddad R, Manne S (1996) New algorithms for gate sizing: a comparative study. In: Proceedings of 33rd IEEE/ACM design automation conference (DAC), pp 734–739, 1996

  • Daems W, Gielen G, Sansen W (2003) Simulation-based generation of posynomial performance models for the sizing of analog integrated circuits. IEEE Trans Comput Aided Des Integr Circuits Syst 22(5):517–534

    Google Scholar 

  • Dantzig G, Johnson S, White W (1958) Shape-preserving properties of univariate cubic L 1 splines. Manag Sci 5(1):38–43

    MATH  MathSciNet  Google Scholar 

  • Dawson J, Boyd S, Hershenson M, Lee T (2001) Optimal allocation of local feedback in multistage amplifiers via geometric programming. IEEE Trans Circuits Syst I Fundam Theory Appl 48(1):1–11

    Google Scholar 

  • Dembo R (1982) Sensitivity analysis in geometric programming. J Optim Theory Appl 37(1):1–21

    MATH  MathSciNet  Google Scholar 

  • Dhillon B, Kuo C (1991) Optimum design of composite hybrid plate girders. J Struct Eng 117(7):2088–2098

    Google Scholar 

  • Dinkel J, Kochenberger M, Wong S (1978) Sensitivity analysis procedures for geometric programs: Computational aspects. ACM Trans Math Softw 4(1):1–14

    MATH  Google Scholar 

  • Duffin R (1970) Linearizing geometric programs. SIAM Rev 12(2):668–675

    MathSciNet  Google Scholar 

  • Duffin R, Peterson E, Zener C (1967) Geometric programming—theory and application. Wiley, New York

    MATH  Google Scholar 

  • Dutta A, Rama D (1992) An optimization model of communications satellite planning. IEEE Trans Commun 40(9):1463–1473

    Google Scholar 

  • Ecker J (1980) Geometric programming: Methods, computations and applications. SIAM Rev 22(3):338–362

    MATH  MathSciNet  Google Scholar 

  • Edvall M, Hellman F (2005) User’s Guide for TOMLAP/GP. Available from http://tomlab.biz/docs/TOMLAB_GP.pdf

  • El Barmi H, Dykstra R (1994) Restricted multinomial maximum likelihood estimation based upon Fenchel duality. Stat Probab Lett 21(2):121–130

    MATH  MathSciNet  Google Scholar 

  • Federowicz A, Rajgopal J (1999) Robustness of posynomial geometric programming optima. Math Program 85(2):421–431

    MathSciNet  Google Scholar 

  • Feigin P, Passy U (1981) The geometric programming dual to the extinction probability problem in simple branching processes. Ann Probab 9(3):498–503

    MATH  MathSciNet  Google Scholar 

  • Fishburn J, Dunlop A (1985) TILOS: a posynomial programming approach to transistor sizing. In: IEEE international conference on computer-aided design: ICCAD-85. Digest of technical papers, pp 326–328. IEEE Computer Society Press

  • Floudas C (1999) Deterministic global optimization: theory, algorithms and applications. Kluwer Academic, Dordrecht

    Google Scholar 

  • Foschini G, Miljanic Z (1993) A simple distributed autonomous power control algorithm and its convergence. IEEE Trans Veh Technol 42(4):641–646

    Google Scholar 

  • Fuh C-D, Hu I (2000) Asymptotically efficient strategies for a stochastic scheduling problem with order constraints. Ann Stat 28(6):1670–1695

    MATH  MathSciNet  Google Scholar 

  • Gao Y, Wong D (1999) Optimal shape function for a bidirectional wire under Elmore delay model. IEEE Trans Comput Aided Des Integr Circuits Syst 18(7):994–999

    Google Scholar 

  • Grant M, Boyd S, Ye Y (2005) cvx: matlab software for disciplined convex programming. Available from www.stanford.edu/~boyd/cvx/

  • Greenberg H (1995) Mathematical programming models for environmental quality control. Oper Res 43(4):578–622

    MATH  Google Scholar 

  • Hajela P (1986) Geometric programming strategies for large-scale structural synthesis. AIAA J 24(7):1173–1178

    MATH  Google Scholar 

  • Hariri A, Abou-El-Ata M (1997) Multi-item production lot-size inventory model with varying order cost under a restriction: a geometric programming approach. Prod Plan Control 8(2):179–182

    Google Scholar 

  • Hassibi A, Hershenson M (2002) Automated optimal design of switched-capacitor filters. In: Proceedings of the 2002 design, automation and test in Europe conference and exhibition (DATE), p 1111, 2002

  • Hastie T, Tibshirani R, Friedman J (2001) The elements of statistical learning. Springer, Berlin

    MATH  Google Scholar 

  • Hershenson M (2002) Design of pipeline analog-to-digital converters via geometric programming. In: Proceedings of the 2002 IEEE/ACM international conference on computer aided design (ICCAD), pp 317–324, San Jose, 2002

  • Hershenson M (2003) Analog design space exploration: efficient description of the design space of analog circuits. In: Proceedings of the 40th design automation conference (DAC), pp 970–973, 2003

  • Hershenson M, Boyd S, Lee T (1998) GPCAD: A tool for CMOS op-amp synthesis. In: Proceedings of the 1998 IEEE/ACM international conference on computer aided design (ICCAD), pp 296–303, 1998

  • Hershenson M, Hajimiri A, Mohan S, Boyd S, Lee T (1999) Design and optimization of LC oscillators. In: Proceedings of the 2000 IEEE/ACM international conference on computer-aided design (ICCAD), pp 65–69, 1999

  • Hershenson M, Boyd S, Lee T (2001) Optimal design of a CMOS op-amp via geometric programming. IEEE Trans Comput Aided Des Integr Circuits Syst 20(1):1–21

    Google Scholar 

  • Hitomi K (1996) Manufacturing systems engineering: a unified approach to manufacturing technology, production management, and industrial economics, 2nd edn. CRC Press, Boca Raton

    Google Scholar 

  • Horowitz M, Alon E, Patil D, Naffziger S, Kumar R, Bernstein K (2005) Scaling, power, and the future of CMOS. In: Proceedings of the 2005 IEEE international electron devices meeting (IEDM), pp 9–15, 2005

  • Hsiung K-L, Kim S-J, Boyd S (2006) Tractable approximate robust geometric programming. Manuscript. Available from www.stanford,edu/~boyd/rgp.html

  • Hu I, Wei C (1989) Irreversible adaptive allocation rules. Ann Stat 17(2):801–823

    MATH  MathSciNet  Google Scholar 

  • Ismail Y, Friedman E, Neves J (2000) Equivalent Elmore delay for RLC trees. IEEE Trans Comput Aided Des Integr Circuits Syst 19(7):83–97

    Google Scholar 

  • Jabr R (2005) Applications of geometric programming to transformer design. IEEE Trans Magn 41(11):4261–4269

    Google Scholar 

  • Jha N (1990) A discrete data base multiple objective optimization of milling operation through geometric programming. ASME J Eng Ind 112(4):368–374

    Article  Google Scholar 

  • Jiang I, Chang Y, Jou J (2000) Crosstalk-driven interconnect optimization by simultaneous gate and wire sizing. IEEE Trans Comput Aided Des Integr Circuits Syst 19(9):999–1010

    Google Scholar 

  • Joshi S, Boyd S, Dutton R (2005) Optimal doping profiles via geometric programming. IEEE Trans Electron Devices 52(12):2660–2675

    Google Scholar 

  • Julian D, Chiang M, O’Neill D, Boyd S (2002) QoS and fairness constrained convex optimization of resource allocation for wireless cellular and ad hoc networks. In: Proceedings of the 21st IEEE INFOCOM, pp 477–486, 2002

  • Jung H, Klein C (2001) Optimal inventory policies under decreasing cost functions via geometric programming. Eur J Oper Res 132(3):628–642

    MATH  Google Scholar 

  • Kandukuri S, Boyd S (2002) Optimal power control in interference-limited fading wireless channels with outage-probability specifications. IEEE Trans Wirel Commun 1(1):46–55

    Google Scholar 

  • Karlof J, Chang Y (1997) Optimal permutation codes for the Gaussian channel. IEEE Trans Inform Theory 43(1):356–358

    MATH  MathSciNet  Google Scholar 

  • Kasamsetty K, Ketkar M, Sapatnekar S (2000) A new class of convex functions for delay modeling and its application to the transistor sizing problem. IEEE Trans Comput Aided Des Integr Circuits Syst 19(7):779–788

    Google Scholar 

  • Kay R, Pileggi L (1998) EWA: Efficient wiring-sizing algorithm for signal nets and clock nets. IEEE Trans Comput Aided Des Integr Circuits Syst 17(1):40–49

    Google Scholar 

  • Kiely T, Gielen G (2004) Performance modeling of analog integrated circuits using least-squares support vector machines. In: Proceedings of the 2004 design, automation and test in Europe conference and exhibition (DATE), pp 16–20, 2004

  • Kim J, Lee J, Vandenberghe L, Yang K (2004) Techniques for improving the accuracy of geometric-programming based analog circuit design optimization. In: Proceedings of the 2004 IEEE/ACM international conference on computer-aided design (ICCAD), pp 863–870, 2004

  • Kim S-J, Boyd S, Yun S, Patil D, Horowitz M (2007) A heuristic for optimizing stochastic activity networks with applications to statistical digital circuit sizing (to appear in Optim Eng). Available from www.stanford.edu/~boyd/heur_san_opt.html

  • Klafszky E, Mayer J, Terlaky T (1992) A geometric programming approach to the channel capacity problem. Eng Optim 19:115–130

    Google Scholar 

  • Kortanek K, Xu X, Ye Y (1996) An infeasible interior-point algorithm for solving primal and dual geometric programs. Math Program 76(1):155–181

    MathSciNet  Google Scholar 

  • Krishnamurthy R, Carley L (1997) Exploring the design space of mixed swing quadrail for low-power digital circuits. IEEE Trans Very Large Scale Integr Syst 5(4):389–400

    Google Scholar 

  • Kyparsis J (1988) Sensitivity analysis in posynomial geometric programming. J Optim Theory Appl 57(1):85–121

    MathSciNet  Google Scholar 

  • Kyparsis J (1990) Sensitivity analysis in geometric programming: theory and computations. Ann Oper Res 27(1):39–64

    MathSciNet  Google Scholar 

  • Lawler E, Wood D (1966) Branch-and-bound methods: a survey. Oper Res 14(4):699–719

    MATH  MathSciNet  Google Scholar 

  • Lee J, Hatcher G, Vandenbergh L, Yang K (2003) Evaluation of fully-integrated switching regulators for CMOS process technologies. In: Proceedings of the 2003 international symposium on system-on-chip, pp 155–158, 2003

  • Lee W, Kim D (1993) Optimal and heuristic decision strategies for integrated production and marketing planning. Decis Sci 24(6):1203–1213

    Google Scholar 

  • Lee Y-M, Chen C, Wong D (2002) Optimal wire-sizing function under the Elmore delay model with bounded wire sizes. IEEE Trans Circuits Syst I Fundam Theory Appl 49(11):1671–1677

    MathSciNet  Google Scholar 

  • Li X, Gopalakrishnan P, Xu Y, Pileggi T (2004) Robust analog/RF circuit design with projection-based posynomial modeling. In: Proceedings of the IEEE/ACM international conference on computer aided design (ICCAD), pp 855–862, 2004

  • Lin T, Pileggi L (2001) RC(L) interconnect sizing with second order considerations via posynomial programming. In: Proceedings of the 2001 ACM/SIGDA international symposium on physical design (ISPD), pp 16–21, 2001

  • Löfberg J (2003) YALMIP. Yet another LMI parser. Version 2.4. Available from http://control.ee.ethz.ch/~joloef/yalmip.php

  • Lou J, Chen W, Pedram M (1999) Concurrent logic restructuring and placement for timing closure. In: Proceedings of the 1999 IEEE/ACM international conference on computer-aided design (ICCAD), pp 31–35, 1999

  • Luenberger D (1984) Linear and nonlinear programming, 2nd edn. Addison–Wesley, Reading

    MATH  Google Scholar 

  • Magnani A, Boyd S (2006) Convex piecewise-linear fitting (submitted to Optim Eng). Available from www.stanford.edu/~boyd/cvx_pwl_fit.html

  • Mandal P, Visvanathan V (2001) CMOS op-amp sizing using a geometric programming formulation. IEEE Trans Comput Aided Des Integr Circuits Syst 20(1):22–38

    Google Scholar 

  • Maranas C, Floudas C (1997) Global optimization in generalized geometric programming. Comput Chem Eng 21(4):351–369

    MathSciNet  Google Scholar 

  • Marković D, Stojanović V, Nikolić B, Horowitz M, Brodersen R (2004) Methods for true energy-performance optimization. IEEE J Solid State Circuits 39(8):1282–1293

    Google Scholar 

  • Matson M, Glasser L (1986) Macromodeling and optimization of digital MOS VLSI circuits. IEEE Trans Comput Aided Des Integr Circuits Syst 5(4):659–678

    Google Scholar 

  • Mazumdar M, Jefferson T (1983) Maximum likelihood estimates for multinomial probabilities via geometric programming. Biometrika 70(1):257–261

    MATH  MathSciNet  Google Scholar 

  • Moh T-S, Chang T-S, Hakimi S (1996) Globally optimal floorplanning for a layout problem. IEEE Trans Circuits Syst I Fundam Theory Appl 43(29):713–720

    MathSciNet  Google Scholar 

  • Mohan S, Hershenson M, Boyd S, Lee T (1999) Simple accurate expressions for planar spiral inductances. IEEE J Solid State Circuit 34(10):1419–1424

    Google Scholar 

  • Mohan S, Hershenson M, Boyd S, Lee T (2000) Bandwidth extension in CMOS with optimized on-chip inductors. IEEE J Solid State Circuits 35(3):346–355

    Google Scholar 

  • Moore R (1991) Global optimization to prescribed accuracy. Comput Math Appl 21(6-7):25–39

    MATH  Google Scholar 

  • MOSEK ApS (2002) The MOSEK optimization tools version 2.5. User’s manual and reference. Available from www.mosek.com

  • Muqattash A, Krunz M, Shu T (2006) Performance enhancement of adaptive orthogonal modulation in wireless CDMA systems. IEEE J Sel Areas Commun 23(3):565–578

    Google Scholar 

  • Mutapcic A, Koh K, Kim S-J, Boyd S (2006) ggplab: a matlab toolbox for geometric programming. Available from www.stanford.edu/~boyd/ggplab/

  • Nesterov Y, Nemirovsky A (1994) Interior-point polynomial methods in convex programming. Studies in applied mathematics, vol 13. SIAM, Philadelphia

    Google Scholar 

  • Nijhamp P (1972) Planning of industrial complexes by means of geometric programming. Rotterdam Univ. Press, Rotterdam

    Google Scholar 

  • Nocedal J, Wright S (1999) Numerical optimization, Springer series in operations research. Springer, New York

    MATH  Google Scholar 

  • Palomar D, Cioffi J, Lagunas M (2003) Joint Tx-Rx beamforming design for multicarrier MIMO channels: a unified framework for convex optimization. IEEE Trans Signal Process 51(9):2381–2401

    Google Scholar 

  • Papalambros P, Wilde D (1988) Principles of optimal design. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Patil D, Yun Y, Kim S-J, Boyd S, Horowitz M (2005) A new method for robust design of digital circuits. In: Proceedings of the 6th international symposium on quality electronic design (ISQED), pp 676–681, 2005

  • Pattanaik M, Banerjee S, Bahinipati B (2003) GP based transistor sizing for optimal design of nanoscale CMOS inverter. In: Proceedings of the 3rd IEEE conference on nanotechnology, pp 524–527, 2003

  • Peterson E (1976) Geometric programming. SIAM Rev 18(1):1–51

    MATH  MathSciNet  Google Scholar 

  • Peterson E (2001) The origins of geometric programming. Ann Oper Res 105(1-4):15–19

    MATH  MathSciNet  Google Scholar 

  • Qin Z, Cheng C-K (2003) Realizable parasitic reduction using generalized Y-Δ transformation. In: Proceedings of the 40th IEEE/ACM design automation conference (DAC), pp, 220–225, 2003

  • Rajasekera J, Yamada M (2001) Estimating the firm value distribution function by entropy optimization and geometric programming. Ann Oper Res 105(1–4):61–75

    MATH  MathSciNet  Google Scholar 

  • Rajpogal J, Bricker D (1990) Posynomial geometric programming as a special case of semi-infinite linear programming. J Optim Theory Appl 66(3):444–475

    Google Scholar 

  • Rao S (1996) Engineering optimization: theory and practice, 3rd edn. Wiley–Interscience, New York

    Google Scholar 

  • Rijckaert M, Walraven E (1985) Geometric programming: estimation of Lagrange multipliers. Oper Res 33(1):85–93

    MATH  MathSciNet  Google Scholar 

  • Rosenberg E (1989) Optimization module sizing in VLSI floorplanning by nonlinear programming. ZOR-Methods Model Oper Res 33(2):131–143

    MATH  Google Scholar 

  • Rubenstein J, Penfield P, Horowitz M (1983) Signal delay in RC tree networks. IEEE Trans Comput Aided Des Integr Circuits Syst 2(3):202–211

    Google Scholar 

  • Sakurai T (1988) Approximation of wiring delay in MOSFET LSI. IEEE J Solid State Circuits 18(4):418–426

    Google Scholar 

  • Salomone H, Iribarren O (1993) Posynomial modeling of batch plants: a procedure to include process decision variables. Comput Chem Eng 16(3):173–184

    Google Scholar 

  • Salomone H, Montagna J, Iribarren O (1994) Dynamic simulations in the design of batch processes. Comput Chem Eng 18(3):191–204

    Google Scholar 

  • Sancheti P, Sapatnekar S (1996) Optimal design of macrocells for low power and high speed. IEEE Trans Comput Aided Des Integr Circuits Syst 15(9):1160–1166

    Google Scholar 

  • Sapatnekar S (1996) Wire sizing as a convex optimization problem: exploring the area-delay tradeoff. IEEE Trans Comput Aided Des Integr Circuits Syst 15(8):1001–1011

    Google Scholar 

  • Sapatnekar S (2004) Timing. Kluwer Academic, Dordrecht

    MATH  Google Scholar 

  • Sapatnekar S, Rao V, Vaidya P, Kang S (1993) An exact solution to the transistor sizing problem for CMOS circuits using convex optimization. IEEE Trans Comput Aided Des Integr Circuits Syst 12(11):1621–1634

    Google Scholar 

  • Sathyamurthy H, Sapatnekar S, Fishburn J (1998) Speeding up pipelined circuits through a combination of gate sizing and clock skew optimization. IEEE Trans Comput Aided Des Integr Circuits Syst 17(2):173–182

    Google Scholar 

  • Satish N, Ravindran K, Moskewicz M, Chinnery D, Keutzer K (2005) Evaluating the effectiveness of statistical gate sizing for power optimization. Technical report ERL memorandum M05/28, University of California at Berkeley, August 2005

  • Scott C, Jefferson T, Jorjani S (2004) Duals for classical inventory models via generalized geometric programming. J Appl Math Decis Sci 8(3):191–200

    MATH  MathSciNet  Google Scholar 

  • Sherali H (1998) Global optimization of nonconvex polynomial programming problems having rational exponents. J Global Optim 12(3):267–283

    MATH  MathSciNet  Google Scholar 

  • Sherwani N (1999) Algorithms for VLSI design automation, 3rd edn. Kluwer Academic, Dordrecht

    MATH  Google Scholar 

  • Shyu J, Sangiovanni-Vincetelli A, Fishburn J, Dunlop A (1988) Optimization-based transistor sizing. IIEEE J Solid State Circuits 23(2):400–409

    Google Scholar 

  • Singh J, Nookala V, Luo Z-Q, Sapatnekar S (2005) Robust gate sizing by geometric programming. In: Proceedings of the 42nd IEEE/ACM design automation conference (DAC), pp 315–320, 2005

  • Smeers Y, Tyteca D (1984) A geometric programming model for the optimal design of wastewater treatment plants. Oper Res 32(2):314–342

    MATH  Google Scholar 

  • Sonmez A, Baykasoglu A, Dereli T, Flz I (1999) Dynamic optimization of multipass milling operations via geometric programming. Int J Mach Tools Manuf 39(2):297–320

    Google Scholar 

  • Stark R, Machida M (1993) Chance design costs-A distributional limit. In: Proceedings of the 2nd international symposium on uncertainty modeling and analysis, pp 269–270, 1993

  • Sutherland I, Sproull B, Harris D (1999) Logical effort: designing fast CMOS circuits. Morgan Kaufmann, San Francisco

    Google Scholar 

  • Swahn B, Hassoun S (2006) Gate sizing: FinFETs vs 32nm bulk MOSFETs. In: Proceedings of the 43rd IEEE/ACM design automation conference (DAC), pp 528–531, 2006

  • Trivedi K, Sigmon T (1981) Optimal design of linear storage hierarchies. J ACM 28(2):270–288

    MATH  MathSciNet  Google Scholar 

  • Tsai J-F, Li H-L, Hu N-Z (2002) Global optimization for signomial discrete programming problems in engineering design. Eng Optim 34(6):613–622

    Google Scholar 

  • Vanderhaegen J, Brodersen R (2004) Automated design of operational transconductance amplifiers using reversed geometric programming. In: Proceedings of the 41st IEEE/ACM design automation conference (DAC), pp 133–138, 2004

  • Vanderplaats G (1984) Numerical optimization techniques for engineering design. McGraw–Hill, New York

    MATH  Google Scholar 

  • Vardi Y (1985) Empirical distributions in selection bias models. Ann Stat 13(1):178–203

    MATH  MathSciNet  Google Scholar 

  • Wall T, Greening D, Woolsey R (1986) Solving complex chemical equilibria using a geometric-programming based technique. Oper Res 34(3):345–355

    Article  MATH  Google Scholar 

  • Wilde D (1978) Globally optimal design. Wiley, New York

    Google Scholar 

  • Wilde D, Beightler C (1967) Foundations of optimization. Prentice Hall, Englewood Cliffs

    MATH  Google Scholar 

  • Wong D (1981) Maximum likelihood, entropy maximization, and the geometric programming approaches to the calibration of trip distribution models. Transp Res Part B Methodol 15(5):329–343

    Google Scholar 

  • Xu Y, Pileggi L, Boyd S (2004) ORACLE: optimization with recourse of analog circuits including layout extraction. In: Proceedings of the 41st IEEE/ACM design automation conference (DAC), pp 151–154, 2004

  • Yang H-H, Bricker D (1997) Investigation of path-following algorithms for signomial geometric programming problems. Eur J Oper Res 103(1):230–241

    MATH  Google Scholar 

  • Yazarel H, Pappas G (2004) Geometric programming relaxations for linear system reachability. In: Proceedings of the 2004 American control conference (ACC), pp 553–559, 2004

  • Young F, Chu C, Luk W, Wong Y (2001) Handling soft modules in general nonslicing floorplan using Lagrangian relaxation. IEEE Trans Comput Aided Des Integr Circuits Syst 20(5):687–629

    Google Scholar 

  • Yun K, Xi C (1997) Second-order method of generalized geometric programming for spatial frame optimization. Comput Methods Appl Mech Eng 141(1-2):117–123

    Google Scholar 

  • Zener C (1971) Eng design by geometric programming. Wiley, New York

    Google Scholar 

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Correspondence to Seung-Jean Kim.

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Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67–127 (2007). https://doi.org/10.1007/s11081-007-9001-7

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