Abstract
Many engineering optimization problems include unavoidable uncertainties in parameters or variables. Ignoring such uncertainties when solving the optimization problems may lead to inferior solutions that may even violate problem constraints. Another challenge in most engineering optimization problems is having different conflicting objectives that cannot be minimized simultaneously. Finding a balanced trade-off between these objectives is a complex and time-consuming task. In this paper, an optimization framework is proposed to address both of these challenges. First, we exploit a self-calibrating multi-objective framework to achieve a balanced trade-off between the conflicting objectives. Then, we develop the robust counterpart of the uncertainty-aware self-calibrating multi-objective optimization framework. The significance of this framework is that it does not need any manual tuning by the designer. We also develop a mathematical demonstration of the objective scale invariance property of the proposed framework. The engineering problem considered in this paper to illustrate the effectiveness of the proposed framework is a popular sizing problem in digital integrated circuit design. However, the proposed framework can be applied to any uncertain multi-objective optimization problem that can be formulated in the geometric programming format. We propose to consider variations in the sizes of circuit elements during the optimization process by employing ellipsoidal uncertainty model. For validation, several industrial clock networks are sized by the proposed framework. The results show a significant reduction in one objective (power, on average 38 %) as well as significant increase in the robustness of solutions to the variations. This is achieved with no significant degradation in the other objective (timing metrics of the circuit) or reduction in its standard deviation which demonstrates a more robust solution.
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References
Andersson A, Thiringer T (2014) Inverter losses minimization using variable switching frequency based on multi-objective optimization. In: Proceedings of ICEM, pp 789–795
Antunes C, Oliveira E, Lima P (2014) A multi-objective GRASP procedure for reactive power compensation planning. Optim Eng 15(1):199–215
Ben-Tal A, Nemirovski A (2000) Robust solutions of linear programming problems contaminated with uncertain data. Math Program 88:411–424
Bertsimas D, Brown D, Caramanis C (2011) Theory and applications of robust optimization. SIAM Rev 53:464–501
Boni O, Ben-Tal A, Nemirovski A (2008) Robust solutions to conic quadratic problems and their applications. Optim Eng 9(1):1–18
Boyd S, Kim S (2005) Geometric programming for circuit optimization. In: Proceedings of ISPD, pp 44–46
Boyd S, Kim S, Patil D, Horowitz M (2005) Digital circuit optimization via geometric programming. Oper Res 53:899–932
Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge
Chang Y, Wang C, Chen H (2012) On construction low power and robust clock tree via slew budgeting. In: Proceedings of ISPD, pp 129–136
Chen J, Tehranipoor M (2013) Critical paths selection and test cost reduction considering process variations. In: Proceedings of ATS, pp 259–264
Chiang M (2005) Geometric programming for communication systems. Commun Inf Theory 2(1/2):1–154
Creese R (2011) Geometric programming for design and cost optimization. Morgan and Claypool Publishers, San Rafael
Doolittle E, Kerivin H, Wiecek M (2009) A robust multiobjective optimization problem with application to internet routing. Technical Report TR2012 11 DKW, Clemson University
Ehrgott M, Ide J, Schobel A (2014) Minmax robustness for multi-objective optimization problems. Eur J Oper Res 239(1):17–31
Ewetz R, Koh C-K (2013) Local merges for effective redundancy in clock networks. In: Proceedings of ISPD, pp 162–167. ACM
Farshidi A, Rakai L, Behjat L, Westwick D (2013) A self-tuning multi-objective optimization framework for geometric programming with gate sizing applications. In: Proceedings of GLSVLSI, pp 305–310
Fliege J, Werner R (2014) Robust multiobjective optimization & applications in portfolio optimization. Eur J Oper Res 40(2–3):422–433
Geoffrion AM (1967) Proper efficiency and the theory of vector maximization. J Math Anal Appl 22(3):618–630
Hsiung K, Kim S, Boyd S (2008) Tractable approximate robust geometric programming. Optim Eng 9(2):95–118
Hu J, Mehrotra S (2012) Robust and stochastically weighted multiobjective optimization models and reformulations. Oper Res 60(4):936–953
ISPD 2010 high performance clock network synthesis contest. http://www.sigda.org/ispd/contests/10/ispd10cns.html. Accessed 4 Mar 2010
Jagarlapudi S, Ben-Tal A, Bhattacharyya C (2013) Robust formulations for clustering-based large-scale classification. Optim Eng 14(2):225–250
Jakobsson S, Saif-Ul-Hasnain M, Rundqvist R, Edelvik F, Andersson B, Patriksson M, Ljungqvist M, Lortet D, Wallesten J (2010) Combustion engine optimization: a multiobjective approach. Optim Eng 11(4):533–554
Kahng AB, Kang S, Lee H (2013) Smart non-default routing for clock power reduction. In: DAC, p 91
Kashfi F, Hatami S, Pedram M (2011) Multi-objective optimization techniques for VLSI circuits. In: Proceedings of ISQED, pp 1–8
Kim J, Joo D, Kim T (2013) An optimal algorithm of adjustable delay buffer insertion for solving clock skew variation problem. In: Proceedings of DAC, pp 1–6
Kuroiwa D, Lee G (2012) On robust multiobjective optimization. Vietnam J Math 234(2):305–317
Lee D, Markov I (2011) Multilevel tree fusion for robust clock networks. In: Proceedings of ICCAD, pp 632–639
Leung S (2007) A non-linear goal programming model and solution method for the multi-objective trip distribution problem in transportation engineering. Optim Eng 8(3):277–298
Lin M (2011) Introduction to VLSI systems: a logic, circuit, and system perspective. CRC Press, Boca Raton
Lorenz R, Boyd S (2005) Robust minimum variance beamforming. IEEE Trans Signal Process 53(5):1684–1696
Miettinen K (1999) Nonlinear multiobjective optimization. Kluwer Academic Publishers, Boston
Mosek 6.0. http://www.mosek.com
Naidu S (2015) Geometric programming formulation for gate sizing with pipelining constraints. In: Proceedings of VLSID, pp 452–457
ngspice 24. http://ngspice.sourceforge.net/
Ny J, Pappas G (2010) Geometric programming and mechanism design for air traffic conflict resolution. In: Proceedings of American control conference, pp 3069–3074
Patil D, Yun S, Kim S, Cheung A, Horowitz M, Boyd S (2005) A new method for design of robust digital circuits. In: Proceedings international symposium on quality electronic design (ISQED), pp 676–681
Rakai L, Farshidi A, Behjat L, Westwick D (2013) Buffer sizing for clock networks using robust geometric programming considering variations in buffer sizes. In: Proceedings of ISPD, pp 154–161
Singh J, Luo Z, Sapatnekar S (2008) A geometric programming-based worst case gate sizing method incorporating spatial correlation. IEEE Trans Comput Aided Des 27(2):295–308
Su P, Li Y (2014) Design optimization of 16-nm bulk FinFET technology via geometric programming. In: Proceedings of IWCE, pp 1–4
Yang K, Huang J, Wu Y, Wang X, Chiang M (2014) Distributed robust optimization (DRO), part i: framework and example. Optim Eng 15(1):35–67
Zhu Q (2003) High-speed clock network design. Kluwer Academic Publishers, Boston
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Farshidi, A., Rakai, L., Behjat, L. et al. Variation-aware clock network buffer sizing using robust multi-objective optimization. Optim Eng 17, 473–500 (2016). https://doi.org/10.1007/s11081-016-9317-2
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DOI: https://doi.org/10.1007/s11081-016-9317-2