Robust optimization: Difference between revisions

Revision as of 07:14, 12 April 2020

Revision as of 23:13, 7 December 2020

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: <math>\max_{x\in X}\min_{u\in U(x)} f(x,u)</math>

where the <math>\max</math> represents the decision maker, the <math>\min</math> represents Nature, namely [[uncertainty]], <math>X</math> represents the decision space and <math>U(x)</math> denotes the set of possible values of <math>u</math> associated with decision <math>x</math>. This is the ''classic'' format of the generic model, and is often referred to as ''minimax'' or ''maximin'' optimization problem. The non-probabilistic ('''deterministic''') model has been and is being extensively used for robust optimization especially in the field of signal processing.<ref>{{cite journal | last1 = Verdu | first1 = S. | last2 = Poor | first2 = H. V. | year = 1984 | title = On Minimax Robustness: A general approach and applications | url = | journal = IEEE Transactions on Information Theory | volume = 30 | issue = 2| pages = 328–340 | doi=10.1109/tit.1984.1056876| citeseerx = 10.1.1.132.837 }}</ref><ref>{{cite journal | last1 = Kassam | first1 = S. A. | last2 = Poor | first2 = H. V. | year = 1985 | title = Robust Techniques for Signal Processing: A Survey | url = | journal = Proceedings of the IEEE | volume = 73 | issue = 3| pages = 433–481 | doi=10.1109/proc.1985.13167| hdl = 2142/74118 | hdl-access = free }}</ref><ref>M. Danish Nisar. [http://www.shaker.eu/shop/978-3-8440-0332-1 "Minimax Robustness in Signal Processing for Communications"], Shaker Verlag, {{ISBN|978-3-8440-0332-1}}, August 2011.</ref>

The equivalent [[mathematical programming]] (MP) of the classic format above is

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== Further reading ==

*H.J. Greenberg. Mathematical Programming Glossary. World Wide Web, http://glossary.computing.society.informs.org/, 1996-2006. Edited by the INFORMS Computing Society.

*{{cite journal | last1 = Ben-Tal | first1 = A. | last2 = Nemirovski | first2 = A. | year = 1998 | title = Robust Convex Optimization | url = | journal = Mathematics of Operations Research | volume = 23 | issue = 4| pages = 769–805 | doi=10.1287/moor.23.4.769| citeseerx = 10.1.1.135.798 }}

*{{cite journal | last1 = Ben-Tal | first1 = A. | last2 = Nemirovski | first2 = A. | year = 1999 | title = Robust solutions to uncertain linear programs | url = | journal = Operations Research Letters | volume = 25 | issue = | pages = 1–13 | doi=10.1016/s0167-6377(99)00016-4| citeseerx = 10.1.1.424.861 }}

*{{cite journal | last1 = Ben-Tal | first1 = A. | last2 = Arkadi Nemirovski | first2 = A. | year = 2002 | title = Robust optimization—methodology and applications | url = | journal = Mathematical Programming, Series B | volume = 92 | issue = 3| pages = 453–480 | doi=10.1007/s101070100286| citeseerx = 10.1.1.298.7965 }}

*Ben-Tal A., El Ghaoui, L. and Nemirovski, A. (2006). ''Mathematical Programming, Special issue on Robust Optimization,'' Volume 107(1-2).

*Ben-Tal A., El Ghaoui, L. and Nemirovski, A. (2009). Robust Optimization. ''Princeton Series in Applied Mathematics,'' Princeton University Press.

*{{cite journal | last1 = Bertsimas | first1 = D. | last2 = Sim | first2 = M. | year = 2003 | title = Robust Discrete Optimization and Network Flows | url = | journal = Mathematical Programming | volume = 98 | issue = 1–3| pages = 49–71 | doi=10.1007/s10107-003-0396-4| citeseerx = 10.1.1.392.4470 }}

*{{cite journal | last1 = Bertsimas | first1 = D. | last2 = Sim | first2 = M. | year = 2006 | title = Tractable Approximations to Robust Conic Optimization Problems Dimitris Bertsimas | url = | journal = Mathematical Programming | volume = 107 | issue = 1| pages = 5–36 | doi=10.1007/s10107-005-0677-1| citeseerx = 10.1.1.207.8378 }}

*{{cite journal | last1 = Chen | first1 = W. | last2 = Sim | first2 = M. | year = 2009 | title = Goal Driven Optimization | url = | journal = Operations Research | volume = 57 | issue = 2| pages = 342–357 | doi=10.1287/opre.1080.0570}}

*{{cite journal | last1 = Chen | first1 = X. | last2 = Sim | first2 = M. | last3 = Sun | first3 = P. | last4 = Zhang | first4 = J. | year = 2008 | title = A Linear-Decision Based Approximation Approach to Stochastic Programming | url = | journal = Operations Research | volume = 56 | issue = 2| pages = 344–357 | doi=10.1287/opre.1070.0457}}

*{{cite journal | last1 = Chen | first1 = X. | last2 = Sim | first2 = M. | last3 = Sun | first3 = P. | year = 2007 | title = A Robust Optimization Perspective on Stochastic Programming | url = | journal = Operations Research | volume = 55 | issue = 6| pages = 1058–1071 | doi=10.1287/opre.1070.0441}}

*{{cite journal | last1 = Dembo | first1 = R | year = 1991 | title = Scenario optimization | url = | journal = Annals of Operations Research | volume = 30 | issue = 1| pages = 63–80 | doi=10.1007/bf02204809}}

* Dodson, B., Hammett, P., and Klerx, R. (2014) ''Probabilistic Design for Optimization and Robustness for Engineers'' John Wiley & Sons, Inc. {{ISBN|978-1-118-79619-1}}

*{{cite journal | last1 = Gupta | first1 = S.K. | last2 = Rosenhead | first2 = J. | year = 1968 | title = Robustness in sequential investment decisions | doi = 10.1287/mnsc.15.2.B18 | journal = Management Science | volume = 15 | issue = 2| pages = 18–29 }}

*Kouvelis P. and Yu G. (1997). ''Robust Discrete Optimization and Its Applications,'' Kluwer.

*{{cite journal | last1 = Mutapcic | first1 = Almir | last2 = Boyd | first2 = Stephen | year = 2009 | title = Cutting-set methods for robust convex optimization with pessimizing oracles | url = | journal = Optimization Methods and Software | volume = 24 | issue = 3| pages = 381–406 | doi=10.1080/10556780802712889| citeseerx = 10.1.1.416.4912 }}

*{{cite journal | last1 = Mulvey | first1 = J.M. | last2 = Vanderbei | first2 = R.J. | last3 = Zenios | first3 = S.A. | year = 1995 | title = Robust Optimization of Large-Scale Systems | url = | journal = Operations Research | volume = 43 | issue = 2| pages = 264–281 | doi=10.1287/opre.43.2.264}}

*{{cite journal | last1 = Rosenblat | first1 = M.J. | year = 1987 | title = A robust approach to facility design | url = | journal = International Journal of Production Research | volume = 25 | issue = 4| pages = 479–486 | doi = 10.1080/00207548708919855 }}

*{{cite journal | last1 = Rosenhead | first1 = M.J | last2 = Elton | first2 = M | last3 = Gupta | first3 = S.K. | year = 1972 | title = Robustness and Optimality as Criteria for Strategic Decisions | url = | journal = Operational Research Quarterly | volume = 23 | issue = 4| pages = 413–430 | doi=10.2307/3007957| jstor = 3007957 }}

*Rustem B. and Howe M. (2002). ''Algorithms for Worst-case Design and Applications to Risk Management,'' Princeton University Press.

*{{cite journal | last1 = Sniedovich | first1 = M | year = 2007 | title = The art and science of modeling decision-making under severe uncertainty | url = | journal = Decision Making in Manufacturing and Services| volume = 1 | issue = 1–2| pages = 111–136 | doi = 10.7494/dmms.2007.1.2.111 | doi-access = free }}

*{{cite journal | last1 = Sniedovich | first1 = M | year = 2008 | title = Wald's Maximin Model: a Treasure in Disguise! | url = | journal = Journal of Risk Finance | volume = 9 | issue = 3| pages = 287–291 | doi=10.1108/15265940810875603}}

*{{cite journal | last1 = Sniedovich | first1 = M | year = 2010 | title = A bird's view of info-gap decision theory | url = | journal = Journal of Risk Finance | volume = 11 | issue = 3| pages = 268–283 | doi=10.1108/15265941011043648}}

*{{cite journal | last1 = Wald | first1 = A | year = 1939 | title = Contributions to the theory of statistical estimation and testing hypotheses | url = | journal = The Annals of Mathematics | volume = 10 | issue = 4| pages = 299–326 | doi=10.1214/aoms/1177732144| doi-access = free }}

*{{cite journal | last1 = Wald | first1 = A | year = 1945 | title = Statistical decision functions which minimize the maximum risk | url = | journal = The Annals of Mathematics | volume = 46 | issue = 2| pages = 265–280 | doi=10.2307/1969022| jstor = 1969022 }}

*Wald, A. (1950). ''Statistical Decision Functions,'' John Wiley, NY.

*{{cite book |doi=10.1109/IranianCEE.2015.7146458|isbn=978-1-4799-1972-7|chapter=Generation Maintenance Scheduling via robust optimization|title=2015 23rd Iranian Conference on Electrical Engineering|year=2015|last1=Shabanzadeh|first1=Morteza|last2=Fattahi|first2=Mohammad|pages=1504–1509}}