Robust optimization: Difference between revisions

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== History ==

The origins of robust optimization date back to the establishment of modern [[decision theory]] in the 1950s and the use of '''worst case analysis''' and [[Wald's maximin model]] as a tool for the treatment of severe uncertainty. It became a discipline of its own in the 1970s with parallel developments in several scientific and technological fields. Over the years, it has been applied in [[statistics]], but also in [[operations research]],<ref>{{cite journal|last=Bertsimas|first=Dimitris|author2=Sim, Melvyn |title=The Price of Robustness|journal=Operations Research|year=2004|volume=52|issue=1|pages=35–53|doi=10.1287/opre.1030.0065}}</ref>[[electrical engineering]],<ref name="VPP Robust 2015">{{Cite journal| title = The design of a risk-hedging tool for virtual power plants via robust optimization approach | journal= Applied Energy | date = October 2015 | doi = 10.1016/j.apenergy.2015.06.059 | author = Shabanzadeh M | volume = 155 | pages = 766–777 | last2 = Sheikh-El-Eslami | first2 = M-K |last3 = Haghifam | first3 = P|last4 = M-R}}</ref><ref name="RO2015">{{Cite book| title = Generation Maintenance Scheduling via robust optimization | journal= 23rd Iranian Conference in Electrical Engineering (ICEE) | pages= 1504–1509 | date = July 2015 | doi = 10.1109/IranianCEE.2015.7146458 | author = Shabanzadeh M | last2 = Fattahi | first2 = M | isbn= 978-1-4799-1972-7 }}</ref> [[control theory]],<ref>{{cite journal|last=Khargonekar|first=P.P.|author2=Petersen, I.R. |author3=Zhou, K. |title=Robust stabilization of uncertain linear systems: quadratic stabilizability and H/sup infinity / control theory|journal=IEEE Transactions on Automatic Control|volume=35|issue=3|pages=356–361|doi=10.1109/9.50357|year=1990}}</ref> [[finance]],<ref>[https://books.google.com/books?id=p6UHHfkQ9Y8C&lpg=PR11&ots=AqlJfX5Z0X&dq=economics%20robust%20optimization&lr&hl=it&pg=PR11#v=onepage&q&f=false%20 Robust portfolio optimization]</ref> [[Investment management|portfolio management]]<ref>Md. Asadujjaman and Kais Zaman, "Robust Portfolio Optimization under Data Uncertainty" 15th National Statistical Conference, December 2014, Dhaka, Bangladesh.</ref> [[logistics]],<ref>{{cite journal|last=Yu|first=Chian-Son|author2=Li, Han-Lin |title=A robust optimization model for stochastic logistic problems|journal=International Journal of Production Economics|volume=64|issue=1–3|pages=385–397|doi=10.1016/S0925-5273(99)00074-2|year=2000}}</ref> [[manufacturing engineering]],<ref>{{cite journal|last=Strano|first=M|title=Optimization under uncertainty of sheet-metal-forming processes by the finite element method|journal=Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture|volume=220|issue=8|pages=1305–1315|doi=10.1243/09544054JEM480|year=2006}}</ref> [[chemical engineering]],<ref>{{cite journal|last=Bernardo|first=Fernando P.|author2=Saraiva, Pedro M. |title=Robust optimization framework for process parameter and tolerance design|journal=AIChE Journal|year=1998|volume=44|issue=9|pages=2007–2017|doi=10.1002/aic.690440908|hdl=10316/8195|hdl-access=free}}</ref> [[medicine]],<ref>{{cite journal|last=Chu|first=Millie|author2=Zinchenko, Yuriy |author3=Henderson, Shane G |author4= Sharpe, Michael B |title=Robust optimization for intensity modulated radiation therapy treatment planning under uncertainty|journal=Physics in Medicine and Biology|year=2005|volume=50|issue=23|pages=5463–5477|doi=10.1088/0031-9155/50/23/003|pmid=16306645}}</ref> and [[computer science]]. In [[engineering]] problems, these formulations often take the name of "Robust Design Optimization", RDO or "Reliability Based Design Optimization", RBDO.

== Example 1==

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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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*{{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.