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|hdl=2268/253225 |s2cid=8946639 }}</ref> [[electrical engineering]],<ref>{{Cite journal |lastlast1=Giraldo |firstfirst1=Juan S. |last2=Castrillon |first2=Jhon A. |last3=Lopez |first3=Juan Camilo |last4=Rider |first4=Marcos J. |last5=Castro |first5=Carlos A. |date=July 2019-07 |title=Microgrids Energy Management Using Robust Convex Programming |url=https://ieeexplore.ieee.org/document/8424876/ |journal=IEEE Transactions on Smart Grid |volume=10 |issue=4 |pages=4520–4530 |doi=10.1109/TSG.2018.2863049 |s2cid=115674048 |issn=1949-3053}}</ref><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 | s2cid= 8774918 }}</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|s2cid=108843522}}</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|s2cid=15713904 }}</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 | 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 | journal = Proceedings of the IEEE | volume = 73 | issue = 3| pages = 433–481 | doi=10.1109/proc.1985.13167| hdl = 2142/74118 | s2cid = 30443041 | 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 = Ben-Tal | first1 = A. | last2 = Nemirovski | first2 = A. | year = 1998 | title = Robust Convex Optimization | 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 | journal = Operations Research Letters | volume = 25 | 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 | journal = Mathematical Programming, Series B | volume = 92 | issue = 3| pages = 453–480 | doi=10.1007/s101070100286| citeseerx = 10.1.1.298.7965 | s2cid = 1429482 }}

*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 | journal = Mathematical Programming | volume = 98 | issue = 1–3| pages = 49–71 | doi=10.1007/s10107-003-0396-4| citeseerx = 10.1.1.392.4470 | s2cid = 1279073 }}

*{{cite journal | last1 = Bertsimas | first1 = D. | last2 = Sim | first2 = M. | year = 2006 | title = Tractable Approximations to Robust Conic Optimization Problems Dimitris Bertsimas | journal = Mathematical Programming | volume = 107 | issue = 1| pages = 5–36 | doi=10.1007/s10107-005-0677-1| citeseerx = 10.1.1.207.8378 | s2cid = 900938 }}

*{{cite journal | last1 = Chen | first1 = W. | last2 = Sim | first2 = M. | year = 2009 | title = Goal Driven Optimization | 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 | 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 | 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 | journal = Annals of Operations Research | volume = 30 | issue = 1| pages = 63–80 | doi=10.1007/bf02204809| s2cid = 44126126 }}

* 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 | journal = Optimization Methods and Software | volume = 24 | issue = 3| pages = 381–406 | doi=10.1080/10556780802712889| citeseerx = 10.1.1.416.4912 | s2cid = 16443437 }}

*{{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 | journal = Operations Research | volume = 43 | issue = 2| pages = 264–281 | doi=10.1287/opre.43.2.264}}

*Nejadseyfi, O., Geijselaers H.J.M, van den Boogaard A.H. (2018). "Robust optimization based on analytical evaluation of uncertainty propagation". ''Engineering Optimization'' '''51''' (9): 1581-1603. doi:[https://www.tandfonline.com/doi/10.1080/0305215X.2018.1536752 10.1080/0305215X.2018.1536752].

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*{{cite journal | last1 = Wald | first1 = A | year = 1945 | title = Statistical decision functions which minimize the maximum risk | 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|s2cid=8774918 }}

==External links==