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{{Short description|Mathematical optimization theory}}

In mathematics, '''robust optimization''' is an approach in [[optimization (mathematics)|optimization]] to deal with uncertainty. It is similar to the recourse model of [[stochastic programming]], in that some of the parameters are [[random variable]]s, except that feasibility for all possible realizations (called scenarios) is replaced by a [[penalty function]] in the objective. As such, the approach integrates [[goal programming]] with a scenario-based description of problem data. To illustrate, consider the LP:

'''Robust optimization''' is a field of [[mathematical optimization]] theory that deals with optimization problems in which a certain measure of robustness is sought against [[uncertainty]] that can be represented as deterministic variability in the value of the parameters of the problem itself and/or its solution. It is related to, but often distinguished from, [[probabilistic optimization]] methods such as chance-constrained optimization.<ref>{{cite journal | doi=10.3390/en15030825 | doi-access=free | title=Probabilistic Optimization Techniques in Smart Power System | date=2022 | last1=Riaz | first1=Muhammad | last2=Ahmad | first2=Sadiq | last3=Hussain | first3=Irshad | last4=Naeem | first4=Muhammad | last5=Mihet-Popa | first5=Lucian | journal=Energies | volume=15 | issue=3 | page=825 | hdl=11250/2988376 | hdl-access=free }}</ref><ref>{{Cite web| title=Robust Optimization: Chance Constraints | date=2008-04-28 | url=https://people.eecs.berkeley.edu/~elghaoui/Teaching/EE227A/lecture24.pdf | archive-url=https://web.archive.org/web/20230605233436/https://people.eecs.berkeley.edu/~elghaoui/Teaching/EE227A/lecture24.pdf | archive-date=2023-06-05}}</ref>

== History ==

:<math>\min cx + dy: Ax=b, Bx + Cy = e, x, y \ge 0,</math>

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 |hdl-access=free}}</ref> [[electrical engineering]],<ref>{{Cite journal |last1=Giraldo |first1=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 |title=Microgrids Energy Management Using Robust Convex Programming |journal=IEEE Transactions on Smart Grid |volume=10 |issue=4 |pages=4520–4530 |doi=10.1109/TSG.2018.2863049 |bibcode=2019ITSG...10.4520G |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| bibcode= 2015ApEn..155..766S }}</ref><ref name="RO2015">{{Cite book| pages= 1504–1509 | date = July 2015 | doi = 10.1109/IranianCEE.2015.7146458 | author = Shabanzadeh M | last2 = Fattahi | first2 = M | title= 2015 23rd Iranian Conference on Electrical Engineering | chapter= Generation Maintenance Scheduling via robust optimization | 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&dq=economics%20robust%20optimization&pg=PR11 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|bibcode=1998AIChE..44.2007B |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|bibcode=2005PMB....50.5463C |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==

where d, B, C and e are random variables with possible realizations <math>{(d(s), B(s), C(s), e(s): s \in \{1,...,N\})}</math>, where N = the number of scenarios. The robust optimization model for this LP is:

Consider the following [[linear programming]] problem

:<math>\min f(x, y(1), ..., y(N)) + wP(z(1), ..., z(N)): Ax=b, x \ge 0,</math>

:<math>\ B(s)\max_{x,y} +\ C(s)y(s)\{3x + z(s)2y\} =\ e(s),</math>\ and\mathrm <math>y(s){ subject \ to }\ \ x,y\ge 0,; cx + dy \le 10, \forall s(c,d)\in =P 1,...,N,</math>

where <math>P</math> is a given subset of <math>\mathbb{R}^{2}</math>.

What makes this a 'robust optimization' problem is the <math>\forall (c,d)\in P</math> clause in the constraints. Its implication is that for a pair <math>(x,y)</math> to be admissible, the constraint <math>cx + dy \le 10</math> must be satisfied by the '''worst''' <math>(c,d)\in P</math> pertaining to <math>(x,y)</math>, namely the pair <math>(c,d)\in P</math> that maximizes the value of <math>cx + dy</math> for the given value of <math>(x,y)</math>.

where f is a function that measures the cost of the policy, P is a penalty function, and w > 0 (a parameter to trade off the cost of infeasibility). One example of f is the expected value: <math>\ f(x, y) = cx + \sum_s{d(s)y(s)p(s)}</math>, where p(s) = probability of scenario s. In a worst-case model, <math>\ f(x,y) = \max_s{d(s)y(s)}</math>. The '''penalty function''' is defined to be zero if (x, y) is feasible (for all scenarios) -- i.e., P(0)=0. In addition, P satisfies a form of monotonicity: worse violations incur greater penalty. This often has the form <math>\ P(z) = U(z^+) + V(-z^-)</math> -- i.e., the "up" and "down" penalties, where U and V are strictly increasing functions.

If the parameter space <math>P</math> is finite (consisting of finitely many elements), then this robust optimization problem itself is a [[linear programming]] problem: for each <math>(c,d)\in P</math> there is a linear constraint <math>cx + dy \le 10</math>.

The above makes robust optimization similar (at least in the model) to a [[goal program]]. Recently, the robust optimization community defines it differently – it optimizes for the worst-case scenario, as in [[minimax]]. Let the uncertain MP be given by

If <math>P</math> is not a finite set, then this problem is a linear [[semi-infinite programming]] problem, namely a linear programming problem with finitely many (2) decision variables and infinitely many constraints.

:<math>\min f(x; s): x \in X(s),</math>

where S is known as the uncertainty set, which is usually a compact convex object as opposed to a small collection of scenarios (like parameter values).

== Classification ==

The robust optimization model (according to this more recent definition) is:

There are a number of classification criteria for robust optimization problems/models. In particular, one can distinguish between problems dealing with '''local''' and '''global''' models of robustness; and between '''probabilistic''' and '''non-probabilistic''' models of robustness. Modern robust optimization deals primarily with non-probabilistic models of robustness that are [[worst case]] oriented and as such usually deploy [[Wald's maximin model]]s.

=== Local robustness ===

:<math>\min_x {\max_{s \in S} f(x; s)}\, x \in X(t)\, \forall t \in S,</math>

The policy (x) is required to be feasible no matter what parameter value (scenario) occurs; hence, it is required to be in the intersection of all possible X(s). The inner maximization yields the worst possible objective value among all scenarios. There are variations, such as "adjustability" (i.e., recourse).

There are cases where robustness is sought against small perturbations in a nominal value of a parameter. A very popular model of local robustness is the [[stability radius|radius of stability]] model:

== See also ==

* [[Info-gap decision theory]]

* [[Minimax]]

* [[Minimax regret]]

* [[Robust statistics]]

: <math>\hat{\rho}(x,\hat{u}):= \max_{\rho\ge 0}\ \{\rho: u\in S(x), \forall u\in B(\rho,\hat{u})\}</math>

==External links==

* [http://www.robustopt.com ROME: Robust Optimization Made Easy]

* [http://www.aimms.com/operations-research/mathematical-programming/robust-optimization AIMMS: Robust Optimization]

* [http://robust.moshe-online.com: Robust Decision-Making Under Severe Uncertainty]

where <math>\hat{u}</math> denotes the nominal value of the parameter, <math>B(\rho,\hat{u})</math> denotes a ball of radius <math>\rho</math> centered at <math>\hat{u}</math> and <math>S(x)</math> denotes the set of values of <math>u</math> that satisfy given stability/performance conditions associated with decision <math>x</math>.

== Bibliography ==

In words, the robustness (radius of stability) of decision <math>x</math> is the radius of the largest ball centered at <math>\hat{u}</math> all of whose elements satisfy the stability requirements imposed on <math>x</math>. The picture is this:

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

[[Image:Local robustness.png|500px]]

Ben-Tal, A., Nemirovski, A. (1998). Robust Convex Optimization. ''Mathematics of Operations Research 23,'' 769-805.

where the rectangle <math>U(x)</math> represents the set of all the values <math>u</math> associated with decision <math>x</math>.

Ben-Tal, A., Nemirovski, A. (1999). Robust solutions to uncertain linear programs. ''Operations Research Letters 25,'' 1-13.

=== Global robustness ===

Ben-Tal, A. and Arkadi Nemirovski, A. (2002). Robust optimization—methodology and applications, ''Mathematical Programming, Series B 92,'' 453-480.

Consider the simple abstract robust optimization problem

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

: <math>\max_{x\in X}\ \{f(x): g(x,u)\le b, \forall u\in U\}</math>

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

where <math>U</math> denotes the set of all ''possible'' values of <math>u</math> under consideration.

Bertsimas, D. and M. Sim. (2003). Robust Discrete Optimization and Network Flows. ''Mathematical Programming,'' 98, 49-71.

This is a ''global'' robust optimization problem in the sense that the robustness constraint <math>g(x,u)\le b, \forall u\in U</math> represents all the ''possible'' values of <math>u</math>.

Bertsimas, D. and M. Sim. (2004). Price of Robustness. ''Operations Research,'' 52(1), 35-53.

The difficulty is that such a "global" constraint can be too demanding in that there is no <math>x\in X</math> that satisfies this constraint. But even if such an <math>x\in X</math> exists, the constraint can be too "conservative" in that it yields a solution <math>x\in X</math> that generates a very small payoff <math>f(x)</math> that is not representative of the performance of other decisions in <math>X</math>. For instance, there could be an <math>x'\in X</math> that only slightly violates the robustness constraint but yields a very large payoff <math>f(x')</math>. In such cases it might be necessary to relax a bit the robustness constraint and/or modify the statement of the problem.

Bertsimas, D. and M. Sim. (2006). Tractable Approximations to Robust Conic Optimization Problems Dimitris Bertsimas. '' Mathematical Programming, '' 107(1), 5 – 36.

==== Example 2====

Chen, W. and M. Sim. (2009). Goal Driven Optimization. ''Operations Research.'' 57(2), 342-357.

Consider the case where the objective is to satisfy a constraint <math>g(x,u)\le b,</math>. where <math>x\in X</math> denotes the decision variable and <math>u</math> is a parameter whose set of possible values in <math>U</math>. If there is no <math>x\in X</math> such that <math>g(x,u)\le b,\forall u\in U</math>, then the following intuitive measure of robustness suggests itself:

: <math>\rho(x):= \max_{Y\subseteq U} \ \{size(Y): g(x,u)\le b, \forall u\in Y\} \ , \ x\in X</math>

Chen, X., M. Sim, P. Sun and J. Zhang. (2008). A Linear-Decision Based Approximation Approach to Stochastic Programming. '' Operations Research '' 56(2), 344-357.

where <math>size(Y)</math> denotes an appropriate measure of the "size" of set <math>Y</math>. For example, if <math>U</math> is a finite set, then <math>size(Y)</math> could be defined as the [[cardinality]] of set <math>Y</math>.

Chen, X., M. Sim and P. Sun (2007). A Robust Optimization Perspective on Stochastic Programming. '' Operations Research, '' 55(6), 1058-1071.

In words, the robustness of decision is the size of the largest subset of <math>U</math> for which the constraint <math>g(x,u)\le b</math> is satisfied for each <math>u</math> in this set. An optimal decision is then a decision whose robustness is the largest.

Dembo, R. (1991). Scenario optimization, ''Annals of Operations Research,'' 30(1), 63-80.

This yields the following robust optimization problem:

Gupta, S.K. and Rosenhead, J. (1968). Robustness in sequential investment decisions, ''Management science,'' 15(2), B-18-29.

: <math>\max_{x\in X, Y\subseteq U} \ \{size(Y): g(x,u) \le b, \forall u\in Y\}</math>

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

This intuitive notion of global robustness is not used often in practice because the robust optimization problems that it induces are usually (not always) very difficult to solve.

Mutapcic, Almir and Boyd, Stephen. (2009). Cutting-set methods for robust convex optimization with pessimizing oracles, ''Optimization Methods and Software,'' 24(3), 381-406.

====Example 3====

Mulvey, J.M., Vanderbei, R.J., Zenios, S.A. (1995). Robust Optimization of Large-Scale Systems

Consider the robust optimization problem

''Operations Research,'' 43(2),264-281.

:<math>z(U):= \max_{x\in X}\ \{f(x): g(x,u)\le b, \forall u\in U\}</math>

where <math>g</math> is a real-valued function on <math>X\times U</math>, and assume that there is no feasible solution to this problem because the robustness constraint <math>g(x,u)\le b, \forall u\in U</math> is too demanding.

To overcome this difficulty, let <math>\mathcal{N}</math> be a relatively small subset of <math>U</math> representing "normal" values of <math>u</math> and consider the following robust optimization problem:

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

:<math>z(\mathcal{N}):= \max_{x\in X}\ \{f(x): g(x,u)\le b, \forall u\in \mathcal{N}\}</math>

Since <math>\mathcal{N}</math> is much smaller than <math>U</math>, its optimal solution may not perform well on a large portion of <math>U</math> and therefore may not be robust against the variability of <math>u</math> over <math>U</math>.

Sniedovich, M. (2007). The art and science of modeling decision-making under severe uncertainty, ''Decision Making in Manufacturing and Services,'' 1(1-2), 111-136.

One way to fix this difficulty is to relax the constraint <math>g(x,u)\le b</math> for values of <math>u</math> outside the set <math>\mathcal{N}</math> in a controlled manner so that larger violations are allowed as the distance of <math>u</math> from <math>\mathcal{N}</math> increases. For instance, consider the relaxed robustness constraint

Sniedovich, M. (2008). Wald's Maximin Model: a Treasure in Disguise!, ''Journal of Risk Finance,'' 9(3), 287-291.

: <math>g(x,u) \le b + \beta \cdot dist(u,\mathcal{N}) \ , \ \forall u\in U</math>

where <math>\beta \ge 0</math> is a control parameter and <math>dist(u,\mathcal{N})</math> denotes the distance of <math>u</math> from <math>\mathcal{N}</math>. Thus, for <math>\beta =0</math> the relaxed robustness constraint reduces back to the original robustness constraint.

Sniedovich, M. (2010). A bird's view of info-gap decision theory, ''Journal of Risk Finance,'' 11(3), 268-283.

This yields the following (relaxed) robust optimization problem:

:<math>z(\mathcal{N},U):= \max_{x\in X}\ \{f(x): g(x,u)\le b + \beta \cdot dist(u,\mathcal{N}) \ , \ \forall u\in U\}</math>

Wald, A. (1939). Contributions to the theory of statistical estimation and testing hypotheses, ''The Annals of Mathematics,'' 10(4), 299-326.

The function <math>dist</math> is defined in such a manner that

Wald, A. (1945). Statistical decision functions which minimize the maximum risk, ''The Annals of Mathematics,'' 46(2), 265-280.

:<math>dist(u,\mathcal{N})\ge 0,\forall u\in U</math>

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

and

[[Category:Mathematical optimization]]

: <math>dist(u,\mathcal{N})= 0,\forall u\in \mathcal{N}</math>

and therefore the optimal solution to the relaxed problem satisfies the original constraint <math>g(x,u)\le b</math> for all values of <math>u</math> in <math>\mathcal{N}</math>. It also satisfies the relaxed constraint

[[sv:Robust optimering]]

: <math>g(x,u)\le b + \beta \cdot dist(u,\mathcal{N})</math>

outside <math>\mathcal{N}</math>.

===Non-probabilistic robust optimization models===

The dominating paradigm in this area of robust optimization is [[Wald's maximin model]], namely

: <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. [https://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

:<math>\max_{x\in X,v\in \mathbb{R}} \ \{v: v\le f(x,u), \forall u\in U(x)\}</math>

Constraints can be incorporated explicitly in these models. The generic constrained classic format is

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

The equivalent constrained MP format is defined as:

:<math>\max_{x\in X,v\in \mathbb{R}} \ \{v: v\le f(x,u), g(x,u)\le b, \forall u\in U(x)\}</math>

===Probabilistically robust optimization models===

These models quantify the uncertainty in the "true" value of the parameter of interest by probability distribution functions. They have been traditionally classified as [[stochastic programming]] and [[stochastic optimization]] models. Recently, probabilistically robust optimization has gained popularity by the introduction of rigorous theories such as [[scenario optimization]] able to quantify the robustness level of solutions obtained by randomization. These methods are also relevant to data-driven optimization methods.

===Robust counterpart===

The solution method to many robust program involves creating a deterministic equivalent, called the robust counterpart. The practical difficulty of a robust program depends on if its robust counterpart is computationally tractable.<ref>Ben-Tal A., El Ghaoui, L. and Nemirovski, A. (2009). Robust Optimization. ''Princeton Series in Applied Mathematics,'' Princeton University Press, 9-16.</ref><ref>[[Sven Leyffer|Leyffer S.]], Menickelly M., Munson T., Vanaret C. and Wild S. M (2020). A survey of nonlinear robust optimization. ''INFOR: Information Systems and Operational Research,'' Taylor \& Francis.</ref>

== See also ==

* [[Stability radius]]

* [[Minimax]]

* [[Minimax estimator]]

* [[Minimax regret]]

* [[Robust statistics]]

* [[Robust decision making]]

* [[Robust fuzzy programming]]

* [[Stochastic programming]]

* [[Stochastic optimization]]

* [[Info-gap decision theory]]

* [[Taguchi methods]]

== References ==

{{Reflist}}

== 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 | 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 | s2cid = 15905691 }}

*{{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 | url = http://scholarbank.nus.edu.sg/handle/10635/43946 }}

*{{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 | url = http://scholarbank.nus.edu.sg/handle/10635/44052 }}

*{{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.

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*{{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==

* [https://www.robustopt.com ROME: Robust Optimization Made Easy]

* [http://robust.moshe-online.com: Robust Decision-Making Under Severe Uncertainty]

* [https://robustimizer.com/ Robustimizer: Robust optimization software]

{{Major subfields of optimization}}

[[Category:Mathematical optimization]]