Editing Mathematical optimization (section)

== Applications ==

===Mechanics===
Problems in [[rigid body dynamics]] (in particular articulated rigid body dynamics) often require mathematical programming techniques, since you can view rigid body dynamics as attempting to solve an [[ordinary differential equation]] on a constraint manifold;<ref>{{cite journal |first=A.F. |last=Vereshchagin |title=Modelling and control of motion of manipulation robots |journal=Soviet Journal of Computer and Systems Sciences |volume=27 |issue=5 |pages=29–38 |year=1989}}</ref> the constraints are various nonlinear geometric constraints such as "these two points must always coincide", "this surface must not penetrate any other", or "this point must always lie somewhere on this curve". Also, the problem of computing contact forces can be done by solving a [[linear complementarity problem]], which can also be viewed as a QP (quadratic programming) problem.

Many design problems can also be expressed as optimization programs. This application is called design optimization. One subset is the [[engineering optimization]], and another recent and growing subset of this field is [[multidisciplinary design optimization]], which, while useful in many problems, has in particular been applied to [[aerospace engineering]] problems.

This approach may be applied in cosmology and astrophysics.<ref>{{cite journal |first1=S. |last1=Haggag|first2=F. |last2=Desokey|first3=M. |last3=Ramadan   |title=A cosmological inflationary model using optimal control |journal= Gravitation and Cosmology|volume=23 |issue=3 |pages=236–239 |year=2017 |issn=1995-0721 | doi=10.1134/S0202289317030069 |bibcode=2017GrCo...23..236H|s2cid=125980981}}</ref>

===Economics and finance===
[[Economics]] is closely enough linked to optimization of [[agent (economics)|agents]] that an influential definition relatedly describes economics ''qua'' science as the "study of human behavior as a relationship between ends and [[scarce]] means" with alternative uses.<ref>[[Lionel Robbins]] (1935, 2nd ed.) ''[[An Essay on the Nature and Significance of Economic Science#Major propositions|An Essay on the Nature and Significance of Economic Science]]'', Macmillan, p. 16.</ref> Modern optimization theory includes traditional optimization theory but also overlaps with [[game theory]] and the study of economic [[equilibrium (economics)|equilibria]]. The ''[[Journal of Economic Literature]]'' [[JEL classification codes|codes]] classify mathematical programming, optimization techniques, and related topics under [[JEL classification codes#Mathematical and quantitative methods JEL: C Subcategories|JEL:C61-C63]].

In microeconomics, the [[utility maximization problem]] and its [[dual problem]], the [[expenditure minimization problem]], are economic optimization problems. Insofar as they behave consistently, [[consumer]]s are assumed to maximize their [[utility]], while [[firm]]s are usually assumed to maximize their [[Profit (economics)|profit]]. Also, agents are often modeled as being [[Risk aversion|risk-averse]], thereby preferring to avoid risk. [[Asset pricing|Asset prices]] are also modeled using optimization theory, though the underlying mathematics relies on optimizing [[stochastic process]]es rather than on static optimization. [[International trade theory]] also uses optimization to explain trade patterns between nations. The optimization of [[Portfolio (finance)|portfolios]] is an example of multi-objective optimization in economics.

Since the 1970s, economists have modeled dynamic decisions over time using [[control theory]].<ref>{{cite journal |first=Robert |last=Dorfman |author-link=Robert Dorfman |title=An Economic Interpretation of Optimal Control Theory |journal=[[American Economic Review]] |volume=59 |issue=5 |year=1969 |pages=817–831 |jstor=1810679 }}</ref> For example, dynamic [[search theory|search models]] are used to study [[labor economics|labor-market behavior]].<ref>{{cite book |first=Thomas J. |last=Sargent |author-link=Thomas J. Sargent |chapter=Search |title=Dynamic Macroeconomic Theory |publisher=Harvard University Press |year=1987 |pages=57–91 |isbn= 9780674043084|chapter-url=https://books.google.com/books?id=nVuyXF8ibeIC&pg=PA57 }}</ref> A crucial distinction is between deterministic and stochastic models.<ref>A.G. Malliaris (2008). "stochastic optimal control," ''The New Palgrave Dictionary of Economics'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_S000269&edition=&field=keyword&q=Taylor's%20th&topicid=&result_number=1 Abstract] {{Webarchive|url=https://web.archive.org/web/20171018182459/http://www.dictionaryofeconomics.com/article?id=pde2008_S000269&edition=&field=keyword&q=Taylor's%20th&topicid=&result_number=1 |date=2017-10-18 }}.</ref> [[Macroeconomics|Macroeconomists]] build [[dynamic stochastic general equilibrium|dynamic stochastic general equilibrium (DSGE)]] models that describe the dynamics of the whole economy as the result of the interdependent optimizing decisions of workers, consumers, investors, and governments.<ref>{{cite journal |first1=Julio |last1=Rotemberg |author-link=Julio Rotemberg |author-link2=Michael Woodford (economist) |first2=Michael |last2=Woodford |year=1997 |title=An Optimization-based Econometric Framework for the Evaluation of Monetary Policy |journal=NBER Macroeconomics Annual |volume=12 |pages=297–346 |doi=10.2307/3585236 |jstor=3585236 |url=http://www.nber.org/chapters/c11041.pdf |doi-access=free }}</ref><ref>From ''[[The New Palgrave Dictionary of Economics]]'' (2008), 2nd Edition with Abstract links:<br />• "[http://www.dictionaryofeconomics.com/article?id=pde2008_N000148&edition=current&q=optimization&topicid=&result_number=1 numerical optimization methods in economics]" by Karl Schmedders<br />• "[http://www.dictionaryofeconomics.com/article?id=pde2008_C000348&edition=current&q=optimization&topicid=&result_number=4 convex programming]" by [[Lawrence E. Blume]]<br />• "[http://www.dictionaryofeconomics.com/article?id=pde2008_A000133&edition=current&q=optimization&topicid=&result_number=20 Arrow–Debreu model of general equilibrium]" by [[John Geanakoplos]].</ref>

===Electrical engineering===
Some common applications of optimization techniques in [[electrical engineering]] include [[active filter]] design,<ref>{{Cite journal|last1=De|first1=Bishnu Prasad|last2=Kar|first2=R.|last3=Mandal|first3=D.|last4=Ghoshal|first4=S.P.|date=2014-09-27|title=Optimal selection of components value for analog active filter design using simplex particle swarm optimization|journal=International Journal of Machine Learning and Cybernetics|volume=6|issue=4|pages=621–636|doi=10.1007/s13042-014-0299-0|s2cid=13071135|issn=1868-8071}}</ref> stray field reduction in superconducting magnetic energy storage systems, [[space mapping]] design of [[microwave]] structures,<ref>{{cite journal |last1=Koziel |first1=Slawomir |last2=Bandler |first2=John W. |title=Space Mapping With Multiple Coarse Models for Optimization of Microwave Components |journal=IEEE Microwave and Wireless Components Letters |date=January 2008 |volume=18 |issue=1 |pages=1–3 |doi=10.1109/LMWC.2007.911969|citeseerx=10.1.1.147.5407 |s2cid=11086218 }}</ref> handset antennas,<ref>{{cite journal |last1=Tu |first1=Sheng |last2=Cheng |first2=Qingsha S. |last3=Zhang |first3=Yifan |last4=Bandler |first4=John W. |last5=Nikolova |first5=Natalia K. |title=Space Mapping Optimization of Handset Antennas Exploiting Thin-Wire Models |journal=IEEE Transactions on Antennas and Propagation |date=July 2013 |volume=61 |issue=7 |pages=3797–3807 |doi=10.1109/TAP.2013.2254695|bibcode=2013ITAP...61.3797T |doi-access=free }}</ref><ref>N. Friedrich, [http://mwrf.com/software/space-mapping-outpaces-em-optimization-handset-antenna-design “Space mapping outpaces EM optimization in handset-antenna design,”] microwaves&rf, August 30, 2013.</ref><ref>{{cite journal |last1=Cervantes-González |first1=Juan C. |last2=Rayas-Sánchez |first2=José E. |last3=López |first3=Carlos A. |last4=Camacho-Pérez |first4=José R. |last5=Brito-Brito |first5=Zabdiel |last6=Chávez-Hurtado |first6=José L. |title=Space mapping optimization of handset antennas considering EM effects of mobile phone components and human body |journal=[[International Journal of RF and Microwave Computer-Aided Engineering]] |date=February 2016 |volume=26 |issue=2 |pages=121–128 |doi=10.1002/mmce.20945|s2cid=110195165 |doi-access=free }}</ref> electromagnetics-based design. Electromagnetically validated design optimization of microwave components and antennas has made extensive use of an appropriate physics-based or empirical [[surrogate model]] and [[space mapping]] methodologies since the discovery of [[space mapping]] in 1993.<ref>{{cite journal |last1=Bandler |first1=J.W. |last2=Biernacki |first2=R.M. |last3=Chen |first3=Shao Hua |last4=Grobelny |first4=P.A. |last5=Hemmers |first5=R.H. |title=Space mapping technique for electromagnetic optimization |journal=IEEE Transactions on Microwave Theory and Techniques |date=1994 |volume=42 |issue=12 |pages=2536–2544 |doi=10.1109/22.339794|bibcode=1994ITMTT..42.2536B }}</ref><ref>{{cite journal |last1=Bandler |first1=J.W. |last2=Biernacki |first2=R.M. |author3=Shao Hua Chen |last4=Hemmers |first4=R.H. |last5=Madsen |first5=K. |title=Electromagnetic optimization exploiting aggressive space mapping |journal=IEEE Transactions on Microwave Theory and Techniques |date=1995 |volume=43 |issue=12 |pages=2874–2882 |doi=10.1109/22.475649|bibcode=1995ITMTT..43.2874B }}</ref> Optimization techniques are also used in [[power-flow analysis]].<ref>{{cite conference|title=Convex relaxation of optimal power flow: A tutorial|conference=2013 iREP Symposium on Bulk Power System Dynamics and Control |url=https://ieeexplore.ieee.org/document/6629391|doi=10.1109/IREP.2013.6629391}}</ref>

===Civil engineering===
Optimization has been widely used in civil engineering. [[Construction management]] and [[transportation engineering]] are among the main branches of civil engineering that heavily rely on optimization. The most common civil engineering problems that are solved by optimization are cut and fill of roads, life-cycle analysis of structures and infrastructures,<ref>{{cite journal |last1=Piryonesi |first1=Sayed Madeh |last2=Tavakolan |first2=Mehdi |title=A mathematical programming model for solving cost-safety optimization (CSO) problems in the maintenance of structures |journal=KSCE Journal of Civil Engineering |date=9 January 2017 |volume=21 |issue=6 |pages=2226–2234 |doi=10.1007/s12205-017-0531-z|bibcode=2017KSJCE..21.2226P |s2cid=113616284 |doi-access=free }}</ref> [[resource leveling]],<ref>{{cite journal |last1=Hegazy |first1=Tarek |title=Optimization of Resource Allocation and Leveling Using Genetic Algorithms |journal=Journal of Construction Engineering and Management |date=June 1999 |volume=125 |issue=3 |pages=167–175 |doi=10.1061/(ASCE)0733-9364(1999)125:3(167) |doi-access=}}</ref><ref name=":0">{{Cite journal|title=Piryonesi, S. M., Nasseri, M., & Ramezani, A. (2018). Resource leveling in construction projects with activity splitting and resource constraints: a simulated annealing optimization. |journal=Canadian Journal of Civil Engineering |date=9 July 2018 |volume=46 |pages=81–86|doi=10.1139/cjce-2017-0670|hdl=1807/93364|hdl-access=free|last1=Piryonesi |first1=S. Madeh |last2=Nasseri |first2=Mehran |last3=Ramezani |first3=Abdollah |s2cid=116480238 }}</ref> [[Hydrological optimization|water resource allocation]], [[traffic]] management<ref>{{Cite journal|last1=Herty|first1=M.|last2=Klar|first2=A.|date=2003-01-01|title=Modeling, Simulation, and Optimization of Traffic Flow Networks|url=https://epubs.siam.org/doi/10.1137/S106482750241459X|journal=SIAM Journal on Scientific Computing|volume=25|issue=3|pages=1066–1087|doi=10.1137/S106482750241459X|bibcode=2003SJSC...25.1066H |issn=1064-8275}}</ref> and schedule optimization.

===Operations research===
Another field that uses optimization techniques extensively is [[operations research]].<ref>{{cite web|title=New force on the political scene: the Seophonisten |url=http://www.seophonist-wahl.de/ |access-date=14 September 2013 |url-status=dead |archive-url=https://web.archive.org/web/20141218090504/http://www.seophonist-wahl.de/ |archive-date=18 December 2014 }}</ref> Operations research also uses stochastic modeling and simulation to support improved decision-making. Increasingly, operations research uses [[stochastic programming]] to model dynamic decisions that adapt to events; such problems can be solved with large-scale optimization and [[stochastic optimization]] methods.

===Control engineering===
Mathematical optimization is used in much modern controller design. High-level controllers such as [[model predictive control]] (MPC) or real-time optimization (RTO) employ mathematical optimization. These algorithms run online and repeatedly determine values for decision variables, such as choke openings in a process plant, by iteratively solving a mathematical optimization problem including constraints and a model of the system to be controlled.

===Geophysics===
Optimization techniques are regularly used in [[geophysics|geophysical]] parameter estimation problems. Given a set of geophysical measurements, e.g. [[seismology|seismic recordings]], it is common to solve for the [[mineral physics|physical properties]] and [[structure of the earth|geometrical shapes]] of the underlying rocks and fluids. The majority of problems in geophysics are nonlinear with both deterministic and stochastic methods being widely used.

===Molecular modeling===
{{main|Molecular modeling}}
Nonlinear optimization methods are widely used in [[conformational analysis]].

===Computational systems biology===
{{main|Computational systems biology}}
Optimization techniques are used in many facets of computational systems biology such as model building, optimal experimental design, metabolic engineering, and synthetic biology.<ref name="Papoutsakis 1984">{{Cite journal|last=Papoutsakis|first=Eleftherios Terry|date=February 1984|title=Equations and calculations for fermentations of butyric acid bacteria|journal=Biotechnology and Bioengineering|volume=26|issue=2|pages=174–187|doi=10.1002/bit.260260210|pmid=18551704|s2cid=25023799|issn=0006-3592}}</ref> [[Linear programming]] has been applied to calculate the maximal possible yields of fermentation products,<ref name="Papoutsakis 1984" /> and to infer gene regulatory networks from multiple microarray datasets<ref>{{Cite journal|last1=Wang|first1=Yong|last2=Joshi|first2=Trupti|last3=Zhang|first3=Xiang-Sun|last4=Xu|first4=Dong|last5=Chen|first5=Luonan|date=2006-07-24|title=Inferring gene regulatory networks from multiple microarray datasets |journal=Bioinformatics|language=en|volume=22|issue=19|pages=2413–2420|doi=10.1093/bioinformatics/btl396|pmid=16864593|issn=1460-2059|doi-access=}}</ref> as well as transcriptional regulatory networks from high-throughput data.<ref>{{Cite journal|last1=Wang|first1=Rui-Sheng|last2=Wang|first2=Yong|last3=Zhang|first3=Xiang-Sun|last4=Chen|first4=Luonan|date=2007-09-22|title=Inferring transcriptional regulatory networks from high-throughput data|journal=Bioinformatics|volume=23|issue=22|pages=3056–3064|doi=10.1093/bioinformatics/btm465|pmid=17890736|issn=1460-2059|doi-access=free}}</ref> [[Nonlinear programming]] has been used to analyze energy metabolism<ref>{{Cite journal|last1=Vo|first1=Thuy D.|last2=Paul Lee|first2=W.N.|last3=Palsson|first3=Bernhard O.|date=May 2007|title=Systems analysis of energy metabolism elucidates the affected respiratory chain complex in Leigh's syndrome|journal=Molecular Genetics and Metabolism|volume=91|issue=1|pages=15–22|doi=10.1016/j.ymgme.2007.01.012|pmid=17336115|issn=1096-7192}}</ref> and has been applied to metabolic engineering and parameter estimation in biochemical pathways.<ref>{{Cite journal|last1=Mendes|first1=P.|author-link1 = Pedro Pedrosa Mendes|last2=Kell|first2=D.|date=1998|title=Non-linear optimization of biochemical pathways: applications to metabolic engineering and parameter estimation|journal=Bioinformatics|volume=14|issue=10|pages=869–883|issn=1367-4803|pmid=9927716|doi=10.1093/bioinformatics/14.10.869|doi-access=free}}</ref>

===Machine learning===
{{main|Machine learning#Optimization}}