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University College London

University College London (branded as UCL  ) is a public research university in London, England. It is a member institution of the federal University of London, and is the second-largest university in the United Kingdom by total enrolment and the largest by postgraduate enrolment.

Established in 1826 as London University (though without university degree-awarding powers) by founders inspired by the radical ideas of Jeremy Bentham, UCL was the first university institution to be established in London, and the first in England to be entirely secular and to admit students regardless of their religion. It was also among the first university colleges to admit women alongside men in 1878, two years after University College, Bristol. Intended by its founders to be England's third university, politics forced it to accept the status of a college in 1836, when it received a royal charter and became one of the two founding colleges of the University of London, although it achieved de facto recognition as a university in the 1990s and formal university status in 2023. It has grown through mergers, including with the Institute of Ophthalmology (in 1995), the Institute of Neurology (in 1997), the Royal Free Hospital Medical School (in 1998), the Eastman Dental Institute (in 1999), the School of Slavonic and East European Studies (in 1999), the School of Pharmacy (in 2012) and the Institute of Education (in 2014).

UCL has its main campus in the Bloomsbury area of central London, with a number of institutes and teaching hospitals elsewhere in central London and has a second campus, UCL East, at Queen Elizabeth Olympic Park in Stratford, East London. UCL is organised into 11 constituent faculties, within which there are over 100 departments, institutes and research centres. UCL operates several museums and collections in a wide range of fields, including the Petrie Museum of Egyptian Archaeology and the Grant Museum of Zoology and Comparative Anatomy, and administers the annual Orwell Prize in political writing. In 2022/23, UCL had a total income of £1.93 billion, of which £527 million was from research grants and contracts. The university generates around £10 billion annually for the UK economy, primarily through the spread of its research and knowledge (£4 billion) and the impact of its own spending (£3 billion).

UCL is a member of numerous academic organisations, including the Russell Group and the League of European Research Universities, and is part of UCL Partners, the world's largest academic health science centre. It is considered part of the "golden triangle" of research-intensive universities in southeast England. UCL has publishing and commercial activities including UCL Press, UCL Business and UCL Consultants.

UCL has many notable alumni, including the founder of Mauritius, the first Prime Minister of Japan, one of the co-discoverers of the structure of DNA, and the members of Coldplay. UCL academics discovered five of the naturally occurring noble gases, discovered hormones, invented the vacuum tube, and made several foundational advances in modern statistics. , 30 Nobel Prize winners and three Fields medallists have been affiliated with UCL as alumni or academic staff.

1826 to 1836 – London University
UCL was founded on 11 February 1826 as an alternative to the Anglican universities of Oxford and Cambridge. It took the form of a joint stock company, with shares sold for £100 to proprietors, under the name of London University, although without legal recognition as a university or the associated right to award degrees. London University's first warden was Leonard Horner, who was the first scientist to head a British university.

Despite the commonly held belief that the philosopher Jeremy Bentham was the founder of UCL, his direct involvement was limited to the purchase of share No. 633, at a cost of £100 paid in nine instalments between December 1826 and January 1830. In 1828, he did nominate a friend to sit on the council, and in 1827, attempted to have his disciple John Bowring appointed as the first professor of English or History, but on both occasions his candidates were unsuccessful. However, Bentham is commonly regarded as the "spiritual father" of UCL, as his ideas on education and society were influential with the institution's founders, particularly James Mill (1773–1836) and Henry Brougham (1778–1868).

In 1828, the chair of political economy at London University was created, with John Ramsay McCulloch as the first incumbent. In 1829, the university appointed the first professor of English in England, although the course concentrated on linguistics and the modern teaching of English – studying English literature – was introduced by King's College London in 1831. In 1830, London University founded the London University School, which would later become University College School. In 1833, the university appointed Alexander Maconochie, secretary to the Royal Geographical Society, as the first professor of geography in Britain. Classes in medicine began at the opening of the college in 1828, and in 1834 University College Hospital (originally North London Hospital) opened as a teaching hospital for these classes, which were organised into a faculty of medicine in 1836.

1836 to 1900 – University College, London
After almost a decade of attempting to win recognition as a university and the right to award degrees, including an Address to the Crown from the House of Commons, the proprietors of London University accepted the government's proposal to establish the University of London as an independent examining body, accepting the status of a college for their institution. As a result, the proprietors of London University were incorporated by royal charter under the name University College, London on 28 November 1836. On the same day, the University of London was created by royal charter as a degree-awarding examining board for students from affiliated schools and colleges, with University College and King's College, London being named in the charter as the first two affiliates. The first students from UCL and King's matriculated as undergraduates in 1838 and the first degrees were awarded to students of the two colleges in 1839.

There had been an intention to establish a course in engineering at the college's opening but no professor was appointed until 1840 or 1841, after engineering courses had started at Durham University (1837) and King's College London (1838). The Slade School of Fine Art was founded as part of University College in 1871, following a bequest from Felix Slade.

In 1878, the University of London gained a supplemental charter making it the first British university to be allowed to award degrees to women. The same year, UCL admitted women to the faculties of Arts and Law and of Science, although women remained barred from the faculties of Engineering and of Medicine (with the exception of courses on public health and hygiene). UCL's admission of women in 1878 came almost three decades after Bedford College became the first institution to offer university-level education for women in Britain, and the establishment of the University of London's General Examination for Women in 1868.

The Ladies' Educational Association held classes for women from 1868, taught by professors from UCL but independently of the college. From 1871 to 1872 these were held inside the college building, although still independently of the college. From 1872, some professors, particularly Edward Poynter of the Slade, started to admit women to their classes. The full opening on the faculties of arts, science and law in 1878 came two years after the admission of women alongside men at the University of Bristol from its foundation (as University College Bristol) in 1876. The first woman to officially enrol in architecture at UCL was Gertrude Leverkus in 1915, although Ethel and Bessie Charles had been allowed to audit classes in the 1890s. Women were finally admitted to medical studies during the First World War in 1917, although limitations were placed on their numbers after the war ended.

A new royal charter granted to the University of London in 1858 effectively removed the affiliation of colleges to the university. Dissatisfaction from the colleges and the desire for a "teaching university" in London led to royal commissions that reported in 1888 and 1892 and the reconstitution of the university under the University of London Act 1898.

1900 to 1976 – University of London, University College
Following the University of London's reconstitution in 1909, transforming it from an examining board to a federal university with constituent "schools", UCL, became a school of the University of London. While most of the colleges that became schools of the university retained their autonomy, UCL chose to be merged into the university in 1907 under the University College London (Transfer) Act 1905 (5 Edw. 7. c. xci) and surrendered its legal independence in return for gaining a greater say in the running of the university. Its formal name became University of London, University College, although for most informal and external purposes the name "University College, London" (or the initialism UCL) was still used. it remains listed as "University of London: University College" on US Federal Student Aid applications.

1900 also saw the decision to appoint a salaried head of the college. The first incumbent was Carey Foster, who served as Principal (as the post was originally titled) from 1900 to 1904. He was succeeded by Gregory Foster (no relation), and in 1906 the title was changed to Provost to avoid confusion with the principal of the University of London. Gregory Foster remained in post until 1929. In 1906, the Cruciform Building was opened as the new home for University College Hospital. UCL opened the first department and chair of chemical engineering in the UK, funded by the Ramsay Memorial Fund, in 1923.

In 1904, Francis Galton donated £1,000 to the University of London for a eugenics laboratory; this transferred to UCL in 1907 with Karl Pearson as its director. UCL apologised for its "fundamental role in the development, propagation and legitimisation of eugenics" in 2021.

In 1911, UCL received an anonymous donation of £30,000 for a building for its school of architecture. In 1919 the donor consented to being named as Herbert Bartlett and the school was renamed in his honour.

UCL sustained considerable bomb damage during the Second World War, including the complete destruction of the Great Hall, the Carey Foster Physics Laboratory and the Ramsay Laboratory. Fires gutted the library and destroyed much of the main building, including the dome; it was not until 1954 that the main building was fully restored. The departments were dispersed across the country to Aberystwyth, Bangor, Gwynedd, Cambridge, Oxford, Rothamsted near Harpenden, Hertfordshire and Sheffield, with the administration at Stanstead Bury near Ware, Hertfordshire. The first UCL student newspaper, Pi, was founded in 1946. The Institute of Jewish Studies relocated from Manchester to UCL in 1959. The Mullard Space Science Laboratory was established in 1967.

In 1973, Peter Kirstein's research group at UCL became one of only two international nodes on the ARPANET, later becoming part of SATNET. UCL's interconnection between the ARPANET and early British academic networks was the first international heterogeneous resource sharing network. UCL played a significant role in the very earliest experimental Internet work and adopted TCP/IP in 1982, ahead of the ARPANET.

The college's senior common room, the Housman Room, remained men-only until 1969. After two unsuccessful attempts, a motion was passed that ended segregation by sex at UCL. This was achieved by Brian Woledge (Fielden Professor of French at UCL from 1939 to 1971) and David Colquhoun, at that time a young lecturer in pharmacology.

1976 to 2005 – University College London
In 1976, a new charter restored UCL's legal independence, although still without the power to award its own degrees. Under this charter the college became formally known as University College London. This name abandoned the comma used in its earlier name of University College, London.

In 1993, a reorganisation of the University of London meant that UCL and other colleges gained direct access to government funding and the right to confer University of London degrees themselves. This led to UCL being regarded as a de facto university in its own right.

Mergers were a major feature of this period of UCL's history. In 1986, the college merged with the Institute of Archaeology. In 1988, UCL merged with the Institute of Laryngology & Otology, the Institute of Orthopaedics, the Institute of Urology & Nephrology and Middlesex Hospital Medical School. Middlesex and University College hospitals, together with the Elizabeth Garrett Anderson Hospital and the Hospital for Tropical Diseases, formed the University College London Hospitals NHS Trust in 1994.

Mergers continued in the 1990s, with the Institute of Child Health joining in 1995, the School of Podiatry in 1996 and the Institute of Neurology in 1997. In 1998, UCL merged with the Royal Free Hospital Medical School to create the Royal Free and University College Medical School (renamed the UCL Medical School in October 2008). In 1999, UCL merged with the School of Slavonic and East European Studies and the Eastman Dental Institute.

Proposals for a merger between UCL and Imperial College London were announced in 2002. The proposal provoked strong opposition from UCL teaching staff and students and the AUT union, which criticised "the indecent haste and lack of consultation", leading to its abandonment by UCL provost Sir Derek Roberts.

From 2005
UCL was granted its own taught and research degree awarding powers in 2005, and all UCL students registered from 2007/08 qualified with UCL degrees. The same year, UCL adopted a new corporate branding under which the name University College London was replaced by the initialism UCL in all external communications.

UCL established the UCL School of Energy & Resources (later UCL Australia) in Adelaide, Australia, in 2008 as the first campus of a British university in the country. The school was based in the historic Torrens Building in Victoria Square. In 2011, the mining company BHP Billiton agreed to donate AU$10 million to UCL to fund the establishment of two energy institutes – the Energy Policy Institute, based in Adelaide, and the Institute for Sustainable Resources, based in London. UCL Australia closed in December 2017, with academic staff and student transferring to the University of South Australia. Since 2018, UCL and the University of South Australia have offered joint master's degrees in data science and in energy systems with study in Adelaide and London.

In 2011, UCL announced plans for a £500 million investment in its main Bloomsbury campus over 10 years, as well as the establishment of a new campus, UCL East, next to the Olympic Park in Stratford in the East End of London. In 2018, UCL opened UCL at Here East, at the Queen Elizabeth Olympic Park, offering courses jointly between the Bartlett Faculty of the Built Environment and the Faculty of Engineering Sciences. The first undergraduate students, on a new Engineering and Architectural Design MEng, started in September 2018. One Pool Street, the first building on the UCL East campus, opened in November 2022. UCL East was officially opened, along with the Marshgate building that completed phase 1 of the development, in September 2023 by Olympic gold medalist and UCL alumna Christine Ohuruogu.

UCL continued to grow through mergers with smaller colleges in the University of London. On 1 January 2012 the School of Pharmacy, University of London merged with UCL, becoming the UCL School of Pharmacy within the Faculty of Life Sciences. UCL and the Institute of Education formed a strategic alliance in October 2012, followed by a full merger in December 2014.

UCL paid tens of thousands of pounds to settle ten sexual harassment claims against staff in the 2017/18 academic year, a rise from four cases the year before. Following pressure from victims, and after physicist Emma Chapman won the legal right to speak freely about her abuse at the university, UCL announced in 2018 that it would abandon non-disclosure settlements in settlements. In 2020, UCL became the first Russell Group university to ban romantic and sexual relationships between lecturers and their students.

It was discovered in 2018 that an annual eugenics conference, the London Conference on Intelligence, had been held at UCL, as an external paid event, between 2014 and 2017. An enquiry found that the organiser, an honorary lecturer, did not correctly follow the room booking procedure, including claiming that no controversial topics would be discussed, leaving the university unaware of the nature of the conference. Following the revelation, UCL announced in 2018 that it would launch an enquiry into the university's historical links with eugenics. This reported in 2020, but covered only historical eugenics and did not address the 2014–17 conferences, leading to a majority of the authors refusing to sign the final report. The Galton Lecture Theatre, Pearson Lecture Theatre and Pearson Building were all renamed in 2020, and in 2021 UCL apologised for its part in promoting eugenics during the first half of the 20th century. UCL was criticised (along with Oxford, Imperial and other London universities) in 2021 for accepting money from the Alexander Mosley Charitable Trust, established to hold the fortune left to Max Mosley by his father, British fascist leader Oswald Mosley. UCL received £500,000 to establish a forensic evidence interpretation laboratory.

Following the passing of the University of London Act 2018, which allowed member institutions to become universities in their own right while remaining part of the University of London, UCL applied for university status in 2019. The application was approved by the Office for Students in 2022 and a supplemental charter was sealed on 17 April 2023, granting UCL university status.

Bloomsbury
UCL is primarily based in the Bloomsbury area of the London Borough of Camden, in Central London. The main campus is located around Gower Street, with many other departments close by in Bloomsbury. Many health institutes are located close to associated hospitals, including the UCL Queen Square Institute of Neurology and the National Hospital for Neurology and Neurosurgery in Queen Square, the UCL Great Ormond Street Institute of Child Health and Great Ormond Street Hospital for Children, and the UCL Eastman Dental Institute and Eastman Dental Hospital.

Historic UCL buildings in Bloomsbury include the grade I listed UCL Main Building, including the original Wilkins building designed by William Wilkins, and, directly opposite on Gower Street, the early 20th century grade II listed Cruciform Building, the last major building designed by Alfred Waterhouse. Nearby are the grade II listed Kathleen Lonsdale Building, UCL's first purpose-built chemistry laboratory, and the grade II listed Rockefeller Building. Elsewhere in Bloomsbury is the 1970s grade II* Institute of Education building by Denys Lasdun and Partners. Much of the estate falls within the Bloomsbury Conservation Area, designated in 1968. Important contemporary buildings include the School of Slavonic and East European Studies building (RIBA Award winner 2006) and the Sainsbury Wellcome Centre for Neural Circuits and Behaviour building (LEAF Award for best façade design and engineering and overall winner 2016).

UCL East
UCL has a second campus, UCL East, at the Queen Elizabeth Olympic Park in Stratford, east London. The first building, with three floors of teaching and research space as well as accommodation for 500 students in two towers, opened in 2022, and the second, with eight floors of teaching and research space, opened in 2023. A further four buildings are planned for construction in the 2030s. UCL also operates a campus within Here East, the former Olympic park media centre.

Other sites

Elsewhere in Central London are the UCL Institute of Ophthalmology adjacent to Moorfields Eye Hospital in Clerkenwell, the Royal Free Hospital and the Whittington Hospital campuses of the UCL Medical School, and a number of other associated teaching hospitals. The UCL School of Management is on levels 38 and 50 (penthouse) of One Canada Square in the financial district of Canary Wharf. The UCL Observatory is in Mill Hill and the Mullard Space Science Laboratory is based in Holmbury St Mary, Surrey. The UCL Athletics Ground is in Shenley, Hertfordshire.

Student housing
UCL owns 26 halls of residence with around 7,000 student beds. The university guarantees accommodation to single full-time first-year undergraduate students who have not previously lived in London while studying at a university, and who make a firm acceptance of a place and apply for accommodation by 10 June each year, and to single overseas first-year postgraduates at UCL who have not previously lived in London while studying at a university, and who make a firm acceptance of a place and apply for accommodation by 30 June each year. Accommodation is also guaranteed for students who are under 18 at the start of the academic year and for students who are care-leavers. There is only limited accommodation available in university halls for returning students and others who do not meet the criteria for a guaranteed place. UCL students are also eligible, as students of a member institution of the University of London, to apply for places in the University of London intercollegiate halls of residence.

In 2013, UCL's newly built New Hall student accommodation building on Caledonian Road, designed by Stephen George and Partners, was awarded the Carbuncle Cup and named the country's worst new building by Building Design magazine, with the comment "this is a building that the jury struggled to see as remotely fit for human occupation". Islington Council had originally turned down planning permission for the building, but this had been overturned on appeal. As it is classified as a hotel or guest house, it was exempt from many of the standards that cover residential buildings, such as having daylight in the rooms.

The UCL East development includes 532 student rooms in One Pool Street, which opened in 2022. Further accommodation will be available in the Marshgate building, expected to open in 2023, and at the second Pool Street site.

Environmental initiatives
UCL's new Student Centre, which opened in 2019, was designed to be environmentally friendly and was one of only 320 buildings worldwide (at the time) to be certified outstanding by BREEAM. This certification requires innovation throughout the design, engineering and construction process, and places the Student Centre among the top 1% of non-domestic buildings in the UK for sustainability. The UCL Student Centre was a finalist at the Green Gown Awards in 2019.

Also in 2019, UCL launched a Strategy for Sustainable UCL 2019–24, including three initiatives to promote sustainability. The Positive Climate initiative saw UCL pledge to have a 40% reduction in energy usage, all energy to come from renewables, and all UCL buildings to be carbon neutral by 2024, along with achieving net zero carbon emissions for UCL by 2030. The Positive Climate initiative was the winner in the "2030 Climate Action" category at the 2020 Green Gown Awards.

A second initiative, The Loop, promotes circular economy. UCL set a target of reducing waste per person by 20% between 2019 and 2024, while aiming for an 85% recycling rate and the elimination of single-use plastics on campus. The third initiative, Wild Bloomsbury, promotes biodiversity. UCL set a target of creating 10000 m2 of biodiverse green space on campus by 2024. The Strategy for a Sustainable UCL was a finalist in the "Sustainable Institution of the Year" category at the 2022 Green Gown Awards.

Governance
The two main bodies in UCL's governance structure are the council and the academic board, both of which are established by the royal charter and with powers defined by the statutes. There is also a University Management Committee, which is the executive committee responsible for the day-to-day operations of the institution. This comprises the President and Provost, the vice-presidents, the vice-provosts, the pro-provost of UCL East, the deans of the faculties, the chief financial officer, chief information officer, and chief people officer, the chief of staff, the general consul, the executive director of media and marketing, and the director of media relations.

The senior leadership team at UCL includes the visitor, a position in English charity law that overseas the operation of the institution. That there shall be a visitor of the college is specified by the royal charter, as is that the position is to be held by the Master of the Rolls, the second most senior judge in England and Wales.

UCL's council comprises 20 members, of whom 11 are members external to UCL; seven are UCL academic staff, including the provost, three UCL professors and three non-professorial staff; and two are UCL students. The chair is appointed by council for a term not normally exceeding five years. The chair is ex officio chair of the honorary degrees and fellowships committee, nominations committee and remuneration and strategy committee. the chair of the council is international businessman and UCL alumnus Victor Chu.

The academic board plays a role similar to the senate in other institutions. It is the senior academic body responsible for advising council on academic matters and also elects academic members to council. It is, however, a much larger body than the senates at many other universities, including all professors as well as elected representatives of other academic and non-academic staff.

UCL's principal academic and administrative officer is the President and Provost, who is also UCL's designated accountable officer for reporting to the Office for Students on behalf of UCL. The provost is appointed by Council after consultation with the academic board, and is ex officio a member of council and chair of the academic board. Michael Spence has been president and provost since January 2021, when he succeeded Michael Arthur.

Vice-provosts are appointed by council on the recommendation of the provost or the academic board, to assist and advise the provost as required. The vice-provosts are members of the provost's senior management team. There are four vice-provosts (for education and student experience; health; research, innovation and global engagement; and faculties). There are also four vice-presidents, who are also members of the senior management team but whose role and manner of appointment is not specified in the statutes, for strategy, external engagement, advancement and operations.

The deans of UCL's faculties are appointed by the council and are members of the provost's senior management team. The deans' principal duties include advising the provost and vice-provosts on academic strategy, staffing matters and resources for academic departments within their faculty; overseeing curricula and programme management at faculty level; liaising with faculty tutors on undergraduate admissions and student academic matters; overseeing examination matters at faculty level; and co-ordinating faculty views on matters relating to education and information support.

Faculties and departments

UCL's research and teaching is organised into eleven faculties, each of which contains a number of schools, departments and institutes. The establishment of faculties and academic departments is formally the responsibility of UCL's council, with advice from the academic board.

There are also two academic units outside the faculty structure:
 * Sainsbury Wellcome Centre for Neural Circuits and Behaviour
 * School of Slavonic and East European Studies (to join the Arts and Humanities faculty from 2023): 193 staff; 559 undergraduate students; 57 taught postgraduate students; 29 research postgraduate students (2022/23)

There are additional staff employed outside the faculty structure in the university administration.

Finances
In the financial year ended 31 July 2020, UCL had a total income (excluding share of joint ventures) of £1.54 billion (2018/19 – £1.49 billion) and a total expenditure of £1.34 billion (2018/19 – £1.67 billion). Key sources of income included £467.7 million from research grants and contracts (2018/19 – £481.1 million), £613.7 million from tuition fees and education contracts (2018/19 – £564.9 million), £227.9 million from funding body grants (2018/19 – £213.5 million) and £26.6 million from donations and endowments (2018/19 – £40.5 million). At year end UCL had endowments of £143.2 million (31 July 2019 – £138.7 million) and total net assets of £1.49 billion (31 July 2019 – £1.29 million).

A report by London Economics in 2022 found that UCL generates around £10 billion annually for the UK economy. The largest contributor to this is through the spread of its research and knowledge, which is worth £4 billion, with another £3 billion being added by the impact of UCL's own spending. Other contributions come from encouraging graduates to create jobs and investment, and from nurturing company spin-offs and start-ups. The report found that in 2018–19, UCL had supported 234 graduate start-ups and 83 spinout companies, with a total turnover of £110 million and employing almost 3,000 people. The report also found that UCL's spending supported 19,000 jobs across the UK, with over 7,000 of these being outside London.

Terms
The UCL academic year is divided into three terms. For most departments, First Term runs from late September to mid December, Second Term from mid January to late March, and Third Term from late April to mid June, with reading weeks in early November and mid February. Certain courses at the medical school, the faculty of education and society, and the school of pharmacy operate on different terms.

Logo and colours
While many universities use their logo for most communications and branding and a coat of arms only for specific ceremonial and official use, UCL exclusively uses a logo and has no coat of arms. The present logo was adopted as part of a rebranding exercise in August 2005. Prior to that date, a different logo was used, in which the letters UCL were incorporated into a stylised representation of the Wilkins Building portico.

A pseudo-heraldic "UCL crest" – a purple shield depicting a raised bent arm dressed in armour between two gold laurel branches holding a green upturned open wreath, with the college motto on a blue celeste ribbon beneath the laurel branches – can be found on the internet. A version of this badge (not on a shield) appears to have been used by UCL Union from shortly after its foundation in 1893. However, the badge has never been the subject of an official grant of arms, and departs from several of the rules and conventions of heraldry. It is not an official logo, although modified forms are used by some by sports teams and societies. The official Team UCL logo, used (with variants) by many sports teams, uses a shield divided into the colours of purple (lower) and blue celeste (upper), but none of the other elements (laurels, wreaths, armoured arm, motto) are present; the only graphic is a depiction of the UCL portico. Students' Union UCL requests teams not to modify this logo, but this is widely ignored.

UCL's motto, "Cuncti adsint meritaeque expectent praemia palmae" is a quotation from Virgil's Aeneid, and translates into English as "Let all come who by merit deserve the most reward".

UCL's traditional sporting and academic colours are purple and blue celeste. UCL uses a palette of 25 colours (including the two traditional colours) in its visual identity; the logo can be used in many different combinations of these colours.

Memberships, affiliations and partnerships
UCL is a member institution of the federal University of London and was one of the two colleges affiliated from the university's founding in 1836 (the other being King's College London). UCL was a founding member of the Russell Group, an association of 24 British research universities established in 1994, and is regarded as forming part of the 'golden triangle', an unofficial term for the research-intensive universities located in the southern English cities of Cambridge, London and Oxford

UCL has been a member of the League of European Research Universities since January 2006. UCL is also a member of the Association of Commonwealth Universities, the European University Association, the global U7+ Alliance and the US Universities Research Association, and has a major collaboration with Yale University, the Yale UCL Collaborative. It also has partnerships with universities in Australia, Canada, China, India, Japan, Singapore and Thailand.

UCL formed the Science and Engineering South engineering and physical sciences research alliance with the universities of Cambridge, Oxford, Southampton and Imperial College London in May 2013. It was also one of the founding members of the Alan Turing Institute, the UK's national institute for data sciences and artificial intelligence, in 2015, with the universities of Cambridge, Edinburgh, Oxford and Warwick.

UCL is a partner in UCLPartners, an academic health science centre, along with multiple NHS trusts, integrated care systems, research and innovation partners, and other universities. UCL is a partner with the National Institute for Health Research, the University College London Hospitals NHS Foundation Trust and UCLPartners in the UCLH Biomedical Research Centre. UCL is also a university partner of the Francis Crick Institute, a major biomedical research centre in London.

UCL offers dual degrees and joint degrees with other universities and institutions, including the University of Cologne, Columbia University, the University of Hong Kong, Imperial College London (ending 2023) and New York University.

UCL is the sponsor of the UCL Academy, a secondary school in the London Borough of Camden. The school opened in September 2012 and was the first in the UK to have a university as sole sponsor. UCL also has a strategic partnership with Newham Collegiate Sixth Form Centre.

Research

In 2021/22, UCL had an income from research grants and contracts of £524.9 million, making up 30% of all revenue. The largest sources of research income were research council grants (£170.4 million) and British charities (£154.4 million). A further £159.3 million of recurrent research funding was allocated to UCL by Research England, making up 9% of income.

UCL submitted 3,432 staff (3,177 FTEs) across 32 units of assessment (areas of research) to the 2021 Research Excellence Framework (REF) assessment. 58% of submitted research was rated 4* ('world leading'), the sixth highest in the REF, and a further 34% as 3* ('internationally excellent'). Overall, UCL was ranked second for both research power and market share by both Times Higher Education and Research Professional News, and sixth on research quality (GPA) by Times Higher Education. UCL submitted more units of assessment to the 2021 REF than any other university. However, UCL's market share (based on the funding formula) declined from 6.23% following the 2014 REF to 5.34%, despite the overall improvement, reflecting increases in research quality across the sector.

Research centres
UCL operates a large number of disciplinary-specific research centres in partnership with other research institutions and private enterprises. Notable examples include:

 * The London Centre for Nanotechnology (LCN) is a multidisciplinary research centre in physical and biomedical nanotechnology based at UCL's campus in Bloomsbury. It is a partnership between UCL, Imperial College London and King's College London. The LCN was established as a joint venture between UCL and Imperial College London in 2003; King's College London joined the LCN in 2018.
 * The Centre for the Study of the Legacies of British Slave-ownership was established at UCL with the support of the Hutchins Center for African and African American Research at Harvard University. It incorporates two earlier projects: the Legacies of British Slave-ownership project (2009–2012) and the Structure and significance of British Caribbean slave-ownership 1763–1833 project (2013–2015).
 * The Sainsbury Wellcome Centre for Neural Circuits and Behaviour (SWC) is a neuroscience research centre established at UCL with funding from the Gatsby Charitable Foundation and Wellcome Trust and opened in 2016.

Publishing and commercialisation
In 2020/21, UCL had an income of £7.3 million from intellectual property and £25.2 million from the sale of shares in spin-off companies. As of 2020/21, UCL had the second largest patent portfolio of any UK university (after Oxford) with 2,391 patents. It granted the third largest number of intellectual property licences (after Oxford and the University of East Anglia), with 2,235.

UCL Business is a technology transfer company which is wholly owned by UCL. It has three main activities: licensing technologies, creating spin-out companies, and project management. UCL Business supports spin-out companies in areas including discovery disclosure, commercialisation, business plan development, contractual advice, incubation support, recruitment of management teams and identification of investors. In the area of licensing technoloiges, the company provides commercial, legal and administrative advice to help companies broker licensing agreements. UCL Business also provides UCL departments and institutes with project management services for single or multi-party collaborative industry projects. The company transferred £4.8 million of royalty income to UCL in 2021/22.

Launched in 2015, UCL Press is a new university press wholly owned by UCL. It was the first fully open access university press in the UK, and publishes monographs, textbooks and other academic books in a wide range of academic areas which are available to download for free, in addition to a number of journals. As of October 2022, UCL Press had had more than 6.5 million downloads of its open access books in 247 countries and territories worldwide. UCL Consultants is an academic consultancy services company which is wholly owned by UCL, which provides four main service offerings: Academic Consultancy, Bespoke Short Courses, Testing & Analysis and Expert Witness.

Libraries
The UCL library system comprises 18 libraries located across the Bloomsbury and UCL East campuses. The libraries contain a total of over 2 million books. The largest library is the UCL Main Library, which is located in the UCL Main Building and contains collections relating to the arts and humanities, economics, history, law and public policy. The second largest library is the UCL Science Library, which is located in the DMS Watson Building on Malet Place and contains collections relating to anthropology, engineering, geography, life sciences, management and the mathematical and physical sciences. The Cruciform Hub contains books and periodicals in the subjects of clinical medicine and medical science. It holds the combined collections of the former Boldero and Clinical Sciences libraries which developed within the Middlesex Hospital, University College Hospital and Royal Free & University College Medical Schools up until their merger in 2005.

Other libraries include the UCL Bartlett Library (architecture and town planning), the UCL Eastman Dental Institute Library (oral health sciences), the UCL Institute of Archaeology Library (archaeology and egyptology), the UCL Institute of Education's Newsam Library (education and related areas of social science), the UCL Institute of Neurology Rockefeller Medical Library (neurosurgery and neuroscience), the Joint Moorfields Eye Hospital & the UCL Institute of Ophthalmology Library (biomedicine, medicine, nursing, ophthalmology and visual science), the UCL Language & Speech Science Library (audiology, communication disorders, linguistics & phonetics, special education, speech & language therapy and voice) and the UCL School of Slavonic and East European Studies Library (the economics, geography, history, languages, literature and politics of Eastern Europe). The newest library is the UCL East Library, currently located in the Learning Hub on the first floor of One Pool Street. Uniquely among UCL libraries, it offers a 'click and collect' service allowing books from any UCL library to be delivered to UCL East rather than having to be picked up from the library that holds them. It is expected to relocate to the new Marshgate building when that opens in September 2023.

UCL staff and students have full access to the main libraries of the University of London – the Senate House Library and the libraries of the institutes of the School of Advanced Study – which are located close to the main UCL campus in Bloomsbury. These libraries contain over 3.7 million books and focus on the arts, humanities and social sciences. The British Library, which contains around 14 million books, is also located close to the main UCL campus and all UCL students and staff can apply for reference access.

UCL's open access institutional repository, UCL Discovery, and UCL Press, UCL's open access academic press are managed by UCL Library Services.

Special collections
UCL's Special Collections contains UCL's collection of historical or culturally significant works. It holds over 150,000 rare books, including 179 incunabula, as well as over 600 collections of archives and manuscripts. The incunabula include a 1477 edition of Dante's Divine Comedy, and a 1493 edition of the Nuremberg Chronicle donated by Jeremy Bentham.

UCL's most significant works are housed in three strong rooms. The special collection includes first editions of Isaac Newton's Principia, Charles Darwin's On the Origin of Species and James Joyce's Ulysses.

Museums
UCL is responsible for several museums and collections in a wide range of fields across the arts and sciences, including:

 * Petrie Museum of Egyptian Archaeology: Founded in 1892 by a donation from Amelia Edwards of several hundred Egyptian items, the museum now contains around 80,000 items and covers the history of the Nile valley from prehistoric times through to the Islamic period. It is named after William Flinders Petrie, the first Edwards Professor at UCL, who excavated dozens of sites in his career and sold his collection to the college in 1913. The Petrie Museum is a designated collection under the Arts Council England Designation Scheme for "pre-eminent collections held in museums, libraries and archives across England".
 * UCL Art Museum: the art collection originated as a teaching and research collection for the Slade, and contains works by women artists dating back to the 1890s. A series of plaster casts of full-size details of sculptures by John Flaxman is located inside the library under the dome of the UCL Main Building.
 * Grant Museum of Zoology and Comparative Anatomy: Established in 1827 by Robert Edmund Grant, UCL's first professor of comparative anatomy and zoology, for teaching purposes. Grant bequeathed his collection of 10,000 specimens to UCL upon his death. With other additions, the museum now contains around 68,000 specimens, including dodo bones and a rare quagga skeleton.

Reputation and rankings
UCL is ranked as one of the top ten universities in all three of the main UK university league tables. These place more emphasis on teaching and student experience than global rankings, using criteria such as teaching quality and learning resources, entry standards, employment prospects, research quality and dropout rates. It went through a dip in rankings in recent years, particularly in The Guardian University Guide, but returned to the top ten in 2022, when its ninth position was its best result in that table since 2014.
 * National

In the 2023 Complete University Guide subject tables, UCL was ranked in the top ten in 34 subjects out of 42 offered (81%). It was ranked top for American studies, linguistics, speech and language therapy, and building.

In the 2023 Guardian University Guide subject tables, UCL is ranked top in construction, surveying and planning. It was ranked in the top ten for 21 of 31 subjects offered (68%).

UCL is ranked top in the 2023 Times/Sunday Times Good University Guide for American studies, building, information systems and management, liberal arts, and town and country planning. It is ranked in the top ten for 31 of 44 subjects offered (70%). The 2023 Good University Guide also ranked UCL 98th in their social inclusion ranking (covering England and Wales). UCL was named University of the Year in the Times/Sunday Times Good University Guide for 2024.

Analysis by the Department for Education in 2018, found that UCL had an impact on earnings of graduates five years after graduation of +15.5% for women (7th highest impact) and +16.2% for men (10th highest impact) compared to average graduates with similar background characteristics (prior attainment, socio-economic status, etc.) and subject choice.

 * Global

UCL has been consistently ranked in the top 25 of the three major global rankings published over 2013 to 2022, including being in the top ten of the QS World University Rankings over the whole of that period.

In the 2022 Academic Ranking of World Universities, UCL was ranked 18th in the world, having been placed between 15th and 21st in the rankings from 2013 to 2022.

In the 2023 QS World University Rankings (published 2022), UCL was ranked 8th in the world. It has ranked between 4th and 10th in the 2014 to 2024 league tables.

In the Times Higher Education World University Rankings 2023 (published 2022), UCL was ranked 22nd in the world, having ranked between 14th and 22nd in the 2014 to 2023 tables. In the Times Higher Education World Reputation Rankings 2022, UCL was ranked 25th, while in the Times Higher Education Impact Rankings 2022 it was ranked 101–200.

In the 2024–2025 USNWR's Best Global Universities, UCL was ranked joint 7th in the world.

Admissions
Admission to UCL is highly selective with an average entry tariff for 2020–21 of 189 UCAS points (approximately equivalent to AAAB at A-level), the 9th highest in the country. According to a Freedom of Information request response, UCL's offer rate for 2021 admission was 36.1% at undergraduate level and 23.5% at postgraduate level across all applicants.

International students have made up the majority of main-scheme applicants to UCL since 2015 and the majority of acceptances since 2017. The ratio of main-scheme applicants to acceptances in 2022 was 10.3 for UK applicants and 9.9 for international applicants. Within the UK, UCL is a local recruiter, with 47.4% of 2022 UK admissions coming from the London region and a further 28.1% from the adjacent East of England and South East regions.

Of UCL's young UK domiciled undergraduates, 32.7% were privately educated in 2019–20, the eighth highest proportion amongst mainstream British universities.

Undergraduate law applicants are required to take the National Admissions Test for Law and undergraduate medical applicants are required to take the BioMedical Admissions Test. Applicants for European Social and Political Studies are required to take the Thinking Skills Assessment (TSA) should they be selected for an assessment day. Medicine, pharmacy and English also interview undergraduate applicants prior to making an offer of admission.

Widening access
UCL runs a contextual offer scheme called Access UCL, whereby eligible applicants can receive conditional offers for courses at UCL that have lower requirements than the standard conditional offers for those courses. Eligibility for Access UCL can be through an applicant living in a deprived areas or an area with low participation in higher education, through having spent time in care, though being a young adult carer, or through being estranged from their family. Except for applicants that have spent time in care, the scheme requires applicants to have attended a state school. Mature applicants are assessed on the same criteria, and are additionally not eligible if they have completed or are on the final year of an undergraduate degree. While the scheme enables applicants from disadvantaged backgrounds to receive contextual offers, it does not guarantee that an offer will be made. Contextual offers vary by course. For example, a contextual offer for the law LLB reduces the requirement from A*AA to BBB at A level, but for the physics MSci from A*AA to AAB.

UCL also runs week-long UCL Summer Schools for high-achieving students from disadvantaged backgrounds in partnership with the Sutton Trust. These give participants the opportunity to explore London, to develop skills in their chosen subject, to improve their university applications through personal statement workshops and talks by admissions tutors, and to take part in social activities.

From the 2023/24 academic year, UCL will be launching an engineering foundation year based at UCL East for students who do not meet the standard entry requirements, who attended UK state schools for A levels or equivalent (unless refugees), and who live in an area with high levels of at least one axis of deprivation.

For international students who do not meet the requirements for admission to the college, it runs intensive one-year foundation courses, called Undergraduate Preparatory Certificates, in either sciences or humanities.

For UK domiciled young full-time undergraduate entrants in 2020/21, 67.6% came from state schools, significantly below the location-adjusted benchmark of 74.5%, and 4.3% came from low participation neighbourhoods, not significantly different from the location-adjusted benchmark of 4.0%. For UK-domiciled undergraduate entrants in 2022/23, UCAS data shows no significant difference in offer rate with ethnicity or gender. Applicants from the 20% of neighbourhoods with the lowest rates of participation in higher education receive offers at a rate 4.9% higher than would be expected based on their subject choice and predicted grades alone, a statistically significant difference, accounting for 6.3% of all offers. The offer rate for applicants from the 20% of neighbourhoods with the lowest rates of participation in higher education is not significantly different from that expected, with applicants from those neighbourhoods accounting for 48.9% of all offers.

Students' union

Founded in 1893, Students' Union UCL, formerly the UCL Union, is one of the oldest students' unions in England, although postdating the Liverpool Guild of Students which formed a student representative council in 1892. Students' Union UCL operates both as the representative voice for UCL students, and as a provider of a wide range of services. It is democratically controlled through General Meetings and referendums, and is run by elected student officers. The union also supports a range of services, including numerous clubs and societies, sports facilities, an advice service, and a number of bars, cafes and shops.

there are over 250 clubs and societies under the umbrella of the UCL Union. These include: UCL Snowsports (one of the largest sports society at UCL, responsible for organising the annual UCL ski trip), Pi Media (responsible for Pi Magazine and Pi Newspaper, UCL's official student publications), the Debating Society (established 1829), and the UCL Union Film Society, with past members including Christopher Nolan.

Faith
From its foundation the college has been deliberately secular; the initial justification for this was that it would enable students of different Christian traditions (specifically Roman Catholics, Anglicans and Nonconformists) to study alongside each other without conflict. In order to cater to people of all faiths, UCL opened a prayer room (with attached ablution facilities) and a silent meditation room in the student centre in February 2019, and there is a quiet contemplation room behind 16–26 Gordon Square. There is also a Christian chaplain (who also serves as interfaith advisor) and there are student societies for most major religions.

Sport
The union runs over 70 sports clubs, including the UCL Cricket Club (Men's and Women's), UCL Boat Club (Men's and Women's clubs), UCL Running, Athletics and Cross Country Club, and UCL Rugby Club (Men's and Women's), as well as RUMS sports clubs for medical students (from Royal Free, University College and Middlesex, the three medical schools that merged into UCL).

UCL clubs compete in inter-university fixtures in the British Universities and Colleges Sport (BUCS) competition in a range of sports, including athletics, basketball, cricket, fencing, football, hockey, netball, rugby union and tennis. In the 2021/22 season, UCL finished in 16th position in the final BUCS rankings.

UCL sports facilities include a fitness centre at the main UCL campus in Bloomsbury and a 90 acre athletics ground in Shenley, Hertfordshire, part of which is used as the Watford Football Club Training Ground. It also exercises effective control over Somers Town Community Sports Centre, with the power to appoint five of the nine trustees. The sports centre includes a six-court sports hall, as well as an activity/dance studio and an all-weather outdoor multi-use games area.

Mascot
The UCL mascot was Phineas MacLino, or Phineas, a wooden tobacconist's sign of a kilted Jacobite Highlander stolen from outside a shop in Tottenham Court Road during the celebrations of the relief of Ladysmith, part of the Second Boer War, in March 1900. In 1922, Phineas was stolen by students from King's, marking the start of 'mascotry', leading to an hour-long battle and the eventual return of Phineas. In 1993, the students' union's centenary year, Phineas was placed in the third floor bar of 25 Gordon Street and the bar named after him. In 2019, the students' union voted to remove the mascot from the bar due to its links to imperialism and British colonialism.

Rivalry with King's College London

UCL has a long-running, mostly friendly rivalry with King's College London, but there were frequent clashes in the interwar period which have historically been known as "rags". UCL students have been referred to by students from King's as the "Godless Scum of Gower Street", in reference to a comment made at the founding of King's, which was based on Christian principles. UCL students in turn referred to King's as "Strand Polytechnic".

In 1922 Phineas, the UCL mascot was kidnapped by King's students, leading to a pitched battle in the King's College quad as UCL students recovered their mascot. Shortly after this, King's adopted their own mascot – initially a large papier-mâché beer bottle, soon replaced by Reggie the Lion. During the 1927 rag, Reggie was captured by UCL students and his body filled with rotten apples. During the same year, an attempt by King's students to capture Phineas led to the "Battle of Gower Street", caught on camera by British Pathé. On another occasion, Reggie was castrated by UCL students.

King's students stole the embalmed head of Jeremy Bentham in October 1975, only returning it after UCL paid a ransom to charity. The head is now kept in the UCL vaults.

Student campaigns
In 1956, UCL students organised a silent march progressing against the Soviet oppression of the Hungarian Revolution of 1956. Around 1,300 students from across institutions in London matches from the Royal Albert Hall to the Soviet Embassy. There were active Campaign for Nuclear Disarmament and anti-apartheid students groups at UCL in the 1960s and a pioneering GaySoc group that helped drive the National Union of Students gay rights campaign in the 1970s. 1977 saw a student occupation of administrative offices and the Slade School in protest against government cuts to higher education.

In 2010, protests by students and staff led UCL to promise to pay a living wage to all UCL staff. As part of the protests against the UK government's plans to increase student fees, around 200 students occupied the Jeremy Bentham Room and part of the Slade School of Fine Art for over two weeks during November and December 2010. The university successfully obtained a court order to evict the students but stated that it did not intend to enforce the order if possible.

The late 2010s saw student campaigns around the cost of university-run accommodation. In 2016, over 1000 students took part in a rent strike in protest against high rents and poor conditions. Organisers said they had won over £1 million in rent cuts, freezes and grants from UCL in the settlement that ended the strike. Another rent strike in 2017 lead to UCL pledging around £1.4 million in bursaries and rent freezes, mostly in the form of accommodation bursaries for less well-off students totalling £600,000 per year for the 2017/18 and 2018/19 academic years. Another rent strike was held at two halls of residence in the third term of the 2017/18 academic year due to complaints over conditions at those halls.

In 2018, Aliza Ayaz, then a first year student founded the university’s first environment student campaign by setting up the Climate Action Society. The student society led pressure on the university to divest from fossil fuels and make the announcement publicly.

In 2024, students protesting against the Israel-Hamas war set up an encampment in the campus, emulating the 2024 pro-Palestinian protests that took place across universities in the United States. The UCL encampment followed similar movements by students in other UK universities, such as the University of Leeds and the University of Bristol.

Student body
In the 2021/22 academic year, UCL had a total of 46,830 students, of whom 23,800 were undergraduates (11,000 UK, 12,800 international), 16,910 were taught postgraduates (8,160 UK, 8,745 international) and 6,120 were research postgraduates (3,520 UK, 2,600 international). In that year, UCL had the second-largest total number of students of any university in the United Kingdom (after the Open University) and the largest number of postgraduate students; however, in terms of UK undergraduates it was 68th by size. It had been the UK university with the highest number of international students every year since 2014/15.

In 2021/22, 87% of UCL's students were full-time and 13% part-time, although among undergraduates only 3% were part-time. The student body was split 60.8% female, 39.1% male and 0.1% other gender identity. 24,145 UCL students (52%) were from outside the UK, of whom 15,795 were from Asia, 4,400 from the European Union, 1,440 from North America, 890 from elsewhere in Europe, 790 from the Middle East, 370 from Africa, 310 from South America, and 155 from Australasia; 45% of overseas students at UCL – 10,785 – came from China. Additionally, UCL had 895 students studying wholly overseas in 2021/22 (10 undergraduate, 785 taught postgraduate and 80 research postgraduate) that are not included in the count of the student population.

For UK domiciled students, UCL's student body in 2021/22 was 52.9% white, 26.4% Asian, 7.6% mixed, 6.4% black and 4.6% other, compared to an average across London institutions of 47.8% white, 22.2% Asian, 6.7% mixed, 15.5% black and 4.9% other. Over the whole student body, 12.5% had a known disability, compared to 15.8% across all institutions.

Diversity
UCL holds an institutional silver Athena SWAN award. It gained its first institutional award (bronze) in 2006 and was promoted to silver in 2015. 21 departments across UCL hold bronze awards, 17 hold silver awards and three hold gold awards. UCL also holds an institutional bronze Race Equality Charter award, which it first gained in 2015.

UCL was formerly a member of Stonewall Diversity Champions scheme, promoting LGBT+ equality. It left in February 2020 as a cost-cutting measure and then controversially decided in late 2021 not to rejoin, against the advice of its equality diversity and inclusion committee, following a vote of the academic board that expressed fears that membership of the scheme could inhibit academic freedom. The decision not to rejoin was strongly opposed by staff and student LGBT+ groups at UCL and by the students' union.

Notable people
UCL alumni include Francis Crick (co-discoverer of the structure of DNA), Lord Herschell (Lord Chancellor of Great Britain), William Stanley Jevons (an early pioneer of modern economics), Charles K. Kao ("Godfather of broadband"), Jomo Kenyatta (considered the "Founding Father" of Kenya) and Joseph Lister (pioneer in the use of antiseptics in surgery). Notable former staff include Hugh Gaitskell (leader of the Labour Party 1955–63), Otto Hahn (pioneer of nuclear chemistry, discoverer of nuclear fusion and Nobel laureate), Peter Higgs (proposer of the Higgs mechanism, which predicted the existence of the Higgs boson, and Nobel laureate), A. E. Housman (classical scholar and poet, who wrote A Shropshire Lad while a professor at UCL), Sir William Ramsay (discoverer of all of the naturally occurring noble gases) and Klaus Roth (mathematician and Field's Medal winner).

Nobel Prizes have been awarded to 30 UCL academics (including visiting academics) and alumni (16 in Physiology or Medicine, seven in Chemistry, five in Physics and one each in Literature and Economic Sciences) as well as three Fields Medals in Mathematics.

In the 19th century UCL operated as a college, with many students taking individual lecture courses rather than studying for degrees. These included well-known alumni such as Mahatma Gandhi, who took English classes with Henry Morley in 1888–89, and John Stuart Mill, who attended lectures on jurisprudence by John Austin.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: University College London


text:
Mathematical optimization

Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.

In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics.

Optimization problems
Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete:
 * An optimization problem with discrete variables is known as a discrete optimization, in which an object such as an integer, permutation or graph must be found from a countable set.
 * A problem with continuous variables is known as a continuous optimization, in which optimal arguments from a continuous set must be found. They can include constrained problems and multimodal problems.

An optimization problem can be represented in the following way:
 * Given: a function $f : A → $\mathbb R$$ from some set $A$ to the real numbers
 * Sought: an element $x_{0} ∈ A$ such that $f(x_{0}) ≤ f(x)$ for all $x ∈ A$ ("minimization") or such that $f(x_{0}) ≥ f(x)$ for all $x ∈ A$ ("maximization").

Such a formulation is called an optimization problem or a mathematical programming problem (a term not directly related to computer programming, but still in use for example in linear programming – see History below). Many real-world and theoretical problems may be modeled in this general framework.

Since the following is valid
 * $$f(\mathbf{x}_{0})\geq f(\mathbf{x}) \Leftrightarrow -f(\mathbf{x}_{0})\leq -f(\mathbf{x}),$$

it suffices to solve only minimization problems. However, the opposite perspective of considering only maximization problems would be valid, too.

Problems formulated using this technique in the fields of physics may refer to the technique as energy minimization, speaking of the value of the function $f$ as representing the energy of the system being modeled. In machine learning, it is always necessary to continuously evaluate the quality of a data model by using a cost function where a minimum implies a set of possibly optimal parameters with an optimal (lowest) error.

Typically, $A$ is some subset of the Euclidean space $$\mathbb R^n$$, often specified by a set of constraints, equalities or inequalities that the members of $A$ have to satisfy. The domain $A$ of $f$ is called the search space or the choice set, while the elements of $A$ are called candidate solutions or feasible solutions.

The function $f$ is called, variously, an objective function, criterion function a loss function or cost function (minimization), a utility function or fitness function (maximization), or, in certain fields, an energy function or energy functional. A feasible solution that minimizes (or maximizes, if that is the goal) the objective function is called an optimal solution.

In mathematics, conventional optimization problems are usually stated in terms of minimization.

A local minimum $x*$ is defined as an element for which there exists some $δ > 0$ such that

 * $$\forall\mathbf{x}\in A \; \text{where} \;\left\Vert\mathbf{x}-\mathbf{x}^{\ast}\right\Vert\leq\delta,\,$$

the expression $f(x*) ≤ f(x)$ holds;

that is to say, on some region around $x*$ all of the function values are greater than or equal to the value at that element. Local maxima are defined similarly.

While a local minimum is at least as good as any nearby elements, a global minimum is at least as good as every feasible element. Generally, unless the objective function is convex in a minimization problem, there may be several local minima. In a convex problem, if there is a local minimum that is interior (not on the edge of the set of feasible elements), it is also the global minimum, but a nonconvex problem may have more than one local minimum not all of which need be global minima.

A large number of algorithms proposed for solving the nonconvex problems – including the majority of commercially available solvers – are not capable of making a distinction between locally optimal solutions and globally optimal solutions, and will treat the former as actual solutions to the original problem. Global optimization is the branch of applied mathematics and numerical analysis that is concerned with the development of deterministic algorithms that are capable of guaranteeing convergence in finite time to the actual optimal solution of a nonconvex problem.

Notation
Optimization problems are often expressed with special notation. Here are some examples:

Minimum and maximum value of a function
Consider the following notation:
 * $$\min_{x\in\mathbb R}\; \left(x^2 + 1\right)$$

This denotes the minimum value of the objective function $x^{2} + 1$, when choosing $x$ from the set of real numbers $$\mathbb{R}$$. The minimum value in this case is 1, occurring at $x = 0$.

Similarly, the notation

 * $$\max_{x\in\mathbb R}\; 2x$$

asks for the maximum value of the objective function $2x$, where $x$ may be any real number. In this case, there is no such maximum as the objective function is unbounded, so the answer is "infinity" or "undefined".

Optimal input arguments
Consider the following notation:
 * $$\underset{x\in(-\infty,-1]}{\operatorname{arg\,min}} \; x^2 + 1,$$

or equivalently
 * $$\underset{x}{\operatorname{arg\,min}} \; x^2 + 1, \; \text{subject to:} \; x\in(-\infty,-1].$$

This represents the value (or values) of the argument $x$ in the interval $(−∞,−1]$ that minimizes (or minimize) the objective function $x^{2} + 1$ (the actual minimum value of that function is not what the problem asks for). In this case, the answer is $x = −1$, since $x = 0$ is infeasible, that is, it does not belong to the feasible set.

Similarly,
 * $$\underset{x\in[-5,5], \; y\in\mathbb R}{\operatorname{arg\,max}} \; x\cos y,$$

or equivalently
 * $$\underset{x, \; y}{\operatorname{arg\,max}} \; x\cos y, \; \text{subject to:} \; x\in[-5,5], \; y\in\mathbb R,$$

represents the ${x, y }$ pair (or pairs) that maximizes (or maximize) the value of the objective function $x cos y$, with the added constraint that $x$ lie in the interval $[−5,5]$ (again, the actual maximum value of the expression does not matter). In this case, the solutions are the pairs of the form ${5, 2k\pi }$ and ${−5, (2k + 1)\pi }$, where $k$ ranges over all integers.

Operators $arg min$ and $arg max$ are sometimes also written as $argmin$ and $argmax$, and stand for argument of the minimum and argument of the maximum.

History
Fermat and Lagrange found calculus-based formulae for identifying optima, while Newton and Gauss proposed iterative methods for moving towards an optimum.

The term "linear programming" for certain optimization cases was due to George B. Dantzig, although much of the theory had been introduced by Leonid Kantorovich in 1939. (Programming in this context does not refer to computer programming, but comes from the use of program by the United States military to refer to proposed training and logistics schedules, which were the problems Dantzig studied at that time.) Dantzig published the Simplex algorithm in 1947, and also John von Neumann and other researches worked on the theoretical aspects of linear programming (like the theory of duality) around the same time.

Other notable researchers in mathematical optimization include the following:

 * Richard Bellman
 * Dimitri Bertsekas
 * Michel Bierlaire
 * Stephen P. Boyd
 * Roger Fletcher
 * Martin Grötschel
 * Ronald A. Howard
 * Fritz John
 * Narendra Karmarkar
 * William Karush
 * Leonid Khachiyan
 * Bernard Koopman
 * Harold Kuhn
 * László Lovász
 * David Luenberger
 * Arkadi Nemirovski
 * Yurii Nesterov
 * Lev Pontryagin
 * R. Tyrrell Rockafellar
 * Naum Z. Shor
 * Albert Tucker

Major subfields

 * Convex programming studies the case when the objective function is convex (minimization) or concave (maximization) and the constraint set is convex. This can be viewed as a particular case of nonlinear programming or as generalization of linear or convex quadratic programming.
 * Linear programming (LP), a type of convex programming, studies the case in which the objective function f is linear and the constraints are specified using only linear equalities and inequalities. Such a constraint set is called a polyhedron or a polytope if it is bounded.
 * Second-order cone programming (SOCP) is a convex program, and includes certain types of quadratic programs.
 * Semidefinite programming (SDP) is a subfield of convex optimization where the underlying variables are semidefinite matrices. It is a generalization of linear and convex quadratic programming.
 * Conic programming is a general form of convex programming. LP, SOCP and SDP can all be viewed as conic programs with the appropriate type of cone.
 * Geometric programming is a technique whereby objective and inequality constraints expressed as posynomials and equality constraints as monomials can be transformed into a convex program.
 * Integer programming studies linear programs in which some or all variables are constrained to take on integer values. This is not convex, and in general much more difficult than regular linear programming.
 * Quadratic programming allows the objective function to have quadratic terms, while the feasible set must be specified with linear equalities and inequalities. For specific forms of the quadratic term, this is a type of convex programming.
 * Fractional programming studies optimization of ratios of two nonlinear functions. The special class of concave fractional programs can be transformed to a convex optimization problem.
 * Nonlinear programming studies the general case in which the objective function or the constraints or both contain nonlinear parts. This may or may not be a convex program. In general, whether the program is convex affects the difficulty of solving it.
 * Stochastic programming studies the case in which some of the constraints or parameters depend on random variables.
 * Robust optimization is, like stochastic programming, an attempt to capture uncertainty in the data underlying the optimization problem. Robust optimization aims to find solutions that are valid under all possible realizations of the uncertainties defined by an uncertainty set.
 * Combinatorial optimization is concerned with problems where the set of feasible solutions is discrete or can be reduced to a discrete one.
 * Stochastic optimization is used with random (noisy) function measurements or random inputs in the search process.
 * Infinite-dimensional optimization studies the case when the set of feasible solutions is a subset of an infinite-dimensional space, such as a space of functions.
 * Heuristics and metaheuristics make few or no assumptions about the problem being optimized. Usually, heuristics do not guarantee that any optimal solution need be found. On the other hand, heuristics are used to find approximate solutions for many complicated optimization problems.
 * Constraint satisfaction studies the case in which the objective function f is constant (this is used in artificial intelligence, particularly in automated reasoning).
 * Constraint programming is a programming paradigm wherein relations between variables are stated in the form of constraints.
 * Disjunctive programming is used where at least one constraint must be satisfied but not all. It is of particular use in scheduling.
 * Space mapping is a concept for modeling and optimization of an engineering system to high-fidelity (fine) model accuracy exploiting a suitable physically meaningful coarse or surrogate model.

In a number of subfields, the techniques are designed primarily for optimization in dynamic contexts (that is, decision making over time):
 * Calculus of variations Is a branch of infinite-dimensional optimization concerned with finding the best way to achieve some goal, such as finding a surface whose boundary is a specific curve, but with the least possible area.
 * Optimal control theory is a generalization of the calculus of variations which introduces control policies.
 * Dynamic programming is the approach to solve the stochastic optimization problem with stochastic, randomness, and unknown model parameters. It studies the case in which the optimization strategy is based on splitting the problem into smaller subproblems. The equation that describes the relationship between these subproblems is called the Bellman equation.
 * Mathematical programming with equilibrium constraints is where the constraints include variational inequalities or complementarities.

Multi-objective optimization
Adding more than one objective to an optimization problem adds complexity. For example, to optimize a structural design, one would desire a design that is both light and rigid. When two objectives conflict, a trade-off must be created. There may be one lightest design, one stiffest design, and an infinite number of designs that are some compromise of weight and rigidity. The set of trade-off designs that improve upon one criterion at the expense of another is known as the Pareto set. The curve created plotting weight against stiffness of the best designs is known as the Pareto frontier.

A design is judged to be "Pareto optimal" (equivalently, "Pareto efficient" or in the Pareto set) if it is not dominated by any other design: If it is worse than another design in some respects and no better in any respect, then it is dominated and is not Pareto optimal.

The choice among "Pareto optimal" solutions to determine the "favorite solution" is delegated to the decision maker. In other words, defining the problem as multi-objective optimization signals that some information is missing: desirable objectives are given but combinations of them are not rated relative to each other. In some cases, the missing information can be derived by interactive sessions with the decision maker.

Multi-objective optimization problems have been generalized further into vector optimization problems where the (partial) ordering is no longer given by the Pareto ordering.

Multi-modal or global optimization
Optimization problems are often multi-modal; that is, they possess multiple good solutions. They could all be globally good (same cost function value) or there could be a mix of globally good and locally good solutions. Obtaining all (or at least some of) the multiple solutions is the goal of a multi-modal optimizer.

Classical optimization techniques due to their iterative approach do not perform satisfactorily when they are used to obtain multiple solutions, since it is not guaranteed that different solutions will be obtained even with different starting points in multiple runs of the algorithm.

Common approaches to global optimization problems, where multiple local extrema may be present include evolutionary algorithms, Bayesian optimization and simulated annealing.

Feasibility problem
The satisfiability problem, also called the feasibility problem, is just the problem of finding any feasible solution at all without regard to objective value. This can be regarded as the special case of mathematical optimization where the objective value is the same for every solution, and thus any solution is optimal.

Many optimization algorithms need to start from a feasible point. One way to obtain such a point is to relax the feasibility conditions using a slack variable; with enough slack, any starting point is feasible. Then, minimize that slack variable until the slack is null or negative.

Existence
The extreme value theorem of Karl Weierstrass states that a continuous real-valued function on a compact set attains its maximum and minimum value. More generally, a lower semi-continuous function on a compact set attains its minimum; an upper semi-continuous function on a compact set attains its maximum point or view.

Necessary conditions for optimality
One of Fermat's theorems states that optima of unconstrained problems are found at stationary points, where the first derivative or the gradient of the objective function is zero (see first derivative test). More generally, they may be found at critical points, where the first derivative or gradient of the objective function is zero or is undefined, or on the boundary of the choice set. An equation (or set of equations) stating that the first derivative(s) equal(s) zero at an interior optimum is called a 'first-order condition' or a set of first-order conditions.

Optima of equality-constrained problems can be found by the Lagrange multiplier method. The optima of problems with equality and/or inequality constraints can be found using the 'Karush–Kuhn–Tucker conditions'.

Sufficient conditions for optimality
While the first derivative test identifies points that might be extrema, this test does not distinguish a point that is a minimum from one that is a maximum or one that is neither. When the objective function is twice differentiable, these cases can be distinguished by checking the second derivative or the matrix of second derivatives (called the Hessian matrix) in unconstrained problems, or the matrix of second derivatives of the objective function and the constraints called the bordered Hessian in constrained problems. The conditions that distinguish maxima, or minima, from other stationary points are called 'second-order conditions' (see 'Second derivative test'). If a candidate solution satisfies the first-order conditions, then the satisfaction of the second-order conditions as well is sufficient to establish at least local optimality.

Sensitivity and continuity of optima
The envelope theorem describes how the value of an optimal solution changes when an underlying parameter changes. The process of computing this change is called comparative statics.

The maximum theorem of Claude Berge (1963) describes the continuity of an optimal solution as a function of underlying parameters.

Calculus of optimization
For unconstrained problems with twice-differentiable functions, some critical points can be found by finding the points where the gradient of the objective function is zero (that is, the stationary points). More generally, a zero subgradient certifies that a local minimum has been found for minimization problems with convex functions and other locally Lipschitz functions, which meet in loss function minimization of the neural network. The positive-negative momentum estimation lets to avoid the local minimum and converges at the objective function global minimum.

Further, critical points can be classified using the definiteness of the Hessian matrix: If the Hessian is positive definite at a critical point, then the point is a local minimum; if the Hessian matrix is negative definite, then the point is a local maximum; finally, if indefinite, then the point is some kind of saddle point.

Constrained problems can often be transformed into unconstrained problems with the help of Lagrange multipliers. Lagrangian relaxation can also provide approximate solutions to difficult constrained problems.

When the objective function is a convex function, then any local minimum will also be a global minimum. There exist efficient numerical techniques for minimizing convex functions, such as interior-point methods.

Global convergence
More generally, if the objective function is not a quadratic function, then many optimization methods use other methods to ensure that some subsequence of iterations converges to an optimal solution. The first and still popular method for ensuring convergence relies on line searches, which optimize a function along one dimension. A second and increasingly popular method for ensuring convergence uses trust regions. Both line searches and trust regions are used in modern methods of non-differentiable optimization. Usually, a global optimizer is much slower than advanced local optimizers (such as BFGS), so often an efficient global optimizer can be constructed by starting the local optimizer from different starting points.

Computational optimization techniques
To solve problems, researchers may use algorithms that terminate in a finite number of steps, or iterative methods that converge to a solution (on some specified class of problems), or heuristics that may provide approximate solutions to some problems (although their iterates need not converge).

Optimization algorithms

 * Simplex algorithm of George Dantzig, designed for linear programming
 * Extensions of the simplex algorithm, designed for quadratic programming and for linear-fractional programming
 * Variants of the simplex algorithm that are especially suited for network optimization
 * Combinatorial algorithms
 * Quantum optimization algorithms

Iterative methods
The iterative methods used to solve problems of nonlinear programming differ according to whether they evaluate Hessians, gradients, or only function values. While evaluating Hessians (H) and gradients (G) improves the rate of convergence, for functions for which these quantities exist and vary sufficiently smoothly, such evaluations increase the computational complexity (or computational cost) of each iteration. In some cases, the computational complexity may be excessively high.

One major criterion for optimizers is just the number of required function evaluations as this often is already a large computational effort, usually much more effort than within the optimizer itself, which mainly has to operate over the N variables. The derivatives provide detailed information for such optimizers, but are even harder to calculate, e.g. approximating the gradient takes at least N+1 function evaluations. For approximations of the 2nd derivatives (collected in the Hessian matrix), the number of function evaluations is in the order of N². Newton's method requires the 2nd-order derivatives, so for each iteration, the number of function calls is in the order of N², but for a simpler pure gradient optimizer it is only N. However, gradient optimizers need usually more iterations than Newton's algorithm. Which one is best with respect to the number of function calls depends on the problem itself.
 * Methods that evaluate Hessians (or approximate Hessians, using finite differences):
 * Newton's method
 * Sequential quadratic programming: A Newton-based method for small-medium scale constrained problems. Some versions can handle large-dimensional problems.
 * Interior point methods: This is a large class of methods for constrained optimization, some of which use only (sub)gradient information and others of which require the evaluation of Hessians.
 * Methods that evaluate gradients, or approximate gradients in some way (or even subgradients):
 * Coordinate descent methods: Algorithms which update a single coordinate in each iteration
 * Conjugate gradient methods: Iterative methods for large problems. (In theory, these methods terminate in a finite number of steps with quadratic objective functions, but this finite termination is not observed in practice on finite–precision computers.)
 * Gradient descent (alternatively, "steepest descent" or "steepest ascent"): A (slow) method of historical and theoretical interest, which has had renewed interest for finding approximate solutions of enormous problems.
 * Subgradient methods: An iterative method for large locally Lipschitz functions using generalized gradients. Following Boris T. Polyak, subgradient–projection methods are similar to conjugate–gradient methods.
 * Bundle method of descent: An iterative method for small–medium-sized problems with locally Lipschitz functions, particularly for convex minimization problems (similar to conjugate gradient methods).
 * Ellipsoid method: An iterative method for small problems with quasiconvex objective functions and of great theoretical interest, particularly in establishing the polynomial time complexity of some combinatorial optimization problems. It has similarities with Quasi-Newton methods.
 * Conditional gradient method (Frank–Wolfe) for approximate minimization of specially structured problems with linear constraints, especially with traffic networks. For general unconstrained problems, this method reduces to the gradient method, which is regarded as obsolete (for almost all problems).
 * Quasi-Newton methods: Iterative methods for medium-large problems (e.g. N<1000).
 * Simultaneous perturbation stochastic approximation (SPSA) method for stochastic optimization; uses random (efficient) gradient approximation.
 * Methods that evaluate only function values: If a problem is continuously differentiable, then gradients can be approximated using finite differences, in which case a gradient-based method can be used.
 * Interpolation methods
 * Pattern search methods, which have better convergence properties than the Nelder–Mead heuristic (with simplices), which is listed below.
 * Mirror descent

Heuristics
Besides (finitely terminating) algorithms and (convergent) iterative methods, there are heuristics. A heuristic is any algorithm which is not guaranteed (mathematically) to find the solution, but which is nevertheless useful in certain practical situations. List of some well-known heuristics:

 * Differential evolution
 * Dynamic relaxation
 * Evolutionary algorithms
 * Genetic algorithms
 * Hill climbing with random restart
 * Memetic algorithm
 * Nelder–Mead simplicial heuristic: A popular heuristic for approximate minimization (without calling gradients)
 * Particle swarm optimization
 * Simulated annealing
 * Stochastic tunneling
 * Tabu search

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; 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.

Economics and finance
Economics is closely enough linked to optimization of 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. Modern optimization theory includes traditional optimization theory but also overlaps with game theory and the study of economic equilibria. The Journal of Economic Literature codes classify mathematical programming, optimization techniques, and related topics under 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, consumers are assumed to maximize their utility, while firms are usually assumed to maximize their profit. Also, agents are often modeled as being risk-averse, thereby preferring to avoid risk. Asset prices are also modeled using optimization theory, though the underlying mathematics relies on optimizing stochastic processes rather than on static optimization. International trade theory also uses optimization to explain trade patterns between nations. The optimization of portfolios is an example of multi-objective optimization in economics.

Since the 1970s, economists have modeled dynamic decisions over time using control theory. For example, dynamic search models are used to study labor-market behavior. A crucial distinction is between deterministic and stochastic models. Macroeconomists build 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.

Electrical engineering
Some common applications of optimization techniques in electrical engineering include active filter design, stray field reduction in superconducting magnetic energy storage systems, space mapping design of microwave structures, handset antennas,  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. Optimization techniques are also used in power-flow analysis.

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, resource leveling, water resource allocation, traffic management and schedule optimization.

Operations research
Another field that uses optimization techniques extensively is operations research. 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 geophysical parameter estimation problems. Given a set of geophysical measurements, e.g. seismic recordings, it is common to solve for the physical properties and 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
Nonlinear optimization methods are widely used in conformational analysis.

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. Linear programming has been applied to calculate the maximal possible yields of fermentation products, and to infer gene regulatory networks from multiple microarray datasets as well as transcriptional regulatory networks from high-throughput data. Nonlinear programming has been used to analyze energy metabolism and has been applied to metabolic engineering and parameter estimation in biochemical pathways.

title: Mathematical optimization