George Bernard Dantzig was an American mathematical scientist who made contributions to industrial engineering, operations research, computer science and statistics. Dantzig is known for his development of the simplex algorithm, an algorithm for solving linear programming problems, for his other work with linear programming. In statistics, Dantzig solved two open problems in statistical theory, which he had mistaken for homework after arriving late to a lecture by Jerzy Neyman. Dantzig was the Professor Emeritus of Transportation Sciences and Professor of Operations Research and of Computer Science at Stanford. Born in Portland, George Bernard Dantzig was named after George Bernard Shaw, the Irish writer, his father, Tobias Dantzig, was a mathematician and linguist, his mother, Anja Dantzig, was a linguist of French Jewish origin. Dantzig's parents met during their study at the University of Paris, where Tobias studied mathematics under Henri Poincaré, after whom Dantzig's brother was named.
The Dantzigs immigrated to the United States, where they settled in Oregon. Early in the 1920s the Dantzig family moved from Baltimore to Washington, his mother became a linguist at the Library of Congress, his father became a math tutor at the University of Maryland, College Park. Dantzig attended Central High School. By the time he reached high school he was fascinated by geometry, this interest was further nurtured by his father, challenging him with complicated problems in projective geometry. George Dantzig received his B. S. from University of Maryland in 1936 in mathematics and physics, part of the University of Maryland College of Computer and Natural Sciences. He earned his master's degree in mathematics from the University of Michigan in 1938. After a two-year period at the Bureau of Labor Statistics, he enrolled in the doctoral program in mathematics at the University of California, where he studied statistics under Jerzy Neyman. With the outbreak of World War II, Dantzig took a leave of absence from the doctoral program at Berkeley to join the U.
S. Air Force Office of Statistical Control. In 1946, he returned to Berkeley to complete the requirements of his program and received his Ph. D. that year. Although he had a faculty offer from Berkeley, he returned to the Air Force as mathematical advisor to the comptroller. In 1952 Dantzig joined the mathematics division of the RAND Corporation. By 1960 he became a professor in the Department of Industrial Engineering at UC Berkeley, where he founded and directed the Operations Research Center. In 1966 he joined the Stanford faculty of Computer Science. A year the Program in Operations Research became a full-fledged department. In 1973 he founded the Systems Optimization Laboratory there. On a sabbatical leave that year, he headed the Methodology Group at the International Institute for Applied Systems Analysis in Laxenburg, Austria, he became the C. A. Criley Professor of Transportation Sciences at Stanford, kept going, well beyond his mandatory retirement in 1985, he was a member of the National Academy of Sciences, the National Academy of Engineering, the American Academy of Arts and Sciences.
Dantzig was the recipient of many honors, including the first John von Neumann Theory Prize in 1974, the National Medal of Science in 1975, an honorary doctorate from the University of Maryland, College Park in 1976. The Mathematical Programming Society honored Dantzig by creating the George B. Dantzig Prize, bestowed every three years since 1982 on one or two people who have made a significant impact in the field of mathematical programming. Dantzig died on May 13, 2005, in his home in Stanford, California, of complications from diabetes and cardiovascular disease, he was 90 years old. Freund wrote further that "through his research in mathematical theory, economic analysis, applications to industrial problems, Dantzig contributed more than any other researcher to the remarkable development of linear programming". Dantzig's work allows the airline industry, for example, to schedule crews and make fleet assignments. Based on his work tools are developed "that shipping companies use to determine how many planes they need and where their delivery trucks should be deployed.
The oil industry long has used linear programming in refinery planning, as it determines how much of its raw product should become different grades of gasoline and how much should be used for petroleum-based byproducts. It is used in manufacturing, revenue management, telecommunications, architecture, circuit design and countless other areas". An event in Dantzig's life became the origin of a famous story in 1939, while he was a graduate student at UC Berkeley. Near the beginning of a class for which Dantzig was late, professor Jerzy Neyman wrote two examples of famously unsolved statistics problems on the blackboard; when Dantzig arrived, he assumed that the two problems were a homework assignment and wrote them down. According to Dantzig, the problems "seemed to be a little harder than usual", but a few days he handed in completed solutions for the two problems, still believing that they were an assignment, overdue. Six weeks Dantzig received a visit from an excited professor Neyman, eager to tell him that the homework problems he had solved were two of the most famous unsolved problems in statistics.
He had prepared one of Dantzig's solutions for publication in a mathematical journal. As Dantzig told it in a 1986 interview in the College Mathematics Journal: A year when I began t
A thesis or dissertation is a document submitted in support of candidature for an academic degree or professional qualification presenting the author's research and findings. In some contexts, the word "thesis" or a cognate is used for part of a bachelor's or master's course, while "dissertation" is applied to a doctorate, while in other contexts, the reverse is true; the term graduate thesis is sometimes used to refer to both master's theses and doctoral dissertations. The required complexity or quality of research of a thesis or dissertation can vary by country, university, or program, the required minimum study period may thus vary in duration; the word "dissertation" can at times be used to describe a treatise without relation to obtaining an academic degree. The term "thesis" is used to refer to the general claim of an essay or similar work; the term "thesis" comes from the Greek θέσις, meaning "something put forth", refers to an intellectual proposition. "Dissertation" comes from the Latin dissertātiō, meaning "discussion".
Aristotle was the first philosopher to define the term thesis. "A'thesis' is a supposition of some eminent philosopher that conflicts with the general opinion...for to take notice when any ordinary person expresses views contrary to men's usual opinions would be silly". For Aristotle, a thesis would therefore be a supposition, stated in contradiction with general opinion or express disagreement with other philosophers. A supposition is a statement or opinion that may or may not be true depending on the evidence and/or proof, offered; the purpose of the dissertation is thus to outline the proofs of why the author disagrees with other philosophers or the general opinion. A thesis may be arranged as a thesis by publication or a monograph, with or without appended papers though many graduate programs allow candidates to submit a curated collection of published papers. An ordinary monograph has a title page, an abstract, a table of contents, comprising the various chapters, a bibliography or a references section.
They differ in their structure in accordance with the many different areas of study and the differences between them. In a thesis by publication, the chapters constitute an introductory and comprehensive review of the appended published and unpublished article documents. Dissertations report on a research project or study, or an extended analysis of a topic; the structure of a thesis or dissertation explains the purpose, the previous research literature impinging on the topic of the study, the methods used, the findings of the project. Most world universities use a multiple chapter format: a) an introduction, which introduces the research topic, the methodology, as well as its scope and significance. Degree-awarding institutions define their own house style that candidates have to follow when preparing a thesis document. In addition to institution-specific house styles, there exist a number of field-specific and international standards and recommendations for the presentation of theses, for instance ISO 7144.
Other applicable international standards include ISO 2145 on section numbers, ISO 690 on bibliographic references, ISO 31 on quantities or units. Some older house styles specify that front matter must use a separate page number sequence from the main text, using Roman numerals; the relevant international standard and many newer style guides recognize that this book design practice can cause confusion where electronic document viewers number all pages of a document continuously from the first page, independent of any printed page numbers. They, avoid the traditional separate number sequence for front matter and require a single sequence of Arabic numerals starting with 1 for the first printed page. Presentation requirements, including pagination, layout and color of paper, use of acid-free paper, paper size, order of components, citation style, will be checked page by page by the accepting officer before the thesis is accepted and a receipt is issued. However, strict standards are not always required.
Most Italian universities, for example, have only general requirements on the character size and the page formatting, leave much freedom for the actual typographic details. A thesis or dissertation committee is a committee. In the US, these committees consist of a primary supervisor or advisor and two or more committee members, who supervise the progress of the dissertation and may act as the examining committee, or jury, at the oral examination of the thesis. At most universities, the committee is chosen by the student in conjunction with his or her primary adviser after completion of the comprehensive examinations or prospectus meeting, may consist of members of the comps committee; the committee members are doctors in their field (whether a PhD or other des
Princeton University is a private Ivy League research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the nine colonial colleges chartered before the American Revolution; the institution moved to Newark in 1747 to the current site nine years and renamed itself Princeton University in 1896. Princeton provides undergraduate and graduate instruction in the humanities, social sciences, natural sciences, engineering, it offers professional degrees through the Woodrow Wilson School of Public and International Affairs, the School of Engineering and Applied Science, the School of Architecture and the Bendheim Center for Finance. The university has ties with the Institute for Advanced Study, Princeton Theological Seminary and the Westminster Choir College of Rider University. Princeton has the largest endowment per student in the United States. From 2001 to 2018, Princeton University was ranked either first or second among national universities by U.
S. News & World Report, holding the top spot for 16 of those 18 years; as of October 2018, 65 Nobel laureates, 15 Fields Medalists and 13 Turing Award laureates have been affiliated with Princeton University as alumni, faculty members or researchers. In addition, Princeton has been associated with 21 National Medal of Science winners, 5 Abel Prize winners, 5 National Humanities Medal recipients, 209 Rhodes Scholars, 139 Gates Cambridge Scholars and 126 Marshall Scholars. Two U. S. Presidents, twelve U. S. Supreme Court Justices and numerous living billionaires and foreign heads of state are all counted among Princeton's alumni body. Princeton has graduated many prominent members of the U. S. Congress and the U. S. Cabinet, including eight Secretaries of State, three Secretaries of Defense and three of the past five Chairs of the Federal Reserve. New Light Presbyterians founded the College of New Jersey in 1746; the college was the religious capital of Scottish Presbyterian America. In 1754, trustees of the College of New Jersey suggested that, in recognition of Governor Jonathan Belcher's interest, Princeton should be named as Belcher College.
Belcher replied: "What a name that would be!" In 1756, the college moved to New Jersey. Its home in Princeton was Nassau Hall, named for the royal House of Orange-Nassau of William III of England. Following the untimely deaths of Princeton's first five presidents, John Witherspoon became president in 1768 and remained in that office until his death in 1794. During his presidency, Witherspoon shifted the college's focus from training ministers to preparing a new generation for secular leadership in the new American nation. To this end, he solicited investment in the college. Witherspoon's presidency constituted a long period of stability for the college, interrupted by the American Revolution and the Battle of Princeton, during which British soldiers occupied Nassau Hall. In 1812, the eighth president of the College of New Jersey, Ashbel Green, helped establish the Princeton Theological Seminary next door; the plan to extend the theological curriculum met with "enthusiastic approval on the part of the authorities at the College of New Jersey".
Today, Princeton University and Princeton Theological Seminary maintain separate institutions with ties that include services such as cross-registration and mutual library access. Before the construction of Stanhope Hall in 1803, Nassau Hall was the college's sole building; the cornerstone of the building was laid on September 17, 1754. During the summer of 1783, the Continental Congress met in Nassau Hall, making Princeton the country's capital for four months. Over the centuries and through two redesigns following major fires, Nassau Hall's role shifted from an all-purpose building, comprising office, dormitory and classroom space; the class of 1879 donated twin lion sculptures that flanked the entrance until 1911, when that same class replaced them with tigers. Nassau Hall's bell rang after the hall's construction; the bell was recast and melted again in the fire of 1855. James McCosh took office as the college's president in 1868 and lifted the institution out of a low period, brought about by the American Civil War.
During his two decades of service, he overhauled the curriculum, oversaw an expansion of inquiry into the sciences, supervised the addition of a number of buildings in the High Victorian Gothic style to the campus. McCosh Hall is named in his honor. In 1879, the first thesis for a Doctor of Philosophy Ph. D. was submitted by James F. Williamson, Class of 1877. In 1896, the college changed its name from the College of New Jersey to Princeton University to honor the town in which it resides. During this year, the college underwent large expansion and became a university. In 1900, the Graduate School was established. In 1902, Woodrow Wilson, graduate of the Class of 1879, was elected the 13th president of the university. Under Wilson, Princeton introduced the preceptorial system in 1905, a then-unique concept in the US that augmented the standard lecture method of teaching with a more personal form in which small groups of students, or precepts, could interact with a single instructor, or preceptor, in their field of interest.
In 1906, the reservoir Lake Carnegie was created by Andrew Carnegie. A collection of historical photographs of the build
Richard M. Karp
Richard Manning Karp is an American computer scientist and computational theorist at the University of California, Berkeley. He is most notable for his research in the theory of algorithms, for which he received a Turing Award in 1985, The Benjamin Franklin Medal in Computer and Cognitive Science in 2004, the Kyoto Prize in 2008. Born to parents Abraham and Rose Karp in Boston, Karp has three younger siblings: Robert and Carolyn, his family was Jewish, he grew up in a small apartment, in a mostly Jewish neighborhood of Dorchester in Boston. Both his parents were Harvard graduates, while his father had had ambitions to go to medical school after Harvard, but became a mathematics teacher as he could not afford the medical school fees, he attended Harvard University, where he received his bachelor's degree in 1955, his master's degree in 1956, his Ph. D. in applied mathematics in 1959. He started working at IBM's Thomas J. Watson Research Center. In 1968, he became Professor of Computer Science and Operations Research at the University of California, Berkeley.
Apart from a 4-year period as a professor at the University of Washington, he has remained at Berkeley. From 1988 to 1995 and 1999 to the present he has been a Research Scientist at the International Computer Science Institute in Berkeley, where he leads the Algorithms Group. Richard Karp was awarded the National Medal of Science, was the recipient of the Harvey Prize of the Technion and the 2004 Benjamin Franklin Medal in Computer and Cognitive Science for his insights into computational complexity. In 1994 he was inducted as a Fellow of the Association for Computing Machinery, he is the recipient of several honorary degrees. In 2012, Karp became the founding director of the Simons Institute for the Theory of Computing at the University of California, Berkeley. Karp has made many important discoveries in computer science, combinatorial algorithms, operations research, his major current research interests include bioinformatics. In 1971 he co-developed with Jack Edmonds the Edmonds–Karp algorithm for solving the maximum flow problem on networks, in 1972 he published a landmark paper in complexity theory, "Reducibility Among Combinatorial Problems", in which he proved 21 problems to be NP-complete.
In 1973 he and John Hopcroft published the Hopcroft–Karp algorithm, the fastest known method for finding maximum cardinality matchings in bipartite graphs. In 1980, along with Richard J. Lipton, Karp proved the Karp-Lipton theorem. In 1987 he co-developed with Michael O. Rabin the Rabin-Karp string search algorithm, his citation for the Turing Award was as follows: For his continuing contributions to the theory of algorithms including the development of efficient algorithms for network flow and other combinatorial optimization problems, the identification of polynomial-time computability with the intuitive notion of algorithmic efficiency, most notably, contributions to the theory of NP-completeness. Karp introduced the now standard methodology for proving problems to be NP-complete which has led to the identification of many theoretical and practical problems as being computationally difficult. ACM Crossroads magazine interview/bio of Richard Karp Karp's Home Page at Berkeley Biography of Richard Karp from the Institute for Operations Research and the Management Sciences
San Francisco the City and County of San Francisco, is the cultural and financial center of Northern California. San Francisco is the 13th-most populous city in the United States, the fourth-most populous in California, with 884,363 residents as of 2017, it covers an area of about 46.89 square miles at the north end of the San Francisco Peninsula in the San Francisco Bay Area, making it the second-most densely populated large US city, the fifth-most densely populated U. S. county, behind only four of the five New York City boroughs. San Francisco is part of the fifth-most populous primary statistical area in the United States, the San Jose–San Francisco–Oakland, CA Combined Statistical Area; as of 2017, it was the seventh-highest income county in the United States, with a per capita personal income of $119,868. As of 2015, San Francisco proper had a GDP of $154.2 billion, a GDP per capita of $177,968. The San Francisco CSA was the country's third-largest urban economy as of 2017, with a GDP of $907 billion.
Of the 500+ primary statistical areas in the US, the San Francisco CSA had among the highest GDP per capita in 2017, at $93,938. San Francisco was ranked 14th in the world and third in the United States on the Global Financial Centres Index as of September 2018. San Francisco was founded on June 29, 1776, when colonists from Spain established Presidio of San Francisco at the Golden Gate and Mission San Francisco de Asís a few miles away, all named for St. Francis of Assisi; the California Gold Rush of 1849 brought rapid growth, making it the largest city on the West Coast at the time. San Francisco became a consolidated city-county in 1856. San Francisco's status as the West Coast's largest city peaked between 1870 and 1900, when around 25% of California's population resided in the city proper. After three-quarters of the city was destroyed by the 1906 earthquake and fire, San Francisco was rebuilt, hosting the Panama-Pacific International Exposition nine years later. In World War II, San Francisco was a major port of embarkation for service members shipping out to the Pacific Theater.
It became the birthplace of the United Nations in 1945. After the war, the confluence of returning servicemen, significant immigration, liberalizing attitudes, along with the rise of the "hippie" counterculture, the Sexual Revolution, the Peace Movement growing from opposition to United States involvement in the Vietnam War, other factors led to the Summer of Love and the gay rights movement, cementing San Francisco as a center of liberal activism in the United States. Politically, the city votes along liberal Democratic Party lines. A popular tourist destination, San Francisco is known for its cool summers, steep rolling hills, eclectic mix of architecture, landmarks, including the Golden Gate Bridge, cable cars, the former Alcatraz Federal Penitentiary, Fisherman's Wharf, its Chinatown district. San Francisco is the headquarters of five major banking institutions and various other companies such as Levi Strauss & Co. Gap Inc. Fitbit, Salesforce.com, Reddit, Inc. Dolby, Weebly, Pacific Gas and Electric Company, Pinterest, Uber, Mozilla, Wikimedia Foundation and Weather Underground.
It is home to a number of educational and cultural institutions, such as the University of San Francisco, University of California, San Francisco, San Francisco State University, the De Young Museum, the San Francisco Museum of Modern Art, the California Academy of Sciences. As of 2019, San Francisco is the highest rated American city on world liveability rankings; the earliest archaeological evidence of human habitation of the territory of the city of San Francisco dates to 3000 BC. The Yelamu group of the Ohlone people resided in a few small villages when an overland Spanish exploration party, led by Don Gaspar de Portolà, arrived on November 2, 1769, the first documented European visit to San Francisco Bay. Seven years on March 28, 1776, the Spanish established the Presidio of San Francisco, followed by a mission, Mission San Francisco de Asís, established by the Spanish explorer Juan Bautista de Anza. Upon independence from Spain in 1821, the area became part of Mexico. Under Mexican rule, the mission system ended, its lands became privatized.
In 1835, Englishman William Richardson erected the first independent homestead, near a boat anchorage around what is today Portsmouth Square. Together with Alcalde Francisco de Haro, he laid out a street plan for the expanded settlement, the town, named Yerba Buena, began to attract American settlers. Commodore John D. Sloat claimed California for the United States on July 7, 1846, during the Mexican–American War, Captain John B. Montgomery arrived to claim Yerba Buena two days later. Yerba Buena was renamed San Francisco on January 30 of the next year, Mexico ceded the territory to the United States at the end of the war. Despite its attractive location as a port and naval base, San Francisco was still a small settlement with inhospitable geography; the California Gold Rush brought a flood of treasure seekers. With their sourdough bread in tow, prospectors accumulated in San Francisco over rival Benicia, raising the population from 1,000 in 1848 to 25,000 by December 1849; the promise of great wealth was so strong that crews on arriving vessels deserted and rushed off to the gold fields, leaving behind a forest of masts in San Francisco harbor.
Some of these 500 abandoned ships were used at times as storeships and hotels.
New York (state)
New York is a state in the Northeastern United States. New York was one of the original thirteen colonies. With an estimated 19.54 million residents in 2018, it is the fourth most populous state. To distinguish the state from the city with the same name, it is sometimes called New York State; the state's most populous city, New York City, makes up over 40% of the state's population. Two-thirds of the state's population lives in the New York metropolitan area, nearly 40% lives on Long Island; the state and city were both named for the 17th century Duke of York, the future King James II of England. With an estimated population of 8.62 million in 2017, New York City is the most populous city in the United States and the premier gateway for legal immigration to the United States. The New York metropolitan area is one of the most populous in the world. New York City is a global city, home to the United Nations Headquarters and has been described as the cultural and media capital of the world, as well as the world's most economically powerful city.
The next four most populous cities in the state are Buffalo, Rochester and Syracuse, while the state capital is Albany. The 27th largest U. S. state in land area, New York has a diverse geography. The state is bordered by New Jersey and Pennsylvania to the south and Connecticut and Vermont to the east; the state has a maritime border with Rhode Island, east of Long Island, as well as an international border with the Canadian provinces of Quebec to the north and Ontario to the northwest. The southern part of the state is in the Atlantic coastal plain and includes Long Island and several smaller associated islands, as well as New York City and the lower Hudson River Valley; the large Upstate New York region comprises several ranges of the wider Appalachian Mountains, the Adirondack Mountains in the Northeastern lobe of the state. Two major river valleys – the north-south Hudson River Valley and the east-west Mohawk River Valley – bisect these more mountainous regions. Western New York is considered part of the Great Lakes region and borders Lake Ontario, Lake Erie, Niagara Falls.
The central part of the state is dominated by the Finger Lakes, a popular vacation and tourist destination. New York had been inhabited by tribes of Algonquian and Iroquoian-speaking Native Americans for several hundred years by the time the earliest Europeans came to New York. French colonists and Jesuit missionaries arrived southward from Montreal for trade and proselytizing. In 1609, the region was visited by Henry Hudson sailing for the Dutch East India Company; the Dutch built Fort Nassau in 1614 at the confluence of the Hudson and Mohawk rivers, where the present-day capital of Albany developed. The Dutch soon settled New Amsterdam and parts of the Hudson Valley, establishing the multicultural colony of New Netherland, a center of trade and immigration. England seized the colony from the Dutch in 1664. During the American Revolutionary War, a group of colonists of the Province of New York attempted to take control of the British colony and succeeded in establishing independence. In the 19th century, New York's development of access to the interior beginning with the Erie Canal, gave it incomparable advantages over other regions of the U.
S. built its political and cultural ascendancy. Many landmarks in New York are well known, including four of the world's ten most-visited tourist attractions in 2013: Times Square, Central Park, Niagara Falls, Grand Central Terminal. New York is home to the Statue of Liberty, a symbol of the United States and its ideals of freedom and opportunity. In the 21st century, New York has emerged as a global node of creativity and entrepreneurship, social tolerance, environmental sustainability. New York's higher education network comprises 200 colleges and universities, including Columbia University, Cornell University, New York University, the United States Military Academy, the United States Merchant Marine Academy, University of Rochester, Rensselaer Polytechnic Institute, Rockefeller University, which have been ranked among the top 40 in the nation and world; the tribes in what is now New York were predominantly Algonquian. Long Island was divided in half between the Wampanoag and Lenape; the Lenape controlled most of the region surrounding New York Harbor.
North of the Lenape was the Mohicans. Starting north of them, from east to west, were three Iroquoian nations: the Mohawk, the original Iroquois and the Petun. South of them, divided along Appalachia, were the Susquehannock and the Erie. Many of the Wampanoag and Mohican peoples were caught up in King Philip's War, a joint effort of many New England tribes to push Europeans off their land. After the death of their leader, Chief Philip Metacomet, most of those peoples fled inland, splitting into the Abenaki and the Schaghticoke. Many of the Mohicans remained in the region until the 1800s, however, a small group known as the Ouabano migrated southwest into West Virginia at an earlier time, they may have merged with the Shawnee. The Mohawk and Susquehannock were the most militaristic. Trying to corner trade with the Europeans, they targeted other tribes; the Mohawk were known for refusing white settlement on their land and banishing any of their people who converted to Christianity. They posed a major threat to the Abenaki and Mohicans, while the Susquehannock conquered the Lenape in the 1600s.
The most devastating event of the century, was the Beaver Wars. From 1640–1680, Iroquoian peoples waged campaigns which extended from modern-day Michigan to Virginia against Algonquian and Siouan tribes, as well as each other; the ai
Dantzig–Wolfe decomposition is an algorithm for solving linear programming problems with special structure. It was developed by George Dantzig and Philip Wolfe and published in 1960. Many texts on linear programming have sections dedicated to discussing this decomposition algorithm. Dantzig–Wolfe decomposition relies on delayed column generation for improving the tractability of large-scale linear programs. For most linear programs solved via the revised simplex algorithm, at each step, most columns are not in the basis. In such a scheme, a master problem containing at least the active columns uses a subproblem or subproblems to generate columns for entry into the basis such that their inclusion improves the objective function. In order to use Dantzig–Wolfe decomposition, the constraint matrix of the linear program must have a specific form. A set of constraints must be identified as "connecting", "coupling", or "complicating" constraints wherein many of the variables contained in the constraints have non-zero coefficients.
The remaining constraints need to be grouped into independent submatrices such that if a variable has a non-zero coefficient within one submatrix, it will not have a non-zero coefficient in another submatrix. This description is visualized below: The D matrix represents the coupling constraints and each Fi represents the independent submatrices. Note that it is possible to run the algorithm when there is only one F submatrix. After identifying the required form, the original problem is reformulated into a master program and n subprograms; this reformulation relies on the fact that a non-empty, bounded convex polyhedron can be represented as a convex combination of its extreme points. Each column in the new master program represents a solution to one of the subproblems; the master program enforces that the coupling constraints are satisfied given the set of subproblem solutions that are available. The master program requests additional solutions from the subproblem such that the overall objective to the original linear program is improved.
While there are several variations regarding implementation, the Dantzig–Wolfe decomposition algorithm can be described as follows: Starting with a feasible solution to the reduced master program, formulate new objective functions for each subproblem such that the subproblems will offer solutions that improve the current objective of the master program. Subproblems are re-solved given their new objective functions. An optimal value for each subproblem is offered to the master program; the master program incorporates one or all of the new columns generated by the solutions to the subproblems based on those columns' respective ability to improve the original problem's objective. Master program performs x iterations of the simplex algorithm, where x is the number of columns incorporated. If objective is improved, goto step 1. Else, continue; the master program cannot be further improved by any new columns from the subproblems, thus return. There are examples of the implementation of Dantzig–Wolfe decomposition available in the AMPL and GAMS mathematical modeling languages.
There is a general, parallel implementation available that leverages the open source GNU Linear Programming Kit. The algorithm can be implemented such that the subproblems are solved in parallel, since their solutions are independent; when this is the case, there are options for the master program as to how the columns should be integrated into the master. The master may wait until each subproblem has completed and incorporate all columns that improve the objective or it may choose a smaller subset of those columns. Another option is that the master may take only the first available column and stop and restart all of the subproblems with new objectives based upon the incorporation of the newest column. Another design choice for implementation involves columns that exit the basis at each iteration of the algorithm; those columns may be retained discarded, or discarded via some policy after future iterations. A recent computational evaluation of Dantzig-Wolfe in general and Dantzig-Wolfe and parallel computation is the PhD thesis by J. R. Tebboth Delayed column generation Benders' decomposition