The Geometry of Numbers and Discrete Production Sets
{'Code': '04050100', 'Directorate': {'Abbreviation': 'SBE', 'LongName': 'Direct For Social, Behav & Economic Scie'}, 'Division': {'Abbreviation': 'SES', 'LongName': 'Divn Of Social and Economic Sciences'}}
{'SignBlockName': 'Daniel H. Newlon', 'PO_EMAI': None, 'PO_PHON': None}
Much of the nation's production process reflects the application                of tools developed from optimization concepts and related                       theories from economics.   One of the most useful of these tools                is the simplex method for solving linear programming problems.                  Linear programming is concerned with the maximization of output                 subject to constraints on the availability of factors of                        production.  The simplex method can be interpreted as a                         decentralized search for prices which equilibrate the demand for                factors of production with their supply.                                                                                                                        This project is concerned with solving maximization problems when               the production set involves indivisibilities.  Indivisibilities                 are introduced into a linear problem by requiring that some, or                 all of the activity levels take on integral values, rather than                 arbitrary real values.  This field of research is called integer                programming.   In contrast to the simplex method, which performs                remarkably well on all linear programming problems, existing                    integer programming algorithms are capricious, unreliable and                   none of these algorithms are capable of being interpreted in any                meaningful economic terms.                                                                                                                                      Under his previous grant the investigator found a new method for                solving integer programming problems that could be a major                      breakthrough in this extremely difficult line of research.  He                  proved that a globally intractable integer programming problem                  can be reduced to a small number of tractable, local integer                    programming problems that eventually converge to the global                     solution.  The sequence of programming problems is a decision                   tree in which the number of branches (the number of possible                    local linear programs) at each step in the process is a small and               declining fraction of the size of the problem.  The project                     examines methods for determining these decision trees explicitly,               explores their economic interpretation, and studies how well                    these algorithms work in practice.                                                                                                                              Most real world problems in such fields as operations research,                 economics and industrial planning involve indivisibilities, so a                new set of powerful mathematical tools to solve these problems                  would be extremely valuable.
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{'Code': '1320', 'Text': 'Economics'}
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