Mathematical Sciences: Continuation and Bifurcations        Investigations In Constrained Optimization
{'Code': '03040100', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'Michael Steuerwalt'}
Constrained optimization problems permeate much of applied                 mathematics, engineering and the sciences ranging from linear and               nonlinear programming, calculus of variations and optimal control               to variational formulations of parameter identification and                     inverse problems, ill-posed problems, and continuum mechanics.                  The primary objective of this research program is the development               of a new class of robust and very fast algorithms for the                       solution of these problems.  These algorithms are based on the                  use of fast numerical continuation techniques to follow smooth                  penalty paths to optimality.  The key to the development of these               algorithms is an expanded Lagrangian system of equations which                  contains the penalty path as a solution, is a perturbation of the               first order necessary conditions, and sometimes, but not always,                provides a basis for finding multiple optima.  The problem                      formulation and continuation methodology provide a natural                      framework for the development and application of Tikhonov                       regularization techniques to the solution of sensitive, ill-                    conditioned, or ill-constrained optimization problems arising                   from model error or uncertainty and for the investigation of the                parametric optimization problem.  Also, in this setting the                     traditional ill-conditioning associated with a sequential                       optimization algorithm is removed for three smooth penalty                      functions:  the quadratic penalty function for equality                         constraints and the logarithmic barrier and quadratic loss                      function for inequality constraints.  In linear programming one                 of the problem formulations and correspondence algorithms is very               similar to the Karmarkar algorithm with similar expectations in                 speed; however, these fast continuation-penalty path algorithms                 are also applicable to the aforementioned constrained                           optimization problems with similar expectations in speed.                            Research of this type aims at developing new computer                      software for optimization of functions of many variables with                   constraints in the most efficient and robust way.  Software of                  this type has a large number of potential applications, ranging                 from management of resources to airline scheduling and machine                  design.
{'FirstName': 'Aubrey', 'LastName': 'Poore', 'EmailAddress': None, 'StartDate': '08/01/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'Colorado State University', 'CityName': 'Fort Collins', 'ZipCode': '805232002', 'PhoneNumber': '9704916355', 'StreetAddress': '601 S Howes St', 'CountryName': 'United States', 'StateName': 'Colorado', 'StateCode': 'CO'}
[{'Code': '1266', 'Text': 'APPLIED MATHEMATICS'}, {'Code': '1271', 'Text': 'COMPUTATIONAL MATHEMATICS'}]
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