Mathematical Sciences: Continuation and Bifurcations Investigations In Constrained Optimization

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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.