Nonlinear Optimization

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Three research projects in nonlinear optimization will be undertaken; they range from theoretical studies of constrained optimization algorithms to the development of a network-based optimization system capable of solving problems automatically over the Internet. The first project is devoted to the design of limited memory approaches for large-scale nonlinear programming. Limited memory methods have proved to be quite useful for solving many classes of large unconstrained problems, but their economy and simplicity have not been fully exploited in the nonlinearly constrained case. The basic guiding principle in this work is to employ only algorithms with modest computational requirements and to represent limited memory matrices in compact form, so that the overall cost of the iteration is only a fraction of that required by the standard Sequential Quadratic Programming iteration. The new algorithms will allow the user to choose between direct and iterative methods for solving the linear subproblems arising in the iteration. The second project is devoted to the design and analysis of interior-point methods for nonlinearly constrained (non-convex) optimization problems. This is a long term research effort that first aims at identifying the crucial ingredients of the algorithms, such as the choice of model, the function, and the numerical techniques for solving the (ill-conditioned) subproblems. The new methods will be formulated in a trust region framework, which allows a uniform treatment of the convex and non-convex cases. The research will be guided by a combination of numerical testing and convergence analysis. An Internet-based optimization system called NEOS has recently been developed by a group of investigators at Northwestern and Argonne. One of the objectives of NEOS is to allow users to solve optimization problems remotely and with minimal effort using modern web browsers. The third project in this proposal considers extending the capabilities of NEOS so that it can interact with modern modeling languages. A new constrained optimization server will also be developed. ***