Practical Techniques for Nonlinear Optimization

Long Hei

doi:https://doi.org/10.21985/N2940P

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Practical Techniques for Nonlinear Optimization

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At the heart of nonlinear optimization methods lies the solution of linear systems of equations. As the size of the problem increases, it is imperative to use iterative methods, such as the conjugate gradient algorithm, to solve these linear systems. In the context of constrained optimization, it has proved to be effective to use the projected conjugate gradient method to solve the equality constrained quadratic subproblems used to generate a step. In the first part of this thesis we present new preconditioners for the projected conjugate gradient method used in interior point methods. They fall under the category of "constraint preconditioners" and make use of the so-called condensed system to exploit the structure arising in interior point methods. We also give attention to the requirements imposed by trust region techniques. In the second part of the thesis we study nonlinear programming formulations and numerical methods for solving a strategic bidding problem in a short term electricity market. This problem can be formulated as a bilevel program whose lower level problem is linear. We analyze the properties of the constraints of the strong duality formulation of this bilevel program and show that the LICQ and MFCQ constraint qualification conditions fail at every feasible point. Therefore, even finding local minimizers is a difficult problem. We then consider the use of nonlinear programming algorithms for solving the bilevel program. They are appealing because they are well suited for large problems, but since the objective function of the strategic bidding problem is discontinuous and has many local minima, we consider heuristics (based on multistart) to attempt to find a solution. We report good solutions in acceptable time. The thesis concludes with some observations about the practical limitations of filter techniques.

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