Nonlinear Programming - Robust Models and Applications

Doctoral thesis, 2009

The major theme of this thesis is nonlinear programming with an emphasis on applications and robust models. The thesis has two parts. The first three papers comprise the first part. Here, we discuss robustness properties of optimal solutions to a variety of models. The first two papers concern optimization models known as Stochastic Mathematical Programs with Equilibrium Constraints (SMPEC). These are stochastic optimization problems that have two levels of ``decisions'': a lower-level one and an upper-level one. The lower-level problem is in the form of a variational inequality, and the upper-level objective function is either the expected value of an objective or the Conditional Value-at-Risk (CVaR). We also consider multiple objective extensions of the SMPEC framework. The stability of optimal solutions due to changes in the underlying probability distribution is analyzed. We also present two applications together with numerical examples: Intensity Modulated Radiation Therapy (IMRT) and traffic network design. In the third paper, we consider a nonlinear program with multiple objectives which are subjected to uncertainty in the variables and in the parameters. Here we do not use a stochastic programming approach, but instead we wish to analyze robustness as a post-process. Given a particular decision maker, we use his or her preferences to assess the robustness of optimal solutions. This is accomplished through the construction of a utility function which reduces the multi-objective problem into a single-objective problem. The second part of the thesis, corresponding to the fourth paper, concerns the problem of numerically folding an airbag. We approximate the airbag geometry by a quasi-cylindrical polyhedron, and we show how Origami mathematics can be used to derive a folding pattern that will collapse the polyhedron. The actual folding problem is solved through the formulation of a nonlinear program whose optimal solution corresponds to the coordinates of the vertices of the flattened polyhedron. The method is demonstrated on a computer model of a passenger airbag.