In finance, the implied volatility surface is plotted against strike price and time to maturity. The shape of this volatility surface can be identified by fitting the model to what is actually observed in the market. The metric that is used to measure the discrepancy between the model and the market is usually defined by a mean squares of error of the model prices to the market prices. A regularization term can be added to this error metric to make the solution possess some desired properties. The discrepancy that we want to minimize is usually a highly nonlinear function of a set of model parameters with the regularization term. Typically monotonic decreasing algorithm is adopted to solve this minimization problem. Steepest descent or Newton type algorithms are two iterative methods but they are local, i.e., they use derivative information around the current iterate to find the next iterate. In order to ensure convergence, line search and trust region methods are two widely used globalization techniques. Motivated by the simplicity of Barzilai-Borwein method and the convergence properties brought by globalization techniques, we propose a new Scaled Gradient (SG) method for minimizing a differentiable function plus an L1-norm. This non-monotone iterative method only requires gradient information and safeguarded Barzilai-Borwein steplength is used in each iteration. An adaptive line search with the Armijo-type condition check is performed in each iteration to ensure convergence. Coleman, Li and Wang proposed another trust region approach in solving the same problem. We give a theoretical proof of the convergence of their algorithm. The objective of this thesis is to numerically investigate the performance of the SG method and establish global and local convergence properties of Coleman, Li and Wang’s trust region method proposed in [26]. Some future research directions are also given at the end of this thesis.

Practical aerodynamic shape design problems must balance performance optimization over a range of on-design operating conditions with constraint satisfaction at off-design operating conditions. A multipoint optimization formulation can be used to represent on-design and off-design conditions with corresponding objective or constraint functions. Two methods are presented for obtaining optimal airfoil designs that satisfy all design objectives and constraints. The first method uses an unconstrained optimization algorithm where optimal design is achieved by minimizing a weighted sum of objective functions at each of the conditions. To address competing design objectives between on-design and off-design conditions, an automated procedure is used to weight off-design objective functions to limit their influence on the overall optimization. The second method uses the constrained optimization algorithm SNOPT, allowing aerodynamic constraints imposed at off-design conditions to be treated explicitly. Both methods are applied to the design of an airfoil for a hypothetical aircraft, which is formulated as an 18-point multipoint optimization.

Practical aerodynamic shape design problems must balance performance optimization over a range of on-design operating conditions with constraint satisfaction at off-design operating conditions. A multipoint optimization formulation can be used to represent on-design and off-design conditions with corresponding objective or constraint functions. Two methods are presented for obtaining optimal airfoil designs that satisfy all design objectives and constraints. The first method uses an unconstrained optimization algorithm where optimal design is achieved by minimizing a weighted sum of objective functions at each of the conditions. To address competing design objectives between on-design and off-design conditions, an automated procedure is used to weight off-design objective functions to limit their influence on the overall optimization. The second method uses the constrained optimization algorithm SNOPT, allowing aerodynamic constraints imposed at off-design conditions to be treated explicitly. Both methods are applied to the design of an airfoil for a hypothetical aircraft, which is formulated as an 18-point multipoint optimization.

Better knowledge in the areas of aerodynamics and optimization has allowed designers to develop efficient wingtip structures in recent years. However, the requirements faced by wingtip devices can be considerably different amongst an aircraft’s flight regimes. Traditional static wingtip devices are then a compromise between conflicting requirements, resulting in less than optimal performance within each regime. Alternatively, a morphing wingtip can reconfigure leading to improved performance over a range of dissimilar flight conditions. Developed within this thesis, is a modular morphing wingtip concept that centers on the use of variable geometry truss mechanisms to permit morphing. A conceptual design framework is established to aid in the development of the concept. The framework uses a metaheuristic optimization procedure to determine optimal continuous wingtip configurations. The configurations are then discretized for the modular concept. The functionality of the framework is demonstrated through a design study on a hypothetical wing/winglet within the thesis.

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