Optimization

In optimization, the main goal is to find the maximum or minimum of an objective function given some constraints and bounds for the input variables. In regards to engineering, the goal may be to minimize the weight of a structure given the constraint that the structure does not fail and is within the design limits.
​
Since carrying out optimization of engineering problems is computationally intensive, our group focuses on using the surrogate models such as Kriging, Radial basis function, polynomial chaos expansion, artificial neural networks, etc. for the objective function and constraints during optimization.

Once the surrogate models are built, the optimization can be run using either gradient-based solver such sequential quadratic programming or heuristic approaches such as genetic algorithm or particle swarm optimization.

Stochastic Optimization

While a huge progress has been in terms of optimization-finding the minimum or maximum of an objective function- considering deterministic parameters and design variables, very few progress has been in terms of design optimization while considering uncertainties. The goal is to consider the unforeseen factors during design optimization that might significantly affect the responses and hence the reliability of the design. Considering a stochastic based optimization also enhances the robustness of the design. However, performing stochastic optimization of complex systems is very challenging due to the requirement of estimation of the probability of failure which requires a substantially large number of function evaluations. The complexity further increases while considering a large number of constraints in the optimization problems.

​To tackle these issues, I have developed surrogate based stochastic optimization framework based on Kriging and support vector machines (SVMs). The Kriging model allows to create an accurate response surface for the objective function whereas the SVMs allows one to construct a single decision boundary based on multiple constraints. The single SVM built on multiple constraints further allow one to carry out sampling for the constraints efficient based on their criticality which lead to additional savings in terms of number of function evaluations.

The importance of stochastic optimization is demonstrated in the figure below where the optimal design (indicated by magenta box) is prone to exceeding the constraints (indicated by the blue circles) due to the presence of uncertainties. The constrains and the objective functions were modeled using the surrogates- Kriging and SVM.

The difference in problem formulation for the deterministic vs stochastic optimization is demonstrated below. The effect of randomness, z is considered during the stochastic optimization and the constraints are formulated in terms of either probability/reliability or robustness (mean and standard deviation) whereas the effect of randomness is ignored during the deterministic optimization.​
Please refer to the following paper on Surrogate based Stochastic Optimization of Wind Turbine Composite Blades for more details:
doi.org/10.1007/s00158-021-03114-8