Gradient based algorithms, in general, tend to converge to local optima. However, the local optima found by these algorithms are not guaranteed to be the best solution (the global optima). There may be more local optima in the design space that are not found by a single run of the gradient based algorithm. Gradient Based Multi-Start is a method to address this problem of multiple local optima.
Description
GBMS runs a gradient based algorithm from multiple starting points to find out if multiple local optima exist.
The GBMS algorithm implemented with Optima2D and Jetstream uses the Sobol sequence to generate multiple starting points that span the design space. The gradient based optimization is run starting from each point to obtain information on:
the number of local optima
the relative success of each optima (value of objective function) and
how likely each local optima is found from a random starting point
The Sobol sequence used allows the results from GBMS to be deterministic (so the results can be reproduced); and allows the method to be extended to a larger sample of starting points without needing to re-run the first few starting points.
Footnotes
The design space refers to the space of all possible air-foil/wing shapes specified by a parametrization. In the case of 2D and 3D shapes, this design space can be represented by bounds on the control points of the 2D b-spline curve or 3D b-spline surface.
The local optima are optimized shapes that correspond to local minima of the objective function (the objective function is a function of the design variables which usually include b-spline control points).
Purpose
Gradient based algorithms, in general, tend to converge to local optima. However, the local optima found by these algorithms are not guaranteed to be the best solution (the global optima). There may be more local optima in the design space that are not found by a single run of the gradient based algorithm. Gradient Based Multi-Start is a method to address this problem of multiple local optima.Description
GBMS runs a gradient based algorithm from multiple starting points to find out if multiple local optima exist.The GBMS algorithm implemented with Optima2D and Jetstream uses the Sobol sequence to generate multiple starting points that span the design space. The gradient based optimization is run starting from each point to obtain information on:
- the number of local optima
- the relative success of each optima (value of objective function) and
- how likely each local optima is found from a random starting point
The Sobol sequence used allows the results from GBMS to be deterministic (so the results can be reproduced); and allows the method to be extended to a larger sample of starting points without needing to re-run the first few starting points.Footnotes