Plane Based Optimization Method: A Method for Better Global Minimum Cost Function Localization

Background: As a popular optimization algorithm, gradient descent is more prone to converge at a local minimum rather than a global one since it is highly dependent on how initial values are instantiated and an iterative process of updates, reducing its efficiency for non-convex cost functions. This limitation is of utmost importance in domains such as medical physics, where there exist many treatment parameters that need to be optimized. Purpose: We introduce the flat plane with maximum perpendicular distance method, a new optimization algorithm which identifies global minima robustly and efficiently in convex cost functions. This facilitates an initialization-free, non-iterative update method that serves as a complementary approach to classic methods. Methods: 1D and 2D convex cost functions, with both global and relative minima, were tested using the flat plane method. Further, the approach was employed to determine optimal locations for radiation beams in a medical physics problem. Results were compared to existing gradient descent methods for effectiveness assessment. Results: The flat plane algorithm found the global minimum in all the scenarios considered while the use of gradient descent would have failed to do so in many cases due to initialization issues. However, due to a less direct but highly parallelizable strategy (i.e. flat plane method), these techniques are certainly scalable for high-dimensional problems with potentially higher expense (e.g. dimensionality reduction, sampling, parallel computing). Conclusion: In low-dimensional problems, the flat plane method provides a strong alternative to gradient descent for the global optimization problem. We are yet to adapt the method for high-dimensional applications but will work in that direction to increase its applicability over more complex optimization problems