Set Separation Problems and Global Optimization.

Area of Science:

• Computational Geometry
• Optimization Theory
• Machine Learning

Background:

• The problem of separating two finite, disjoint sets A and B in R^n is fundamental with broad applications.
• While simple separation is always possible (e.g., using Voronoi partitions), linear or quadratic separability is often preferred.
• Existing methods may yield numerous separators when sets are separable, necessitating a 'robust' or 'best' separator.

Purpose of the Study:

• To examine optimization problems associated with the set separation problem.
• To characterize these optimization problems as convex or non-convex.
• To propose algorithms for solving set separation and robust separation problems.

Main Methods:

• Analysis of optimization problem formulations for set separation.
• Characterization of problem convexity (convex or non-convex).
• Development and suggestion of algorithms for finding optimal separators.

Main Results:

• Identified and characterized the optimization problems inherent in set separation.
• Provided a framework for determining separability (linear, quadratic, or otherwise).
• Proposed algorithmic approaches for finding robust separators or best approximations.

Conclusions:

• The study provides a comprehensive examination of set separation optimization.
• It offers methods for both finding optimal separators when sets are separable and approximating them when they are not.
• The proposed algorithms aim to address the need for robust solutions in geometric set separation tasks.