Extracting Optimal Solution Manifolds using Constrained Neural Optimization
This addresses the need for more flexible and interpretable solutions in domains like hyperspectral unmixing and Pareto fronts, though it appears incremental as it builds on existing neural optimization approaches.
The paper tackles the problem of constrained optimization with non-convex objectives and constraints, which often yields multiple optimal solutions, by proposing a neural method to extract approximate manifolds of optimal sets, validated with synthetic and realistic cases for accuracy and efficiency.
Constrained Optimization solution algorithms are restricted to point based solutions. In practice, single or multiple objectives must be satisfied, wherein both the objective function and constraints can be non-convex resulting in multiple optimal solutions. Real world scenarios include intersecting surfaces as Implicit Functions, Hyperspectral Unmixing and Pareto Optimal fronts. Local or global convexification is a common workaround when faced with non-convex forms. However, such an approach is often restricted to a strict class of functions, deviation from which results in sub-optimal solution to the original problem. We present neural solutions for extracting optimal sets as approximate manifolds, where unmodified, non-convex objectives and constraints are defined as modeler guided, domain-informed $L_2$ loss function. This promotes interpretability since modelers can confirm the results against known analytical forms in their specific domains. We present synthetic and realistic cases to validate our approach and compare against known solvers for bench-marking in terms of accuracy and computational efficiency.