Extracting Optimal Solution Manifolds using Constrained Neural Optimization

TL;DR

This paper introduces neural methods to extract approximate solution manifolds for constrained optimization problems with non-convex objectives and constraints, enhancing interpretability and efficiency over traditional solvers.

Contribution

It proposes a novel neural approach that directly models non-convex objectives and constraints as loss functions to extract solution manifolds, bypassing convexification limitations.

Findings

Effective in synthetic and real-world scenarios

Achieves competitive accuracy with improved efficiency

Provides interpretable solutions aligned with analytical forms

Abstract

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.