First-Order Geometry, Spectral Compression, and Structural Compatibility under Bounded Computation

TL;DR

This paper introduces an operator-theoretic framework for constrained optimization, revealing how geometric constraints influence dynamics, spectral compression, and multi-objective compatibility in a unified manner.

Contribution

It proposes a novel operator-based approach to analyze constrained optimization, connecting geometric, spectral, and multi-objective aspects within a single framework.

Findings

Optimal first-order directions are pseudoinverse-weighted gradients.

Effective dynamics are concentrated on dominant spectral modes.

The framework unifies gradient projection, spectral truncation, and multi-objective feasibility.

Abstract

Optimization under structural constraints is typically analyzed through projection or penalty methods, obscuring the geometric mechanism by which constraints shape admissible dynamics. We propose an operator-theoretic formulation in which computational or feasibility limitations are encoded by self-adjoint operators defining locally reachable subspaces. In this setting, the optimal first-order improvement direction emerges as a pseudoinverse-weighted gradient, revealing how constraints induce a distorted ascent geometry. We further demonstrate that effective dynamics concentrate along dominant spectral modes, yielding a principled notion of spectral compression, and establish a compatibility principle that characterizes the existence of common admissible directions across multiple objectives. The resulting framework unifies gradient projection, spectral truncation, and multi-objective feasibility within a single geometric structure.