🤖 AI Summary
This work addresses the challenge of characterizing how constraints shape the geometric structure of feasible regions in constrained optimization. The authors propose a geometric framework grounded in self-adjoint operators, which constructs local reachable subspaces by encoding computational or feasibility constraints and yields a pseudoinverse-weighted gradient as the optimal first-order update direction. This approach unifies the treatment of projection, spectral truncation, and multi-objective feasibility, revealing a constraint-induced warped ascent geometry. It establishes a compatibility principle between spectral compression and multi-objective structures, with the algorithm dynamically focusing on dominant spectral modes to provide a unified geometric characterization of gradient projection, spectral compression, and multi-objective feasible directions.
📝 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.