arXiv:2609.08736v1 Announce Type: new
Abstract: We study certificates of positivity and optimality for learning problems whose objectives and constraints need not be polynomial. We isolate an axiomatic core of Fischer's constructive strict and weak Positivstellens\"{a}tze and prove the resulting theorems for abstract function algebras over ordered fields. The framework separates two roles that can otherwise be conflated: objective and constraint functions may be built from broad classes of continuous or definable operations, while the auxiliary primitives used to construct a certificate satisfy explicit scalar and closure axioms. We give instances over continuous and definable function algebras, including ordered fields not closed under square roots, derive lower-bound and global-optimality certificates, and analyze both expanded term length and shared computation-graph complexity.
Abstract: We study certificates of positivity and optimality for learning problems whose objectives and constraints need not be polynomial. We isolate an axiomatic core of Fischer's constructive strict and weak Positivstellens\"{a}tze and prove the resulting theorems for abstract function algebras over ordered fields. The framework separates two roles that can otherwise be conflated: objective and constraint functions may be built from broad classes of continuous or definable operations, while the auxiliary primitives used to construct a certificate satisfy explicit scalar and closure axioms. We give instances over continuous and definable function algebras, including ordered fields not closed under square roots, derive lower-bound and global-optimality certificates, and analyze both expanded term length and shared computation-graph complexity.
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