发表机构
Czech Technical University in Prague（布拉格捷克技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提炼Fischer Positivstellensätze的公理核心，在有序域抽象函数代数上证明正性与最优性证书定理，适用于非多项式目标与约束，并给出实例及复杂度分析。
AI 中文摘要
我们研究学习问题中的正性和最优性证书，其中目标函数和约束函数不必是多项式。我们提炼了Fischer构造性严格与弱Positivstellensätze的公理核心，并在有序域上的抽象函数代数上证明了所得定理。该框架区分了可能被混淆的两个角色：目标函数和约束函数可由广泛的连续或可定义运算构建，而用于构造证书的辅助原语则满足显式的标量和闭包公理。我们给出了连续和可定义函数代数上的实例，包括对平方根不封闭的有序域，推导出下界和全局最优性证书，并分析了扩展项长度和共享计算图复杂度。
英文摘要
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 Positivstellensätze 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.
