When Can One Obtain Certificates of Optimality Using Positivstellensaetze?
Quick summary
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 satisf
Key takeaways
- 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 satisf
Why it matters
“When Can One Obtain Certificates of Optimality Using Positivstellensaetze?” highlights the need for repeatable measurement rather than a single impressive demonstration. Independent validation across datasets and clearly stated limitations determine whether a result can guide product decisions.
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