Algorithms in Real Algebraic Geometry

Abstract

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Generic Optimality Conditions for Semialgebraic Convex Programs

“…The Tarski-Seidenberg Theorem [2,8] shows thatΦ is semi-algebraic. Let L denote the subset of R n asserted in Corollary 6.…”

“…In particular, every semi-algebraic set can be written as a finite disjoint union of manifolds (or "strata") that fit together in a regular "stratification": see [2], for example, or the exposition in [42, §4.2]. In particular, the dimension of a semi-algebraic set is the maximum of the dimensions of the strata, a number independent of the stratification: see [8, Definition 9.14] for more details.…”

“…The fact that the map c → x c is semi-algebraic follows by a standard argument using the Tarski-Seidenberg Theorem [2,8]. The existence of a dense open semi-algebraic set G ⊂ R n on which this map is C k follows by applying the stratification result.…”

“…For example, the feasible region of any semidefinite program is semi-algebraic. A good resource on semi-algebraic geometry is [2].…”

Reasoning About States of Probabilistic Sequential Programs

“…In order to achieve recursive axiomatization for EPPL, we also assume that the measures take values from an arbitrary real closed field instead of the set of real numbers. The first order theory of such fields is decidable [12,3], and this technique of achieving decidability was inspired by other work in probabilistic reasoning [7,1].…”