Polyhedral Approximation of Multivariate Polynomials Using Handelman’s Theorem

Abstract

Convex polyhedra are commonly used in the static analysis of programs to represent over-approximations of sets of reachable states of numerical program variables. When the analyzed programs contain nonlinear instructions, they do not directly map to standard polyhedral operations: some kind of linearization is needed. Convex polyhedra are also used in satisfiability modulo theory solvers which combine a propositional satisfiability solver with a fast emptiness check for polyhedra. Existing decision procedures become expensive when nonlinear constraints are involved: a fast procedure to ensure emptiness of systems of nonlinear constraints is needed. We present a new linearization algorithm based on Handelman's representation of positive polynomials. Given a polyhedron and a polynomial (in)equality, we compute a polyhedron enclosing their intersection as the solution of a parametric linear programming problem. To get a scalable algorithm, we provide several heuristics that guide the construction of the Handelman's representation. To ensure the correctness of our polyhedral approximation, our ocaml implementation generates certificates verified by a checker certified in coq.

Scalable Minimizing-Operators on Polyhedra via Parametric Linear Programming

“…Naive Fourier-Motzkin elimination produces O ( 12 Removing the redundant constraints is costly, even though there exists improved algorithms [21].…”

“…Our linearization operator for computing a polyhedral over-approximation of a conjunctions of linear and polynomial constraints i g i (x) ≥ 0 is also implemented in the VPL via PLP [21]. However, it does not prevent redundancies as we do not know how to provide a normalization point satisfying i g i (x) ≥ 0.…”

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A CDCL-Style Calculus for Solving Non-linear Constraints

“…The theory of computable functions has been developed in Computable Analysis [32] with implementations provided by exact real arithmetic [24]. Linearisations have been employed in different SMT theories before, including NRA and a recently considered one with transcendental functions [21,29,4], however, not for the broad class we consider here. We define a general class of functions called functions with decidable rational approximations to which our approach is applicable.…”