Numerical invariants through convex relaxation and max-strategy iteration

@article{Gawlitza2014NumericalIT,
  title={Numerical invariants through convex relaxation and max-strategy iteration},
  author={Thomas Gawlitza and Helmut Seidl},
  journal={Formal Methods in System Design},
  year={2014},
  volume={44},
  pages={101-148}
}
We present an algorithm for computing the uniquely determined least fixpoints of self-maps on $\overline{\mathbb{R}}^{n}$ (with $\overline{\mathbb{R}} = \mathbb{R} \cup\{ \pm\infty\}$) that are point-wise maximums of finitely many monotone and order-concave self-maps. This natural problem occurs in the context of systems analysis and verification. As an example application we discuss how our method can be used to compute template-based quadratic invariants for linear systems with guards. The… 
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References

SHOWING 1-10 OF 34 REFERENCES
Coupling policy iteration with semi-definite relaxation to compute accurate numerical invariants in static analysis
TLDR
A new domain for finding precise numerical invariants of programs by abstract interpretation is introduced, which consists of level sets of non-linear functions and it is shown that the abstract fixpoint equation can be solved accurately by coupling policy iteration and semi-definite programming.
Precise Relational Invariants Through Strategy Iteration
TLDR
A practical algorithm is presented for computing exact least solutions of systems of equations over the rationals with addition, multiplication with positive constants, minimum and maximum and applied to compute the abstract least fixpoint semantics of affine programs over the relational template constraint matrix domain.
Solving systems of rational equations through strategy iteration
TLDR
The present article applies techniques for computing abstract least fixpoint semantics of affine programs over the relational template polyhedra domain to practical algorithms for computing exact least solutions of equation systems over the reals with addition, multiplication by positive constants, minimum and maximum.
A Policy Iteration Technique for Time Elapse over Template Polyhedra
TLDR
This work proposes a policy iteration technique that iterates over the space of invariant certificates to converge onto a solution that is close to the least solution, and incorporates the ideas in the prototype tool TimePass for safety verification of affine hybrid systems.
Precise Fixpoint Computation Through Strategy Iteration
TLDR
A practical algorithm for computing least solutions of systems of equations over the integers with addition, multiplication with positive constants, maximum and minimum, based on strategy iteration is presented.
Computing Relaxed Abstract Semantics w.r.t. Quadratic Zones Precisely
TLDR
A relaxed abstract semantics is used and a practical strategy improvement algorithm is presented for precisely computing least solutions of fixpoint equation systems, whose right-hand sides use order-concave operators and the maximum operator.
Scalable Analysis of Linear Systems Using Mathematical Programming
TLDR
The method generalizes similar analyses in the interval, octagon, and octahedra domains, without resorting to polyhedral manipulations, and demonstrates the performance of the method on some benchmark programs.
Abstract interpretation meets convex optimization
A New Numerical Abstract Domain Based on Difference-Bound Matrices
TLDR
This paper presents a new numerical abstract domain for static analysis by abstract interpretation that allows for invariants of the form (x - y ≤ c)an d (±x ≤ c), where x and y are variables values and c is an integer or real constant.
Iterative solution of nonlinear equations in several variables
TLDR
Convergence of Minimization Methods An Annotated List of Basic Reference Books Bibliography Author Index Subject Index.
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