77 research outputs found
Sharper and Simpler Nonlinear Interpolants for Program Verification
Interpolation of jointly infeasible predicates plays important roles in
various program verification techniques such as invariant synthesis and CEGAR.
Intrigued by the recent result by Dai et al.\ that combines real algebraic
geometry and SDP optimization in synthesis of polynomial interpolants, the
current paper contributes its enhancement that yields sharper and simpler
interpolants. The enhancement is made possible by: theoretical observations in
real algebraic geometry; and our continued fraction-based algorithm that rounds
off (potentially erroneous) numerical solutions of SDP solvers. Experiment
results support our tool's effectiveness; we also demonstrate the benefit of
sharp and simple interpolants in program verification examples
Encoding inductive invariants as barrier certificates: synthesis via difference-of-convex programming
A barrier certificate often serves as an inductive invariant that isolates an
unsafe region from the reachable set of states, and hence is widely used in
proving safety of hybrid systems possibly over an infinite time horizon. We
present a novel condition on barrier certificates, termed the invariant
barrier-certificate condition, that witnesses unbounded-time safety of
differential dynamical systems. The proposed condition is the weakest possible
one to attain inductive invariance. We show that discharging the invariant
barrier-certificate condition -- thereby synthesizing invariant barrier
certificates -- can be encoded as solving an optimization problem subject to
bilinear matrix inequalities (BMIs). We further propose a synthesis algorithm
based on difference-of-convex programming, which approaches a local optimum of
the BMI problem via solving a series of convex optimization problems. This
algorithm is incorporated in a branch-and-bound framework that searches for the
global optimum in a divide-and-conquer fashion. We present a weak completeness
result of our method, namely, a barrier certificate is guaranteed to be found
(under some mild assumptions) whenever there exists an inductive invariant (in
the form of a given template) that suffices to certify safety of the system.
Experimental results on benchmarks demonstrate the effectiveness and efficiency
of our approach.Comment: To be published in Inf. Comput. arXiv admin note: substantial text
overlap with arXiv:2105.1431
A Non-linear Arithmetic Procedure for Control-Command Software Verification
International audienceState-of-the-art (semi-)decision procedures for non-linear real arithmetic address polynomial inequalities by mean of symbolic methods, such as quantifier elimination, or numerical approaches such as interval arithmetic. Although (some of) these methods offer nice completeness properties, their high complexity remains a limit, despite the impressive efficiency of modern implementations. This appears to be an obstacle to the use of SMT solvers when verifying, for instance, functional properties of control-command programs. Using off-the-shelf convex optimization solvers is known to constitute an appealing alternative. However, these solvers only deliver approximate solutions, which means they do not readily provide the soundness expected for applications such as software verification. We thus investigate a-posteriori validation methods and their integration in the SMT framework. Although our early prototype, implemented in the Alt-Ergo SMT solver, often does not prove competitive with state of the art solvers, it already gives some interesting results, particularly on control-command programs
Validation of Convex Optimization Algorithms and Credible Implementation for Model Predictive Control
Advanced real-time embedded algorithms are growing in complexity and length, related to the growth in autonomy, which allows vehicles to plan paths of their own. However,
this promise cannot happen without proper attention to the considerably stronger operational constraints that real time, safety-critical applications must meet. This paper discusses the formal verification for optimization algorithms with a particular emphasis on receding-horizon controllers. Following a brief historical overview, a prototype autocoder for embedded convex optimization algorithms is discussed. Options for encoding code properties and proofs, and their applicability and limitations is detailed as well
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Non-Convex Optimization and Applications to Bilinear Programming and Super-Resolution Imaging
Bilinear programs and Phase Retrieval are two instances of nonconvex problems that arise in engineering and physical applications, and both occur with their fundamental difficulties. In this thesis, we consider various methods and algorithms for tackling these challenging problems and discuss their effectiveness. Bilinear programs (BLPs) are ubiquitous in engineering applications, economics, and operations research, and have a natural encoding to quadratic programs. They appear in the study of Lyapunov functions used to deduce the stability of solutions to differential equations describing dynamical systems. For multivariate dynamical systems, the problem formulation for computing an appropriate Lyapunov function is a BLP. In electric power systems engineering, one of the most practically important and well-researched subfields of constrained nonlinear optimization is Optimal Power Flow wherein one attempts to optimize an electric power system subject to physical constraints imposed by electrical laws and engineering limits, which can be naturally formulated as a quadratic program. In a recent publication, we studied the relationship between data flow constraints for numerical domains such as polyhedra and bilinear constraints. The problem of recovering an image from its Fourier modulus, or intensity, measurements emerges in many physical and engineering applications. The problem is known as Fourier phase retrieval wherein one attempts to recover the phase information of a signal in order to accurately reconstruct it from estimated intensity measurements by applying the inverse Fourier transform. The problem of recovering phase information from a set of measurements can be formulated as a quadratic program. This problem is well-studied but still presents many challenges. The resolution of an optical device is defined as the smallest distance between two objects such that the two objects can still be recognized as separate entities. Due to the physics of diffraction, and the way that light bends around an obstacle, the resolving power of an optical system is limited. This limit, known as the diffraction limit, was first introduced by Ernst Abbe in 1873. Obtaining the complete phase information would enable one to perfectly reconstruct an image; however, the problem is severely ill-posed and the leads to a specialized type of quadratic program, known as super-resolution imaging, wherein one attempts to learn phase information beyond the limits of diffraction and the limitations imposed by the imaging device
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