A Sums-of-Squares Extension of Policy Iterations

In order to address the imprecision often introduced by widening operators in static analysis, policy iteration based on min-computations amounts to considering the characterization of reachable value set of a program as an iterative computation of policies, starting from a post-fixpoint. Computing each policy and the associated invariant relies on a sequence of numerical optimizations. While the early research efforts relied on linear programming (LP) to address linear properties of linear programs, the current state of the art is still limited to the analysis of linear programs with at most quadratic invariants, relying on semidefinite programming (SDP) solvers to compute policies, and LP solvers to refine invariants. We propose here to extend the class of programs considered through the use of Sums-of-Squares (SOS) based optimization. Our approach enables the precise analysis of switched systems with polynomial updates and guards. The analysis presented has been implemented in Matlab and applied on existing programs coming from the system control literature, improving both the range of analyzable systems and the precision of previously handled ones.Comment: 29 pages, 4 figure

Certification of Bounds of Non-linear Functions: the Templates Method

The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of inequalities. We introduce an approximation algorithm, which combines ideas of the max-plus basis method (in optimal control) and of the linear templates method developed by Manna et al. (in static analysis). This algorithm consists in bounding some of the constituents of the function by suprema of quadratic forms with a well chosen curvature. This leads to semialgebraic optimization problems, solved by sum-of-squares relaxations. Templates limit the blow up of these relaxations at the price of coarsening the approximation. We illustrate the efficiency of our framework with various examples from the literature and discuss the interfacing with Coq.Comment: 16 pages, 3 figures, 2 table

Improving Strategies via SMT Solving

We consider the problem of computing numerical invariants of programs by abstract interpretation. Our method eschews two traditional sources of imprecision: (i) the use of widening operators for enforcing convergence within a finite number of iterations (ii) the use of merge operations (often, convex hulls) at the merge points of the control flow graph. It instead computes the least inductive invariant expressible in the domain at a restricted set of program points, and analyzes the rest of the code en bloc. We emphasize that we compute this inductive invariant precisely. For that we extend the strategy improvement algorithm of [Gawlitza and Seidl, 2007]. If we applied their method directly, we would have to solve an exponentially sized system of abstract semantic equations, resulting in memory exhaustion. Instead, we keep the system implicit and discover strategy improvements using SAT modulo real linear arithmetic (SMT). For evaluating strategies we use linear programming. Our algorithm has low polynomial space complexity and performs for contrived examples in the worst case exponentially many strategy improvement steps; this is unsurprising, since we show that the associated abstract reachability problem is Pi-p-2-complete

Certification of Real Inequalities -- Templates and Sums of Squares

We consider the problem of certifying lower bounds for real-valued multivariate transcendental functions. The functions we are dealing with are nonlinear and involve semialgebraic operations as well as some transcendental functions like $\cos$, $\arctan$, $\exp$, etc. Our general framework is to use different approximation methods to relax the original problem into polynomial optimization problems, which we solve by sparse sums of squares relaxations. In particular, we combine the ideas of the maxplus estimators (originally introduced in optimal control) and of the linear templates (originally introduced in static analysis by abstract interpretation). The nonlinear templates control the complexity of the semialgebraic relaxations at the price of coarsening the maxplus approximations. In that way, we arrive at a new - template based - certified global optimization method, which exploits both the precision of sums of squares relaxations and the scalability of abstraction methods. We analyze the performance of the method on problems from the global optimization literature, as well as medium-size inequalities issued from the Flyspeck project.Comment: 27 pages, 3 figures, 4 table

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

Formal Verification for Autopilot - Preliminary state of the art

This document is a preliminary state of the art for the formal verification of the autopilot of an Unmanned Air Vehicle (UAV). We will first present UAV autopilots and more specifically the Paparazzi autopilot developed at ENAC which will be our case study. We then present which properties could be verified and on which representation of the autopilot (source code, model). A more complete state of the art of current formal methods will be then detailed and focus on deductive methods, abstract interpretation, model checking and proof assistants. Finally, some immediate perspective for the thesis are proposed