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

Formal Proofs for Nonlinear Optimization

We present a formally verified global optimization framework. Given a semialgebraic or transcendental function $f$ and a compact semialgebraic domain $K$, we use the nonlinear maxplus template approximation algorithm to provide a certified lower bound of $f$ over $K$. This method allows to bound in a modular way some of the constituents of $f$ by suprema of quadratic forms with a well chosen curvature. Thus, we reduce the initial goal to a hierarchy of semialgebraic optimization problems, solved by sums of squares relaxations. Our implementation tool interleaves semialgebraic approximations with sums of squares witnesses to form certificates. It is interfaced with Coq and thus benefits from the trusted arithmetic available inside the proof assistant. This feature is used to produce, from the certificates, both valid underestimators and lower bounds for each approximated constituent. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of multivariate transcendental inequalities. We illustrate the performance of our formal framework on some of these inequalities as well as on examples from the global optimization literature.Comment: 24 pages, 2 figures, 3 table

A Logical Product Approach to Zonotope Intersection

We define and study a new abstract domain which is a fine-grained combination of zonotopes with polyhedric domains such as the interval, octagon, linear templates or polyhedron domain. While abstract transfer functions are still rather inexpensive and accurate even for interpreting non-linear computations, we are able to also interpret tests (i.e. intersections) efficiently. This fixes a known drawback of zonotopic methods, as used for reachability analysis for hybrid sys- tems as well as for invariant generation in abstract interpretation: intersection of zonotopes are not always zonotopes, and there is not even a best zonotopic over-approximation of the intersection. We describe some examples and an im- plementation of our method in the APRON library, and discuss some further in- teresting combinations of zonotopes with non-linear or non-convex domains such as quadratic templates and maxplus polyhedra

The Traffic Phases of Road Networks

We study the relation between the average traffic flow and the vehicle density on road networks that we call 2D-traffic fundamental diagram. We show that this diagram presents mainly four phases. We analyze different cases. First, the case of a junction managed with a priority rule is presented, four traffic phases are identified and described, and a good analytic approximation of the fundamental diagram is obtained by computing a generalized eigenvalue of the dynamics of the system. Then, the model is extended to the case of two junctions, and finally to a regular city. The system still presents mainly four phases. The role of a critical circuit of non-priority roads appears clearly in the two junctions case. In Section 4, we use traffic light controls to improve the traffic diagram. We present the improvements obtained by open-loop, local feedback, and global feedback strategies. A comparison based on the response times to reach the stationary regime is also given. Finally, we show the importance of the design of the junction. It appears that if the junction is enough large, the traffic is almost not slowed down by the junction.Comment: 37 page

The tropical double description method

We develop a tropical analogue of the classical double description method allowing one to compute an internal representation (in terms of vertices) of a polyhedron defined externally (by inequalities). The heart of the tropical algorithm is a characterization of the extreme points of a polyhedron in terms of a system of constraints which define it. We show that checking the extremality of a point reduces to checking whether there is only one minimal strongly connected component in an hypergraph. The latter problem can be solved in almost linear time, which allows us to eliminate quickly redundant generators. We report extensive tests (including benchmarks from an application to static analysis) showing that the method outperforms experimentally the previous ones by orders of magnitude. The present tools also lead to worst case bounds which improve the ones provided by previous methods.Comment: 12 pages, prepared for the Proceedings of the Symposium on Theoretical Aspects of Computer Science, 2010, Nancy, Franc

Towards a generalisation of formal concept analysis for data mining purposes

In this paper we justify the need for a generalisation of Formal Concept Analysis for the purpose of data mining and begin the synthesis of such theory. For that purpose, we first review semirings and semimodules over semirings as the appropriate objects to use in abstracting the Boolean algebra and the notion of extents and intents, respectively. We later bring to bear powerful theorems developed in the field of linear algebra over idempotent semimodules to try to build a Fundamental Theorem for K-Formal Concept Analysis, where K is a type of idempotent semiring. Finally, we try to put Formal Concept Analysis in new perspective by considering it as a concrete instance of the theory developed

About Dynamical Systems Appearing in the Microscopic Traffic Modeling

Motivated by microscopic traffic modeling, we analyze dynamical systems which have a piecewise linear concave dynamics not necessarily monotonic. We introduce a deterministic Petri net extension where edges may have negative weights. The dynamics of these Petri nets are well-defined and may be described by a generalized matrix with a submatrix in the standard algebra with possibly negative entries, and another submatrix in the minplus algebra. When the dynamics is additively homogeneous, a generalized additive eigenvalue may be introduced, and the ergodic theory may be used to define a growth rate under additional technical assumptions. In the traffic example of two roads with one junction, we compute explicitly the eigenvalue and we show, by numerical simulations, that these two quantities (the additive eigenvalue and the growth rate) are not equal, but are close to each other. With this result, we are able to extend the well-studied notion of fundamental traffic diagram (the average flow as a function of the car density on a road) to the case of two roads with one junction and give a very simple analytic approximation of this diagram where four phases appear with clear traffic interpretations. Simulations show that the fundamental diagram shape obtained is also valid for systems with many junctions. To simulate these systems, we have to compute their dynamics, which are not quite simple. For building them in a modular way, we introduce generalized parallel, series and feedback compositions of piecewise linear concave dynamics.Comment: PDF 38 page