73 research outputs found
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
Tropical polar cones, hypergraph transversals, and mean payoff games
We discuss the tropical analogues of several basic questions of convex
duality. In particular, the polar of a tropical polyhedral cone represents the
set of linear inequalities that its elements satisfy. We characterize the
extreme rays of the polar in terms of certain minimal set covers which may be
thought of as weighted generalizations of minimal transversals in hypergraphs.
We also give a tropical analogue of Farkas lemma, which allows one to check
whether a linear inequality is implied by a finite family of linear
inequalities. Here, the certificate is a strategy of a mean payoff game. We
discuss examples, showing that the number of extreme rays of the polar of the
tropical cyclic polyhedral cone is polynomially bounded, and that there is no
unique minimal system of inequalities defining a given tropical polyhedral
cone.Comment: 27 pages, 6 figures, revised versio
Certification of inequalities involving transcendental functions: combining SDP and max-plus approximation
We consider the problem of certifying an inequality of the form ,
, where is a multivariate transcendental function, and
is a compact semialgebraic set. We introduce a certification method, combining
semialgebraic optimization and max-plus approximation. We assume that is
given by a syntaxic tree, the constituents of which involve semialgebraic
operations as well as some transcendental functions like , ,
, etc. We bound some of these constituents by suprema or infima of
quadratic forms (max-plus approximation method, initially introduced in optimal
control), leading to semialgebraic optimization problems which we solve by
semidefinite relaxations. The max-plus approximation is iteratively refined and
combined with branch and bound techniques to reduce the relaxation gap.
Illustrative examples of application of this algorithm are provided, explaining
how we solved tight inequalities issued from the Flyspeck project (one of the
main purposes of which is to certify numerical inequalities used in the proof
of the Kepler conjecture by Thomas Hales).Comment: 7 pages, 3 figures, 3 tables, Appears in the Proceedings of the
European Control Conference ECC'13, July 17-19, 2013, Zurich, pp. 2244--2250,
copyright EUCA 201
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 , , , 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 and a compact semialgebraic domain
, we use the nonlinear maxplus template approximation algorithm to provide a
certified lower bound of over . This method allows to bound in a modular
way some of the constituents of 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
Tropicalizing the simplex algorithm
We develop a tropical analog of the simplex algorithm for linear programming.
In particular, we obtain a combinatorial algorithm to perform one tropical
pivoting step, including the computation of reduced costs, in O(n(m+n)) time,
where m is the number of constraints and n is the dimension.Comment: v1: 35 pages, 7 figures, 4 algorithms; v2: improved presentation, 39
pages, 9 figures, 4 algorithm
Minimal external representations of tropical polyhedra
Tropical polyhedra are known to be representable externally, as intersections
of finitely many tropical half-spaces. However, unlike in the classical case,
the extreme rays of their polar cones provide external representations
containing in general superfluous half-spaces. In this paper, we prove that any
tropical polyhedral cone in R^n (also known as "tropical polytope" in the
literature) admits an essentially unique minimal external representation. The
result is obtained by establishing a (partial) anti-exchange property of
half-spaces. Moreover, we show that the apices of the half-spaces appearing in
such non-redundant external representations are vertices of the cell complex
associated with the polyhedral cone. We also establish a necessary condition
for a vertex of this cell complex to be the apex of a non-redundant half-space.
It is shown that this condition is sufficient for a dense class of polyhedral
cones having "generic extremities".Comment: v1: 32 pages, 10 figures; v2: minor revision, 34 pages, 10 figure
Tropical complementarity problems and Nash equilibria
Linear complementarity programming is a generalization of linear programming
which encompasses the computation of Nash equilibria for bimatrix games. While
the latter problem is PPAD-complete, we show that the tropical analogue of the
complementarity problem associated with Nash equilibria can be solved in
polynomial time. Moreover, we prove that the Lemke--Howson algorithm carries
over the tropical setting and performs a linear number of pivots in the worst
case. A consequence of this result is a new class of (classical) bimatrix games
for which Nash equilibria computation can be done in polynomial time
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