17,866 research outputs found
A new look at nonnegativity on closed sets and polynomial optimization
We first show that a continuous function f is nonnegative on a closed set
if and only if (countably many) moment matrices of some signed
measure with support equal to K, are all positive semidefinite
(if is compact is an arbitrary finite Borel measure with support
equal to K. In particular, we obtain a convergent explicit hierarchy of
semidefinite (outer) approximations with {\it no} lifting, of the cone of
nonnegative polynomials of degree at most . Wen used in polynomial
optimization on certain simple closed sets \K (like e.g., the whole space
, the positive orthant, a box, a simplex, or the vertices of the
hypercube), it provides a nonincreasing sequence of upper bounds which
converges to the global minimum by solving a hierarchy of semidefinite programs
with only one variable. This convergent sequence of upper bounds complements
the convergent sequence of lower bounds obtained by solving a hierarchy of
semidefinite relaxations
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
Linearly Solvable Stochastic Control Lyapunov Functions
This paper presents a new method for synthesizing stochastic control Lyapunov
functions for a class of nonlinear stochastic control systems. The technique
relies on a transformation of the classical nonlinear Hamilton-Jacobi-Bellman
partial differential equation to a linear partial differential equation for a
class of problems with a particular constraint on the stochastic forcing. This
linear partial differential equation can then be relaxed to a linear
differential inclusion, allowing for relaxed solutions to be generated using
sum of squares programming. The resulting relaxed solutions are in fact
viscosity super/subsolutions, and by the maximum principle are pointwise upper
and lower bounds to the underlying value function, even for coarse polynomial
approximations. Furthermore, the pointwise upper bound is shown to be a
stochastic control Lyapunov function, yielding a method for generating
nonlinear controllers with pointwise bounded distance from the optimal cost
when using the optimal controller. These approximate solutions may be computed
with non-increasing error via a hierarchy of semidefinite optimization
problems. Finally, this paper develops a-priori bounds on trajectory
suboptimality when using these approximate value functions, as well as
demonstrates that these methods, and bounds, can be applied to a more general
class of nonlinear systems not obeying the constraint on stochastic forcing.
Simulated examples illustrate the methodology.Comment: Published in SIAM Journal of Control and Optimizatio
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