42,349 research outputs found
Shallow Circuits with High-Powered Inputs
A polynomial identity testing algorithm must determine whether an input
polynomial (given for instance by an arithmetic circuit) is identically equal
to 0. In this paper, we show that a deterministic black-box identity testing
algorithm for (high-degree) univariate polynomials would imply a lower bound on
the arithmetic complexity of the permanent. The lower bounds that are known to
follow from derandomization of (low-degree) multivariate identity testing are
weaker. To obtain our lower bound it would be sufficient to derandomize
identity testing for polynomials of a very specific norm: sums of products of
sparse polynomials with sparse coefficients. This observation leads to new
versions of the Shub-Smale tau-conjecture on integer roots of univariate
polynomials. In particular, we show that a lower bound for the permanent would
follow if one could give a good enough bound on the number of real roots of
sums of products of sparse polynomials (Descartes' rule of signs gives such a
bound for sparse polynomials and products thereof). In this third version of
our paper we show that the same lower bound would follow even if one could only
prove a slightly superpolynomial upper bound on the number of real roots. This
is a consequence of a new result on reduction to depth 4 for arithmetic
circuits which we establish in a companion paper. We also show that an even
weaker bound on the number of real roots would suffice to obtain a lower bound
on the size of depth 4 circuits computing the permanent.Comment: A few typos correcte
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
Recent Advances in Computational Methods for the Power Flow Equations
The power flow equations are at the core of most of the computations for
designing and operating electric power systems. The power flow equations are a
system of multivariate nonlinear equations which relate the power injections
and voltages in a power system. A plethora of methods have been devised to
solve these equations, starting from Newton-based methods to homotopy
continuation and other optimization-based methods. While many of these methods
often efficiently find a high-voltage, stable solution due to its large basin
of attraction, most of the methods struggle to find low-voltage solutions which
play significant role in certain stability-related computations. While we do
not claim to have exhausted the existing literature on all related methods,
this tutorial paper introduces some of the recent advances in methods for
solving power flow equations to the wider power systems community as well as
bringing attention from the computational mathematics and optimization
communities to the power systems problems. After briefly reviewing some of the
traditional computational methods used to solve the power flow equations, we
focus on three emerging methods: the numerical polynomial homotopy continuation
method, Groebner basis techniques, and moment/sum-of-squares relaxations using
semidefinite programming. In passing, we also emphasize the importance of an
upper bound on the number of solutions of the power flow equations and review
the current status of research in this direction.Comment: 13 pages, 2 figures. Submitted to the Tutorial Session at IEEE 2016
American Control Conferenc
Polynomial Optimization with Real Varieties
We consider the optimization problem of minimizing a polynomial f(x) subject
to polynomial constraints h(x)=0, g(x)>=0. Lasserre's hierarchy is a sequence
of sum of squares relaxations for finding the global minimum. Let K be the
feasible set. We prove the following results: i) If the real variety V_R(h) is
finite, then Lasserre's hierarchy has finite convergence, no matter the complex
variety V_C(h) is finite or not. This solves an open question in Laurent's
survey. ii) If K and V_R(h) have the same vanishing ideal, then the finite
convergence of Lasserre's hierarchy is independent of the choice of defining
polynomials for the real variety V_R(h). iii) When K is finite, a refined
version of Lasserre's hierarchy (using the preordering of g) has finite
convergence.Comment: 12 page
Relative Entropy Relaxations for Signomial Optimization
Signomial programs (SPs) are optimization problems specified in terms of
signomials, which are weighted sums of exponentials composed with linear
functionals of a decision variable. SPs are non-convex optimization problems in
general, and families of NP-hard problems can be reduced to SPs. In this paper
we describe a hierarchy of convex relaxations to obtain successively tighter
lower bounds of the optimal value of SPs. This sequence of lower bounds is
computed by solving increasingly larger-sized relative entropy optimization
problems, which are convex programs specified in terms of linear and relative
entropy functions. Our approach relies crucially on the observation that the
relative entropy function -- by virtue of its joint convexity with respect to
both arguments -- provides a convex parametrization of certain sets of globally
nonnegative signomials with efficiently computable nonnegativity certificates
via the arithmetic-geometric-mean inequality. By appealing to representation
theorems from real algebraic geometry, we show that our sequences of lower
bounds converge to the global optima for broad classes of SPs. Finally, we also
demonstrate the effectiveness of our methods via numerical experiments
- …