40 research outputs found
On the Complexity of Optimization over the Standard Simplex
We review complexity results for minimizing polynomials over the standard simplex and unit hypercube.In addition, we show that there exists a polynomial time approximation scheme (PTAS) for minimizing Lipschitz continuous functions and functions with uniformly bounded Hessians over the standard simplex.This extends an earlier result by De Klerk, Laurent and Parrilo [A PTAS for the minimization of polynomials of fixed degree over the simplex, Theoretical Computer Science, to appear.]global optimization;standard simplex;PTAS;multivariate Bernstein approximation;semidefinite programming
On the Complexity of Optimization over the Standard Simplex
We review complexity results for minimizing polynomials over the standard simplex and unit hypercube.In addition, we show that there exists a polynomial time approximation scheme (PTAS) for minimizing Lipschitz continuous functions and functions with uniformly bounded Hessians over the standard simplex.This extends an earlier result by De Klerk, Laurent and Parrilo [A PTAS for the minimization of polynomials of fixed degree over the simplex, Theoretical Computer Science, to appear.]
Estimating operator norms using covering nets
We present several polynomial- and quasipolynomial-time approximation schemes
for a large class of generalized operator norms. Special cases include the
norm of matrices for , the support function of the set of
separable quantum states, finding the least noisy output of
entanglement-breaking quantum channels, and approximating the injective tensor
norm for a map between two Banach spaces whose factorization norm through
is bounded.
These reproduce and in some cases improve upon the performance of previous
algorithms by Brand\~ao-Christandl-Yard and followup work, which were based on
the Sum-of-Squares hierarchy and whose analysis used techniques from quantum
information such as the monogamy principle of entanglement. Our algorithms, by
contrast, are based on brute force enumeration over carefully chosen covering
nets. These have the advantage of using less memory, having much simpler proofs
and giving new geometric insights into the problem. Net-based algorithms for
similar problems were also presented by Shi-Wu and Barak-Kelner-Steurer, but in
each case with a run-time that is exponential in the rank of some matrix. We
achieve polynomial or quasipolynomial runtimes by using the much smaller nets
that exist in spaces. This principle has been used in learning theory,
where it is known as Maurey's empirical method.Comment: 24 page
Integer Polynomial Optimization in Fixed Dimension
We classify, according to their computational complexity, integer
optimization problems whose constraints and objective functions are polynomials
with integer coefficients and the number of variables is fixed. For the
optimization of an integer polynomial over the lattice points of a convex
polytope, we show an algorithm to compute lower and upper bounds for the
optimal value. For polynomials that are non-negative over the polytope, these
sequences of bounds lead to a fully polynomial-time approximation scheme for
the optimization problem.Comment: In this revised version we include a stronger complexity bound on our
algorithm. Our algorithm is in fact an FPTAS (fully polynomial-time
approximation scheme) to maximize a non-negative integer polynomial over the
lattice points of a polytop
A refined error analysis for fixed-degree polynomial optimization over the simplex
We consider the problem of minimizing a fixed-degree polynomial over the
standard simplex. This problem is well known to be NP-hard, since it contains
the maximum stable set problem in combinatorial optimization as a special case.
In this paper, we revisit a known upper bound obtained by taking the minimum
value on a regular grid, and a known lower bound based on P\'olya's
representation theorem. More precisely, we consider the difference between
these two bounds and we provide upper bounds for this difference in terms of
the range of function values. Our results refine the known upper bounds in the
quadratic and cubic cases, and they asymptotically refine the known upper bound
in the general case.Comment: 13 page
Integration and Optimization of Multivariate Polynomials by Restriction onto a Random Subspace
We consider the problem of efficient integration of an n-variate polynomial
with respect to the Gaussian measure in R^n and related problems of complex
integration and optimization of a polynomial on the unit sphere. We identify a
class of n-variate polynomials f for which the integral of any positive integer
power f^p over the whole space is well-approximated by a properly scaled
integral over a random subspace of dimension O(log n). Consequently, the
maximum of f on the unit sphere is well-approximated by a properly scaled
maximum on the unit sphere in a random subspace of dimension O(log n). We
discuss connections with problems of combinatorial counting and applications to
efficient approximation of a hafnian of a positive matrix.Comment: 15 page
A linear programming reformulation of the standard quadratic optimization problem
The problem of minimizing a quadratic form over the standard simplex is known as the standard quadratic optimization problem (SQO). It is NP-hard, and contains the maximum stable set problem in graphs as a special case. In this note,
we show that the SQO problem may be reformulated as an (exponentially sized) linear program (LP). This reformulation also suggests a hierarchy of polynomial-time solvable LPās whose optimal values converge finitely to the optimal value of the SQO problem. The hierarchies of LP relaxations from the literature do not share this finite convergence property for SQO, and we review the relevant counterexamples.Accepted versio