Related papers: Finite de Finetti for convex bodies and Polynomial…
We construct a convergent family of outer approximations for the problem of optimizing polynomial functions over convex bodies subject to polynomial constraints. This is achieved by generalizing the polarization hierarchy, which has…
Polynomial optimization problems often arise in sequences indexed by dimension, and it is of interest to compute bounds on the optimal values of all problems in the sequence. Examples include certifying inequalities between symmetric…
A finite form of de Finetti's representation theorem is established using elementary information-theoretic tools: The distribution of the first $k$ random variables in an exchangeable binary vector of length $n\geq k$ is close to a mixture…
We show how to bound the accuracy of a family of semi-definite programming relaxations for the problem of polynomial optimization on the hypersphere. Our method is inspired by a set of results from quantum information known as quantum de…
We show that $\varepsilon$-additive approximations of the optimal value of fixed-size two-player free games with fixed-dimensional entanglement assistance can be computed in time $\mathrm{poly}(1/\varepsilon)$. This stands in contrast to…
We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial…
Convex optimization problems arise naturally in quantum information theory, often in terms of minimizing a convex function over a convex subset of the space of hermitian matrices. In most cases, finding exact solutions to these problems is…
The de Finetti theorem and its extensions concern the structure of multipartite probability distributions with certain symmetry properties, the paradigmatic original example being permutation symmetry. These theorems assert that such…
This thesis explores algorithmic applications and limitations of convex relaxation hierarchies for approximating some discrete and continuous optimization problems. - We show a dichotomy of approximability of constraint satisfaction…
We study stochastic team (known also as decentralized stochastic control or identical interest stochastic dynamic game) problems with large or countably infinite number of decision makers, and characterize existence and structural…
We show the existence of a fully polynomial-time approximation scheme (FPTAS) for the problem of maximizing a non-negative polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed. Moreover, using a…
In multi-objective optimization, computing the entire non-dominated set (also known as the Pareto front or the Pareto frontier) is often intractable. However, for any multiplicative factor greater than one, an approximation set can be…
Given an affine variety X and a finite dimensional vector space of regular functions L on X, we associate a convex body to (X, L) such that its volume is responsible for the number of solutions of a generic system of functions from L. This…
We present a novel analogue for finite exchangeable sequences of the de Finetti, Hewitt and Savage theorem and investigate its implications for multi-marginal optimal transport (MMOT) and Bayesian statistics. If $(Z_1,...,Z_N)$ is a…
We propose an algorithm to generate inner and outer polyhedral approximations to the upper image of a bounded convex vector optimization problem. It is an outer approximation algorithm and is based on solving norm-minimizing scalarizations.…
One of the most fruitful results from Minkowski's geometric viewpoint on number theory is his so called 1st Fundamental Theorem. It provides an optimal upper bound for the volume of an o-symmetric convex body whose only interior lattice…
We extend Fukushima's result on the finite convergence of an algorithm for the global convex feasibility problem to the local nonconvex case.
In this paper the problem of maximizing the distance to a given fixed point over an intersection of balls is considered. It is known that this problem is NP complete in the general case, since any subset sum problem can be solved upon…
We investigate a class of composite nonconvex functions, where the outer function is the sum of univariate extended-real-valued convex functions and the inner function is the limit of difference-of-convex functions. A notable feature of…
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…