Related papers: Tractable semidefinite bounds of positive maximal …
This paper introduces and develops the algebraic framework of moment polynomials, which are polynomial expressions in commuting variables and their formal mixed moments. Their positivity and optimization over probability measures supported…
We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and…
We study QPT (quasi-polynomial tractability) in the worst case setting for linear tensor product problems defined over Hilbert spaces. We assume that the domain space is a reproducing kernel Hilbert space so that function values are well…
This paper investigates the minimization of the expectation of piecewise polynomial loss functions over Wasserstein balls. This optimization problem often appears as a key sub-problem of distributionally robust optimization problems. We…
We consider the optimization of pairwise objective functions, i.e., objective functions of the form $H(\mathbf{x}) = H(x_1,\ldots,x_N) = \sum_{1\leq i<j \leq N} H_{ij}(x_i,x_j)$ for $x_i$ in some continuous state spaces $\mathcal{X}_i$.…
A common optimization problem is the minimization of a symmetric positive definite quadratic form $< x,Tx >$ under linear constrains. The solution to this problem may be given using the Moore-Penrose inverse matrix. In this work we extend…
This paper studies how to compute all real eigenvalues of a symmetric tensor. As is well known, the largest or smallest eigenvalue can be found by solving a polynomial optimization problem, while the other middle eigenvalues can not. We…
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…
In this paper, we propose some new semidefinite relaxations for a class of nonconvex complex quadratic programming problems, which widely appear in the areas of signal processing and power system. By deriving new valid constraints to the…
This two-part paper is concerned with the problem of minimizing a linear objective function subject to a bilinear matrix inequality (BMI) constraint. In this part, we first consider a family of convex relaxations which transform BMI…
Valued constraint satisfaction problems (VCSPs) are a large class of combinatorial optimisation problems. It is desirable to classify the computational complexity of VCSPs depending on a fixed set of allowed cost functions in the input.…
Certificates of polynomial nonnegativity can be used to obtain tight dual bounds for polynomial optimization problems. We consider Sums of Nonnegative Circuit (SONC) polynomials certificates, which are well suited for sparse problems since…
We consider the problems of finding the lexicographically minimal (or maximal) satisfying assignment of propositional formulae for different restricted formula classes. It turns out that for each class from our framework, the above problem…
We study the approximation of high-dimensional rank one tensors using point evaluations and consider deterministic as well as randomized algorithms. We prove that for certain parameters (smoothness and norm of the $r$th derivative) this…
The values of the determinant of Vandermonde matrices with real elements are analyzed both visually and analytically over the unit sphere in various dimensions. For three dimensions some generalized Vandermonde matrices are analyzed…
We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several…
We study one-dimensional Schr\"odinger operators $\operatorname{H} = -\partial_x^2 + V$ with unbounded complex potentials $V$ and derive asymptotic estimates for the norm of the resolvent, $\Psi(\lambda) := \| (\operatorname{H} -…
This paper focuses on the study of a mathematical program with equilibrium constraints, where the objective and the constraint functions are all polynomials. We present a method for finding its global minimizers and global minimum using a…
A new technique is presented to solve a class of linear boundary value problems (BVP). Technique is primarily based on an operational matrix developed from a set of modified Bernoulli polynomials. The new set of polynomials is an…
Seeking tighter relaxations of combinatorial optimization problems, semidefinite programming is a generalization of linear programming that offers better bounds and is still polynomially solvable. Yet, in practice, a semidefinite program is…