Related papers: A certificate for semidefinite relaxations in comp…
This paper studies random operator-valued positive definite (p.d.) kernels and their connection to moment dilations. A class of random p.d. kernels is introduced in which the positivity requirement is imposed only in expectation, extending…
The relaxation complexity $\mathrm{rc}(X)$ of the set of integer points $X$ contained in a polyhedron is the smallest number of facets of any polyhedron $P$ such that the integer points in $P$ coincide with $X$. It is a useful tool to…
In recent years, semidefinite relaxations of common optimization problems in robotics have attracted growing attention due to their ability to provide globally optimal solutions. In many cases, it was shown that specific handcrafted…
We propose a numerical linear algebra based method to find the multiplication operators of the quotient ring $\mathbb{C}[x]/I$ associated to a zero-dimensional ideal $I$ generated by $n$ $\mathbb{C}$-polynomials in $n$ variables. We assume…
Solving a polynomial system, or computing an associated Gr\"obner basis, has been a fundamental task in computational algebra. However, it is also known for its notorious doubly exponential time complexity in the number of variables in the…
Semidefinite relaxations are widely used to compute upper bounds on the objective of optimization problems involving noncommutative polynomials. Such optimization problems are prevalent in quantum information. We present an algorithm able…
We introduce a method to construct general multivariate positive definite kernels on a nonempty set $X$ that employs a prescribed bounded completely monotone function and special multivariate functions on $X$.\ The method is consistent with…
We present a hierarchy of tractable relaxations to obtain lower bounds on the minimum value of a polynomial over a constraint set defined by polynomial equations. In contrast to previous convex relaxation techniques for this problem, our…
Positive semidefiniteness, recursiveness, and the variety condition of a moment matrix are necessary and sufficient conditions to solve the quadratic and quartic moment problems. Also, positive semidefiniteness, combined with another…
Given standard angular momentum and boost matrices, the commutation rules for vector and momentum matrices are solved. The resulting matrix components are displayed as detailed functions of spin with factors such as the square root of…
In this paper we present a unified framework for solving a general class of problems arising in the context of set-membership estimation/identification theory. More precisely, the paper aims at providing an original approach for the…
Pseudo-hermitian matrices are matrices hermitian with respect to an indefinite metric. They can be thought of as the truncation of pseudo-hermitian operators, defined over some Krein space, together with the associated metric, to a finite…
Standard noncommutative Gr\"obner basis procedures are used for computing ideals of free noncommutative polynomial rings over fields. This paper describes Gr\"obner basis procedures for one-sided ideals in finitely presented noncommutative…
We consider a certain class of Schubert varieties of the affine Grassmannian of type A. By embedding a Schubert variety into a finite-dimensional Grassmannian, we construct an explicit basis of sections of the basic line bundle by…
In this paper we study the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its Semidefinite Programming (SDP) relaxation. We consider two quadratic optimization models: (1) $\min \{x^* C x…
The classic integrated conditional moment test is a promising method for testing regression model misspecification. However, it severely suffers from the curse of dimensionality. To extend it to handle the testing problem for parametric…
Conjugation covariants of matrices are applied to study the real algebraic variety consisting of complex Hermitian matrices with a bounded number of distinct eigenvalues. A minimal generating system of the vanishing ideal of degenerate…
We will explore some properties of minimal graded free resolutions of $R/I$, where $R$ is a trivariate polynomial ring over a field and $I$ is a monomial ideal. Our focus will be to consider a specific form of the resolutions when $I$ is…
Solutions to many-body problem instances often involve an intractable number of degrees of freedom and admit no known approximations in general form. In practice, representing quantum-mechanical states of a given Hamiltonian using available…
In this work we study convex relaxations of quadratic optimisation problems over permutation matrices. While existing semidefinite programming approaches can achieve remarkably tight relaxations, they have the strong disadvantage that they…