Related papers: Semidefinite programming for permutation codes
We show that the spectral embeddings of all known triangle-free strongly regular graphs are optimal spherical codes (the new cases are $56$ points in $20$ dimensions, $50$ points in $21$ dimensions, and $77$ points in $21$ dimensions), as…
In this paper, "chance optimization" problems are introduced, where one aims at maximizing the probability of a set defined by polynomial inequalities. These problems are, in general, nonconvex and computationally hard. With the objective…
This paper investigates the Terwilliger algebra of some group association schemes related to codes. In addition, it also shows the generators of invariant rings appearing by E-polynomials.
We develop a practical approach to semidefinite programming (SDP) that includes the von Neumann entropy, or an appropriate variant, as a regularization term. In particular we solve the dual of the regularized program, demonstrating how a…
We survey recent generalizations and improvements of the linear programming method that involve semidefinite programming. A general framework using group representations and tools from graph theory is provided.
Symmetric extensions are essential in quantum mechanics, providing a lens to investigate the correlations of entangled quantum systems and to address challenges like the quantum marginal problem. Though semi-definite programming (SDP) is a…
We develop a general framework for finding approximately-optimal preconditioners for solving linear systems. Leveraging this framework we obtain improved runtimes for fundamental preconditioning and linear system solving problems including…
Polynomial optimization problems are infinite-dimensional, nonconvex, NP-hard, and are often handled in practice with the moment-sums of squares hierarchy of semidefinite programming bounds. We consider problems where the objective function…
In this thesis we present several results in coding theory, concerning error-correcting codes and the Shannon capacity. 1. We give a general symmetry reduction of matrices occuring in semidefinite programs in coding theory. 2. We apply the…
The completely bounded trace and spectral norms, for finite-dimensional spaces, are known to be efficiently expressible by semidefinite programs (J. Watrous, Theory of Computing 5: 11, 2009). This paper presents two new, and arguably much…
We introduce the concept of $\delta$-sequence. A $\delta$-sequence $\Delta$ generates a well-ordered semigroup $S$ in $\mathbb{Z}^2$ or $\mathbb{R}$. We show how to construct (and compute parameters) for the dual code of any evaluation code…
The question whether there exists an integral solution to the system of linear equations with non-negative constraints, $A\x = \b, \, \x \ge 0$, where $A \in \Z^{m\times n}$ and ${\mathbf b} \in \Z^m$, finds its applications in many areas,…
Finding cliques in random graphs and the closely related "planted" clique variant, where a clique of size t is planted in a random G(n,1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best…
Let $\mathscr{Q}(m,q)$ and $\mathscr{S}(m,q)$ be the sets of quadratic forms and symmetric bilinear forms on an $m$-dimensional vector space over $\mathbb{F}_q$, respectively. The orbits of $\mathscr{Q}(m,q)$ and $\mathscr{S}(m,q)$ under a…
We present exact mixed-integer linear programming formulations for verifying the performance of first-order methods for parametric quadratic optimization. We formulate the verification problem as a mixed-integer linear program where the…
We explore some connections between association schemes and the analyses of the semidefinite programming (SDP) based convex relaxations of combinatorial optimization problems in the Lov\'{a}sz--Schrijver lift-and-project hierarchy. Our…
Let $q$ be an odd prime power and let $X(m,q)$ be the set of symmetric bilinear forms on an $m$-dimensional vector space over $\mathbb{F}_q$. The partition of $X(m,q)$ induced by the action of the general linear group gives rise to a…
Given symmetric matrices $A_0, A_1, \ldots, A_n$ of size $m$ with rational entries, the set of real vectors $x = (x_1, \ldots, x_n)$ such that the matrix $A_0 + x_1 A_1 + \cdots + x_n A_n$ has non-negative eigenvalues is called a…
This paper presents a comprehensive exploration of semi-definite programming (SDP) techniques within the context of quantum information. It examines the mathematical foundations of convex optimization, duality, and SDP formulations,…
We study the ternary quadratic problem (TQP), a quadratic optimization problem with linear constraints where the variables take values in $\{0, \pm 1\}$. While semidefinite programming (SDP) techniques are well established for $\{0,1\}$-…