Related papers: Border Basis relaxation for polynomial optimizatio…
We consider clustering problems where the goal is to determine an optimal partition of a given point set in Euclidean space in terms of a collection of affine subspaces. While there is vast literature on heuristics for this kind of problem,…
The Lasserre hierarchy of semidefinite programming (SDP) relaxations is an effective scheme for finding computationally feasible SDP approximations of polynomial optimization over compact semi-algebraic sets. In this paper, we show that,…
Motivated by a geometric problem, we introduce a new non-convex graph partitioning objective where the optimality criterion is given by the sum of the Dirichlet eigenvalues of the partition components. A relaxed formulation is identified…
We consider the computation of averaged coefficients for the homogenization of elliptic partial differential equations. In this problem, like in many multiscale problems, a large number of similar computations parametrized by the…
The Alternating Direction Method of Multipliers (ADMM) has gained significant attention across a broad spectrum of machine learning applications. Incorporating the over-relaxation technique shows potential for enhancing the convergence rate…
We consider a new hierarchy of semidefinite relaxations for the general polynomial optimization problem $(P):\:f^{\ast}=\min \{\,f(x):x\in K\,\}$ on a compact basic semi-algebraic set $K\subset\R^n$. This hierarchy combines some advantages…
Motivated by relativistic materials, we develop a numerical scheme to support existing or state new conjectures in the spectral optimisation of eigenvalues of the Dirac operator, subject to infinite-mass boundary conditions. We study the…
We propose a relax-and-round approach combined with a greedy search strategy for performing complex lattice basis reduction. Taking an optimization perspective, we introduce a relaxed version of the problem that, while still nonconvex, has…
This work establishes a novel, unified theoretical framework for a class of high order embedded boundary methods, revealing that the Reconstruction for Off-site Data (ROD) treatment shares a fundamental structure with the recently developed…
We consider the problem of finding the global optimum of a real-valued complex polynomial on a compact set defined by real-valued complex polynomial inequalities. It reduces to solving a sequence of complex semidefinite programming…
In this paper, we generalize the notion of border bases of zero-dimensional polynomial ideals to the module setting. To this end, we introduce order modules as a generalization of order ideals and module border bases of submodules with…
In this paper, we analyze the convergence %semi-convergence properties of projected non-stationary block iterative methods (P-BIM) aiming to find a constrained solution to large linear, usually both noisy and ill-conditioned, systems of…
Solving systems of polynomial equations, particularly those with finitely many solutions, is a crucial challenge across many scientific fields. Traditional methods like Gr\"obner and Border bases are fundamental but suffer from high…
The recovery of signals with finite-valued components from few linear measurements is a problem with widespread applications and interesting mathematical characteristics. In the compressed sensing framework, tailored methods have been…
In this paper, by means of a standard model problem, we devise an approach to computing approximate dual bounds for use in global optimization of coefficient identification in partial differential equations (PDEs) by, e.g., (spatial)…
This paper considers a quadratically-constrained cardinality minimization problem with applications to digital filter design, subset selection for linear regression, and portfolio selection. Two relaxations are investigated: the continuous…
This paper studies how to certify the convergence of Lasserre's hierarchy of semidefinite programming relaxations for solving multivariate polynomial optimization. We propose flat truncation as a general certificate for this purpose. Assume…
Many computer vision applications require robust and efficient estimation of camera geometry. The robust estimation is usually based on solving camera geometry problems from a minimal number of input data measurements, i.e., solving minimal…
Lagrangian decomposition (LD) is a relaxation method that provides a dual bound for constrained optimization problems by decomposing them into more manageable sub-problems. This bound can be used in branch-and-bound algorithms to prune the…
The Lasserre hierarchy is a systematic procedure for constructing a sequence of increasingly tight relaxations that capture the convex formulations used in the best available approximation algorithms for a wide variety of optimization…