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We develop a new algorithmic framework for designing approximation algorithms for cut-based optimization problems on capacitated undirected graphs that undergo edge insertions and deletions. Specifically, our framework dynamically maintains…
When solving partial differential equations (PDEs) using finite difference or finite element methods, efficient solvers are required for handling large sparse linear systems. In this paper, a recursive sparse LU decomposition for matrices…
Upper bound limit analysis allows one to evaluate directly the ultimate load of structures without performing a cumbersome incremental analysis. In order to numerically apply this method to thin plates in bending, several authors have…
When factorizing binary matrices, we often have to make a choice between using expensive combinatorial methods that retain the discrete nature of the data and using continuous methods that can be more efficient but destroy the discrete…
In this paper, we study the mathematical structure and numerical approximation of elliptic problems posed in a (3D) domain $\Omega$ when the right-hand side is a (1D) line source $\Lambda$. The analysis and approximation of such problems is…
In spectral clustering, one defines a similarity matrix for a collection of data points, transforms the matrix to get the Laplacian matrix, finds the eigenvectors of the Laplacian matrix, and obtains a partition of the data using the…
This work is devoted to the development and analysis of a linearization algorithm for microscopic elliptic equations, with scaled degenerate production, posed in a perforated medium and constrained by the homogeneous Neumann-Dirichlet…
We develop two adaptive discretization algorithms for convex semi-infinite optimization, which terminate after finitely many iterations at approximate solutions of arbitrary precision. In particular, they terminate at a feasible point of…
Rounding linear programs using techniques from discrepancy is a recent approach that has been very successful in certain settings. However this method also has some limitations when compared to approaches such as randomized and iterative…
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…
Data-driven algorithm design is a paradigm that uses statistical and machine learning techniques to select from a class of algorithms for a computational problem an algorithm that has the best expected performance with respect to some…
We propose a successive generation of cutting inequalities for binary quadratic optimization problems. Multiple cutting inequalities are successively generated for the convex hull of the set of the optimal solutions $\subset \{0, 1\}^n$,…
The perfect matching polytope, i.e. the convex hull of (incidence vectors of) perfect matchings of a graph is used in many combinatorial algorithms. Kotzig, Lov\'asz and Plummer developed a decomposition theory for graphs with perfect…
We study a nonlinear decomposition of a positive definite matrix into two components: the inverse of another positive definite matrix and a symmetric matrix constrained to lie in a prescribed linear subspace. Equivalently, the inverse…
Machine learning is increasingly used to guide branch-and-cut (B&C) for mixed-integer linear programming by learning score-based policies for selecting branching variables and cutting planes. Many approaches train on local signals from…
Defect-adaptive surface-code methods have substantially advanced the construction of valid logical patches on imperfect hardware, but fault-tolerant computation also requires executable logical oper ations on the resulting irregular…
This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its…
Optimization problems with rank constraints arise in many applications, including matrix regression, structured PCA, matrix completion and matrix decomposition problems. An attractive heuristic for solving such problems is to factorize the…
In this paper, we develop a new decomposition technique for solving bi-objective linear programming problems. The proposed methodology combines the bi-objective simplex algorithm with Benders decomposition and can be used to obtain a…
We study the Sparse Plus Low-Rank decomposition problem (SLR), which is the problem of decomposing a corrupted data matrix into a sparse matrix of perturbations plus a low-rank matrix containing the ground truth. SLR is a fundamental…