Related papers: Finding Maximum Determinant Principal Submatrices …
Given vectors $v_1,\dots,v_n\in\mathbb{R}^d$ and a matroid $M=([n],I)$, we study the problem of finding a basis $S$ of $M$ such that $\det(\sum_{i \in S}v_i v_i^\top)$ is maximized. This problem appears in a diverse set of areas such as…
We study the convex relaxation of a polynomial optimization problem, maximizing a product of linear forms over the complex sphere. We show that this convex program is also a relaxation of the permanent of Hermitian positive semidefinite…
A multidimensional optimization problem is formulated in the tropical mathematics setting as to maximize a nonlinear objective function, which is defined through a multiplicative conjugate transposition operator on vectors in a…
We address the problem of estimating a high-dimensional matrix from linear measurements, with a focus on designing optimal rank-adaptive algorithms. These algorithms infer the matrix by estimating its singular values and the corresponding…
The classical problem of maximizing a submodular function under a matroid constraint is considered. Defining a new measure for the increments made by the greedy algorithm at each step, called the discriminant, improved approximation ratio…
Finding tight bounds on the optimal solution is a critical element of practical solution methods for discrete optimization problems. In the last decade, decision diagrams (DDs) have brought a new perspective on obtaining upper and lower…
We show how the numerical range of a matrix can be used to bound the optimal value of certain optimization problems over real tensor product vectors. Our bound is stronger than the trivial bounds based on eigenvalues, and can be computed…
The control and sensing of large-scale systems results in combinatorial problems not only for sensor and actuator placement but also for scheduling or observability/controllability. Such combinatorial constraints in system design and…
This work presents a hybrid approach to solve the maximum stable set problem, using constraint and semidefinite programming. The approach consists of two steps: subproblem generation and subproblem solution. First we rank the variable…
Linear fixed point equations in Hilbert spaces arise in a variety of settings, including reinforcement learning, and computational methods for solving differential and integral equations. We study methods that use a collection of random…
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…
In many contexts one encounters Hermitian operators $M$ on a Hilbert space whose dimension is so large that it is impossible to write down all matrix entries in an orthonormal basis. How does one determine whether such $M$ is positive…
Forward regression is a classical and effective tool for variable screening in ultra-high dimensional linear models, but its standard projection-based implementation can be computationally costly and numerically unstable when predictors are…
Considering a 2D matrix of positive and negative numbers, how might one draw a rectangle within it whose contents sum higher than all other rectangles'? This fundamental problem, commonly known the maximum rectangle problem or subwindow…
We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem, which models linear classification. Both are fundamental problems in computational geometry and machine learning in their exact and…
This paper studies the problem of selecting a submatrix of a positive definite matrix in order to achieve a desired bound on the smallest eigenvalue of the submatrix. Maximizing this smallest eigenvalue has applications to selecting input…
The determinants of $\{\pm 1\}$-matrices are calculated by via the oriented hypergraphic Laplacian and summing over an incidence generalization of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on…
We consider a class of optimization problems that involve determining the maximum value that a function in a particular class can attain subject to a collection of difference constraints. We show that a particular linear programming…
The problem of finding a $k \times k$ submatrix of maximum volume of a matrix $A$ is of interest in a variety of applications. For example, it yields a quasi-best low-rank approximation constructed from the rows and columns of $A$. We show…
The Maximum Balanced Biclique Problem (MBBP) is a prominent model with numerous applications. Yet, the problem is NP-hard and thus computationally challenging. We propose novel ideas for designing effective exact algorithms for MBBP.…