Related papers: Accuracy and componentwise accuracy in multilinear…
In this paper, we propose a quasi-Newton method for solving smooth and monotone nonlinear equations, including unconstrained minimization and minimax optimization as special cases. For the strongly monotone setting, we establish two global…
A commonly used approach to study stability in a complex system is by analyzing the Jacobian matrix at an equilibrium point of a dynamical system. The equilibrium point is stable if all eigenvalues have negative real parts. Here, by…
An effective numerical method is presented for optimizing model parameters that can be applied to any type of system of non-linear equations and any number of data-points, which does not require explicit formulation of the objective…
In this paper, we propose three methods to solve the PageRank problem for the transition matrices with both row and column sparsity. Our methods reduce the PageRank problem to the convex optimization problem over the simplex. The first…
We propose a method to enhance the stability of a neural ordinary differential equation (neural ODE) by reducing the maximum error growth subsequent to a perturbation of the initial value. Since the stability depends on the logarithmic norm…
We develop a computationally efficient algorithm for the automatic regularization of nonlinear inverse problems based on the discrepancy principle. We formulate the problem as an equality constrained optimization problem, where the…
In this work, we consider ranking problems among a finite set of candidates: for instance, selecting the top-$k$ items among a larger list of candidates or obtaining the full ranking of all items in the set. These problems are often…
Rank minimization is of interest in machine learning applications such as recommender systems and robust principal component analysis. Minimizing the convex relaxation to the rank minimization problem, the nuclear norm, is an effective…
Numerous applications in data mining and machine learning require recovering a matrix of minimal rank. Robust principal component analysis (RPCA) is a general framework for handling this kind of problems. Nuclear norm based convex surrogate…
We study robust convex quadratic programs where the uncertain problem parameters can contain both continuous and integer components. Under the natural boundedness assumption on the uncertainty set, we show that the generic problems are…
We investigate the computation of stable fracture paths in brittle thin films using one-dimensional damage models with an elastic foundation. The underlying variational formulation is non-convex, making the evolution path sensitive to…
Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the…
Although it is relatively easy to apply, the gradient method often displays a disappointingly slow rate of convergence. Its convergence is specially based on the structure of the matrix of the algebraic linear system, and on the choice of…
The linearly constrained matrix rank minimization problem is widely applicable in many fields such as control, signal processing and system identification. The tightest convex relaxation of this problem is the linearly constrained nuclear…
Modeling complex multiway relationships in large-scale networks is becoming more and more challenging in data science. The multilinear PageRank problem, arising naturally in the study of higher-order Markov chains, is a powerful framework…
After the phenomenal success of the PageRank algorithm, many researchers have extended the PageRank approach to ranking graphs with richer structures beside the simple linkage structure. In some scenarios we have to deal with…
It is common in stability analysis to linearize a system and investigate the spectrum of the Jacobian matrix. This approach faces the challenge of determining the matrix spectrum when the coefficients depend on parameters or when the…
We present a derivative-based algorithm for nonlinearly constrained optimization problems that is tolerant of inaccuracies in the data. The algorithm solves a semi-smooth set of nonlinear equations that are equivalent to the first-order…
The sparse nonlinear programming (SNP) problem has wide applications in signal and image processing, machine learning, pattern recognition, finance and management, etc. However, the computational challenge posed by SNP has not yet been well…
Motivated by recent applications of the Lyapunov's method in artificial neural networks, which could be considered as dynamical systems for which the convergence of the system trajectories to equilibrium states is a necessity. We re-look at…