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Constrained optimization problems appear in a wide variety of challenging real-world problems, where constraints often capture the physics of the underlying system. Classic methods for solving these problems rely on iterative algorithms…

We study the implicit regularization effects of deep learning in tensor factorization. While implicit regularization in deep matrix and 'shallow' tensor factorization via linear and certain type of non-linear neural networks promotes…

Optimization problems pervade essentially every scientific discipline and industry. Many such problems require finding a solution that maximizes the number of constraints satisfied. Often, these problems are particularly difficult to solve…

Estimating equations arise in a wide range of statistical applications, including longitudinal and clustered data analysis, survival analysis, econometrics, and semiparametric inference. In high-dimensional settings, adding…

In the pursuit of explaining implicit regularization in deep learning, prominent focus was given to matrix and tensor factorizations, which correspond to simplified neural networks. It was shown that these models exhibit an implicit…

For solving linear ill-posed problems regularization methods are required when the right hand side is with some noise. In the present paper regularized solutions are obtained by implicit iteration methods in Hilbert scales. % By exploiting…

Functional iterations such as Newton's are a popular tool for polynomial root-finding. We consider realistic situation where some (e.g., better-conditioned) roots have already been approximated and where further computations is directed to…

In recent years many efforts have been devoted to finding bidiagonal factorizations of nonsingular totally positive matrices, since their accurate computation allows to numerically solve several important algebraic problems with great…

Trace norm regularization is a widely used approach for learning low rank matrices. A standard optimization strategy is based on formulating the problem as one of low rank matrix factorization which, however, leads to a non-convex problem.…

In this paper we consider the application of direct methods for solving a sequence of saddle-point systems. Our goal is to design a method that reuses information from one factorization and applies it to the next one. In more detail, when…

We introduce the dual-path fixing strategy to exploit dual algorithms for solving relaxations of mixed-integer nonlinear-optimization problems. Such dual algorithms are naturally applied in the context of branch-and-bound, and eventual…

This paper considers the regularization continuation method and the trust-region updating strategy for the optimization problem with linear equality constraints.The proposed method utilizes the linear conservation law of the regularization…

This paper presents a customized second-order cone programming (SOCP) solver tailored for embedded real-time optimization, which frequently arises in modern guidance and control (G&C) applications. The solver employs a practically efficient…

In many practical applications of constrained optimization, scale and solving time limits make traditional optimization solvers prohibitively slow. Thus, the research question of how to design optimization proxies -- machine learning models…

In this paper, we consider a nonconvex optimization problem with nonlinear equality constraints. We assume that both, the objective function and the functional constraints are locally smooth. For solving this problem, we propose a…

Max-norm regularizer has been extensively studied in the last decade as it promotes an effective low-rank estimation for the underlying data. However, such max-norm regularized problems are typically formulated and solved in a batch manner,…

This paper develops the proximal method of multipliers for a class of nonsmooth convex optimization. The method generates a sequence of minimization problems (subproblems). We show that the sequence of approximations to the solutions of the…

In this paper, we consider a regularization strategy for the factorization method when there is noise added to the data operator. The factorization method is a qualitative method used in shape reconstruction problems. These methods are…

Large sparse linear algebraic systems can be found in a variety of scientific and engineering fields, and many scientists strive to solve them in an efficient and robust manner. In this paper, we propose an interpretable neural solver, the…

This paper consists of four general parts: convex sets; convex functions; convex optimization; and the interior-point algorithm. I will start by introducing the definition of convex sets and give three common convex set examples which will…