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We study a family of (potentially non-convex) constrained optimization problems with convex composite structure. Through a novel analysis of non-smooth geometry, we show that proximal-type algorithms applied to exact penalty formulations of…
The paper addresses general constrained and non-linear optimization problems. For some of these notoriously hard problems, there exists a reformulation as an unconstrained, global optimization problem. We illustrate the transformation, and…
The non-negative matrix factorization (NMF) model with an additional orthogonality constraint on one of the factor matrices, called the orthogonal NMF (ONMF), has been found a promising clustering model and can outperform the classical…
Flexible sparsity regularization means stably approximating sparse solutions of operator equations by using coefficient-dependent penalizations. We propose and analyse a general nonconvex approach in this respect, from both theoretical and…
We study a new penalty reformulation of constrained convex optimization based on the softplus penalty function. We develop novel and tight upper bounds on the objective value gap and the violation of constraints for the solutions to the…
In this paper we consider the problem of distributed nonlinear optimisation of a separable convex cost function over a graph subject to cone constraints. We show how to generalise, using convex analysis, monotone operator theory and…
Nonlinear matrix equations arise in many practical contexts related to control theory, dynamical programming and finite element methods for solving some partial differential equations. In most of these applications, it is needed to compute…
Ordinary differential equations (ODEs) are commonly used to model dynamic behavior of a system. Because many parameters are unknown and have to be estimated from the observed data, there is growing interest in statistics to develop…
We present a simple transformation of any linear program or semidefinite program into an equivalent convex optimization problem whose only constraints are linear equations. The objective function is defined on the whole space, making…
Traditional mathematical programming solvers require long computational times to solve constrained minimization problems of complex and large-scale physical systems. Therefore, these problems are often transformed into unconstrained ones,…
We present a new optimization-theoretic approach to analyzing Follow-the-Leader style algorithms, particularly in the setting where perturbations are used as a tool for regularization. We show that adding a strongly convex penalty function…
The problem of finding global minima of nonlinear discrete functions arises in many fields of practical matters. In recent years, methods based on discrete filled functions become popular as ways of solving these sort of problems. However,…
In this paper, we propose a proximal splitting methodology with a non-convex penalty function based on the heavy-tailed Cauchy distribution. We first suggest a closed-form expression for calculating the proximal operator of the Cauchy…
Image smoothing is a fundamental procedure in applications of both computer vision and graphics. The required smoothing properties can be different or even contradictive among different tasks. Nevertheless, the inherent smoothing nature of…
This paper is concerned with solving nonconvex learning problems with folded concave penalty. Despite that their global solutions entail desirable statistical properties, they lack optimization techniques that guarantee global optimality in…
In this paper we consider general rank minimization problems with rank appearing in either objective function or constraint. We first establish that a class of special rank minimization problems has closed-form solutions. Using this result,…
In this paper, we study two general classes of optimization algorithms for kernel methods with convex loss function and quadratic norm regularization, and analyze their convergence. The first approach, based on fixed-point iterations, is…
For some typical and widely used non-convex half-quadratic regularization models and the Ambrosio-Tortorelli approximate Mumford-Shah model, based on the Kurdyka-\L ojasiewicz analysis and the recent nonconvex proximal algorithms, we…
Despite recent advances in algorithmic fairness, methodologies for achieving fairness with generalized linear models (GLMs) have yet to be explored in general, despite GLMs being widely used in practice. In this paper we introduce two…
Positive linear programs (LP), also known as packing and covering linear programs, are an important class of problems that bridges computer science, operations research, and optimization. Despite the consistent efforts on this problem, all…