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The area distance to a convex plane curve is an important concept in computer vision. In this paper we describe a strong link between area distances and improper affine spheres. This link makes possible a better understanding of both…
We propose a general convex optimization problem for computing regularized geodesic distances. We show that under mild conditions on the regularizer the problem is well posed. We propose three different regularizers and provide analytical…
Our work focuses on stochastic gradient methods for optimizing a smooth non-convex loss function with a non-smooth non-convex regularizer. Research on this class of problem is quite limited, and until recently no non-asymptotic convergence…
The authors in (Banjac et al., 2019) recently showed that the Douglas-Rachford algorithm provides certificates of infeasibility for a class of convex optimization problems. In particular, they showed that the difference between consecutive…
Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization…
We propose a new decomposition framework for continuous nonlinear constrained two-stage optimization, where both first- and second-stage problems can be nonconvex. A smoothing technique based on an interior-point formulation renders the…
In this paper, we study the performance of a large family of SGD variants in the smooth nonconvex regime. To this end, we propose a generic and flexible assumption capable of accurate modeling of the second moment of the stochastic…
We propose a novel adaptive, accelerated algorithm for the stochastic constrained convex optimization setting. Our method, which is inspired by the Mirror-Prox method, \emph{simultaneously} achieves the optimal rates for smooth/non-smooth…
Parallel and cyclic projection algorithms are proposed for minimizing the sum of a finite family of convex functions over the intersection of a finite family of closed convex subsets of a Hilbert space. These algorithms are of…
To facilitate widespread adoption of automated engineering design techniques, existing methods must become more efficient and generalizable. In the field of topology optimization, this requires the coupling of modern optimization methods…
In this paper, we propose a unified convergence analysis for a class of generic shuffling-type gradient methods for solving finite-sum optimization problems. Our analysis works with any sampling without replacement strategy and covers many…
In imaging modalities recording diffraction data, the original image can be reconstructed assuming known phases. When phases are unknown, oversampling and a constraint on the support region in the original object can be used to solve a…
We adapt the Douglas-Rachford (DR) splitting method to solve nonconvex feasibility problems by studying this method for a class of nonconvex optimization problem. While the convergence properties of the method for convex problems have been…
We present a comprehensive framework that unifies several research areas within the context of vertex-weighted bipartite graphs, providing deeper insights and improved solutions. The fundamental solution concept for each problem involves…
The Douglas-Rachford algorithm is a very popular splitting technique for finding a zero of the sum of two maximally monotone operators. However, the behaviour of the algorithm remains mysterious in the general inconsistent case, i.e., when…
The orienteering problem is a route optimization problem which consists in finding a simple cycle that maximizes the total collected profit subject to a maximum distance limitation. In the last few decades, the occurrence of this problem in…
Due to their wide appearance in environmental settings as well as industrial and medical applications, the Stokes-Darcy problems with different sets of interface conditions establish an active research area in the community of mathematical…
Distributed stochastic optimization has drawn great attention recently due to its effectiveness in solving large-scale machine learning problems. Though numerous algorithms have been proposed and successfully applied to general practical…
Aerodynamic shape optimization has many industrial applications. Existing methods, however, are so computationally demanding that typical engineering practices are to either simply try a limited number of hand-designed shapes or restrict…
An optimization algorithm for nonsmooth nonconvex constrained optimization problems with upper-C2 objective functions is proposed and analyzed. Upper-C2 is a weakly concave property that exists in difference of convex (DC) functions and…