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This paper investigates the box-constrained $\ell_0$-regularized sparse optimization problem. We introduce the concept of a $\tau$-stationary point and establish its connection to the local and global minima of the box-constrained…

We study the performance of stochastic first-order methods for finding saddle points of convex-concave functions. A notorious challenge faced by such methods is that the gradients can grow arbitrarily large during optimization, which may…

We consider a class of nonconvex nonsmooth multicomposite optimization problems where the objective function consists of a Tikhonov regularizer and a composition of multiple nonconvex nonsmooth component functions. Such optimization…

In this paper, we construct local Morse homology in a new way and compute the local Morse homology of the critical points of the Lagrange problem. As a corollary, we prove for the first time that each of the linear critical points is either…

Crafting adversarial examples can be formulated as an optimization problem. While sign-based optimizers such as I-FGSM and MI-FGSM have become the de facto standard for the induced optimization problems, there still exist several unsolved…

Large-scale non-convex optimization problems are expensive to solve due to computational and memory costs. To reduce the costs, first-order (computationally efficient) and asynchronous-parallel (memory efficient) algorithms are necessary to…

A basic idea in optimal transport is that optimizers can be characterized through a geometric property of their support sets called cyclical monotonicity. In recent years, similar "monotonicity principles" have found applications in other…

This paper settles the computational complexity of model checking of several extensions of the monadic second order (MSO) logic on two classes of graphs: graphs of bounded treewidth and graphs of bounded neighborhood diversity. A classical…

We propose a regularized saddle-point algorithm for convex networked optimization problems with resource allocation constraints. Standard distributed gradient methods suffer from slow convergence and require excessive communication when…

We study solution methods for (strongly-)convex-(strongly)-concave Saddle-Point Problems (SPPs) over networks of two type - master/workers (thus centralized) architectures and meshed (thus decentralized) networks. The local functions at…

This paper investigates the stability properties of neural operators through the structured representation offered by the Hybrid B-spline Deep Neural Operator (HBDNO). While existing stability-aware architectures typically enforce…

Many real-world optimization problems can be stated in terms of submodular functions. Furthermore, these real-world problems often involve uncertainties which may lead to the violation of given constraints. A lot of evolutionary…

We study the optimization landscape of deep linear neural networks with the square loss. It is known that, under weak assumptions, there are no spurious local minima and no local maxima. However, the existence and diversity of non-strict…

The Sinkhorn algorithm is the most popular method for solving the entropy minimization problem called the Schr\"odinger problem: in the non-degenerate cases, the latter admits a unique solution towards which the algorithm converges…

The multi-objective optimization is to optimize several objective functions over a common feasible set. Since the objectives usually do not share a common optimizer, people often consider (weakly) Pareto points. This paper studies…

The Monge-Kantorovich problem is revisited by means of a variant of the saddle-point method without appealing to $c$-conjugates. A new abstract characterization of the optimal plans is obtained in the case where the cost function takes…

Stochastic approximation (SA) that involves multiple coupled sequences, known as multiple-sequence SA (MSSA), finds diverse applications in the fields of signal processing and machine learning. However, existing theoretical understandings…

Many machine learning algorithms minimize a regularized risk, and stochastic optimization is widely used for this task. When working with massive data, it is desirable to perform stochastic optimization in parallel. Unfortunately, many…

We consider optimization problems with a disjunctive structure of the constraints. Prominent examples of such problems are mathematical programs with equilibrium constraints or vanishing constraints. Based on the concepts of directional…

In optimization routines used for on-line Model Predictive Control (MPC), linear systems of equations are usually solved in each iteration. This is true both for Active Set (AS) methods as well as for Interior Point (IP) methods, and for…