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Submodular function minimization is a fundamental optimization problem that arises in several applications in machine learning and computer vision. The problem is known to be solvable in polynomial time, but general purpose algorithms have…
We introduce a new convex optimization problem, termed quadratic decomposable submodular function minimization. The problem is closely related to decomposable submodular function minimization and arises in many learning on graphs and…
We study distributed composite optimization over networks: agents minimize a sum of smooth (strongly) convex functions, the agents' sum-utility, plus a nonsmooth (extended-valued) convex one. We propose a general unified algorithmic…
The stochastic gradient descent has been widely used for solving composite optimization problems in big data analyses. Many algorithms and convergence properties have been developed. The composite functions were convex primarily and…
This paper develops a novel approach to necessary optimality conditions for constrained variational problems defined in generally incomplete subspaces of absolutely continuous functions. Our approach involves reducing a variational problem…
Nonconvex optimization problems arise in many areas of computational science and engineering and are (approximately) solved by a variety of algorithms. Existing algorithms usually only have local convergence or subsequence convergence of…
A general class of nonconvex optimization problems is considered, where the penalty is the composition of a linear operator with a nonsmooth nonconvex mapping, which is concave on the positive real line. The necessary optimality condition…
We develop a rigorous framework for global non-convex optimization by reformulating the minimization problem as a discounted infinite-horizon optimal control problem. For non-convex, continuous, and possibly non-smooth objective functions…
Functional constrained optimization is becoming more and more important in machine learning and operations research. Such problems have potential applications in risk-averse machine learning, semisupervised learning, and robust optimization…
We study the set of continuous functions that admit no spurious local optima (i.e. local minima that are not global minima) which we term \textit{global functions}. They satisfy various powerful properties for analyzing nonconvex and…
Composite minimization is a powerful framework in large-scale convex optimization, based on decoupling of the objective function into terms with structurally different properties and allowing for more flexible algorithmic design. We…
This paper proposes a novel technique called "successive stochastic smoothing" that optimizes nonsmooth and discontinuous functions while considering various constraints. Our methodology enables local and global optimization, making it a…
In this paper we consider convex optimization problems with stochastic composite objective function subject to (possibly) infinite intersection of constraints. The objective function is expressed in terms of expectation operator over a sum…
In this paper, we present a generic framework to extend existing uniformly optimal convex programming algorithms to solve more general nonlinear, possibly nonconvex, optimization problems. The basic idea is to incorporate a local search…
We extend the standard notion of self-concordance to non-convex optimization and develop a family of second-order algorithms with global convergence guarantees. In particular, two function classes -- \textit{weakly self-concordant}…
Decomposition techniques for linear programming are difficult to extend to conic optimization problems with general non-polyhedral convex cones because the conic inequalities introduce an additional nonlinear coupling between the variables.…
There is a recent surge of interest in nonconvex reformulations via low-rank factorization for stochastic convex semidefinite optimization problem in the purpose of efficiency and scalability. Compared with the original convex formulations,…
Continuous optimization is an important problem in many areas of AI, including vision, robotics, probabilistic inference, and machine learning. Unfortunately, most real-world optimization problems are nonconvex, causing standard convex…
We consider the task of minimizing the sum of convex functions stored in a decentralized manner across the nodes of a communication network. This problem is relatively well-studied in the scenario when the objective functions are smooth, or…
We investigate finite-dimensional constrained structured optimization problems, featuring composite objective functions and set-membership constraints. Offering an expressive yet simple language, this problem class provides a modeling…