To construct a parallel approach for solving optimization problems with orthogonality constraints is usually regarded as an extremely difficult mission, due to the low scalability of the orthonormalization procedure. However, such demand is particularly huge in some application areas such as materials computation. In this paper, we propose a proximal linearized augmented Lagrangian algorithm (PLAM) … Read more
We consider distributed smooth nonconvex unconstrained optimization over networks, modeled as a connected graph. We examine the behavior of distributed gradient-based algorithms near strict saddle points. Specifically, we establish that (i) the renowned Distributed Gradient Descent (DGD) algorithm likely converges to a neighborhood of a Second-order Stationary (SoS) solution; and (ii) the more recent class … Read more
The role of the steplength selection strategies in gradient methods has been widely investigated in the last decades. Starting from the work of Barzilai and Borwein (1988), many efficient steplength rules have been designed, that contributed to make the gradient approaches an effective tool for the large-scale optimization problems arising in important real-world applications. Most … Read more
We consider a scalar objective minimization problem over the solution set of another optimization problem. This problem is known as simple bilevel optimization problem and has drawn a significant attention in the last few years. Our inner problem consists of minimizing the sum of smooth and nonsmooth functions while the outer one is the minimization … Read more
Recent papers indicate that some algorithms for constrained optimization may exhibit worst-case complexity bounds that are very similar to those of unconstrained optimization algorithms. A natural question is whether well established practical algorithms, perhaps with small variations, may enjoy analogous complexity results. In the present paper we show that the answer is positive with respect … Read more
The primary focus of this paper is on designing inexact first-order methods for solving large-scale constrained nonlinear optimization problems. By controlling the inexactness of the subproblem solution, we can significantly reduce the computational cost needed for each iteration. A penalty parameter updating strategy during the subproblem solve enables the algorithm to automatically detect infeasibility. Global … Read more
This paper shows that the self-concordance parameter of the universal barrier on any n-dimensional proper convex domain is upper bounded by n. This bound is tight and improves the previous O(n) bound by Nesterov and Nemirovski. The key to our main result is a pair of new, sharp moment inequalities for s-concave distributions, which could … Read more
In the last two decades, many descent methods for multiobjective optimization problems were proposed. In particular, the steepest descent and the Newton methods were studied for the unconstrained case. In both methods, the search directions are computed by solving convex subproblems, and the stepsizes are obtained by an Armijo-type line search. As a consequence, the … Read more
In this paper, we discuss input modeling and solution techniques for several classes of chance constrained programs (CCPs). We propose to use Gaussian mixture models (GMM), a semi-parametric approach, to fit the data available and to model the randomness. We demonstrate the merits of using GMM. We consider several scenarios that arise from practical applications … Read more
The alternating direction method of multipliers(ADMM) has been proved to be effective for solving two-block separable convex optimization subject to linear constraints. However, it is not necessarily convergent when it is extended to multiple-block case directly. One remedy could be regrouping multiple-block variables into two groups firstly and then adopting the classic ADMM to the … Read more