This paper is devoted to studying the global and finite convergence of the semi-smooth Newton method for solving a piecewise linear system that arises in cone-constrained quadratic programming problems and absolute value equations. We first provide a negative answer via a counterexample to a conjecture on the global and finite convergence of the Newton iteration … Read more
In this paper, we show enhanced upper bounds of the nontrivial n_1 × n_2 × n_3 points problem for every n_1
Motivated by the prediction-correction framework constructed by He and Yuan [SIAM J. Numer. Anal. 50: 700-709, 2012], we propose a unified prediction-correction framework to accelerate Lagrangian-based methods. More precisely, for strongly convex optimization, general linearized Lagrangian method with indefinite proximal term, alternating direction method of multipliers (ADMM) with the step size of Lagrangian multiplier not … Read more
Federated learning (FL) has found many important applications in smart-phone-APP based machine learning applications. Although many algorithms have been studied for FL, to the best of our knowledge, algorithms for FL with nonconvex constraints have not been studied. This paper studies FL over Riemannian manifolds, which finds important applications such as federated PCA and federated … Read more
Bilevel optimization has been successfully applied to many important machine learning problems. Algorithms for solving bilevel optimization have been studied under various settings. In this paper, we study the nonconvex-strongly-convex bilevel optimization under a decentralized setting. We design decentralized algorithms for both deterministic and stochastic bilevel optimization problems. Moreover, we analyze the convergence rates of … Read more
In deterministic optimization, it is typically assumed that all parameters of the problem are fixed and known. In practice, however, some parameters may be a priori unknown but can be estimated from historical data. A typical predict-then-optimize approach separates predictions and optimization into two stages. Recently, end-to-end predict-then-optimize has become an attractive alternative. In this … Read more
We study a pessimistic stochastic bilevel program in the context of sequential two-player games, where the leader makes a binary here-and-now decision, and the follower responds a continuous wait-and-see decision after observing the leader’s action and revelation of uncertainty. Only the information of the mean, covariance, and support is known. We formulate the problem as … Read more
We consider the two-bar charts packing (2-BCPP), a recent combinatorial optimization problem whose aim is to pack a set of one-dimensional items into the minimum number of bins. As opposed to the well-known bin packing problem, pairs of items are grouped to form bar charts, and a solution is only feasible if the first and … Read more
We consider using gradient descent to minimize the nonconvex function $f(X)=\phi(XX^{T})$ over an $n\times r$ factor matrix $X$, in which $\phi$ is an underlying smooth convex cost function defined over $n\times n$ matrices. While only a second-order stationary point $X$ can be provably found in reasonable time, if $X$ is additionally \emph{rank deficient}, then its … Read more
We introduce a class of algorithms, called Trajectory Following Dynamic Programming (TFDP) algorithms, that iteratively refines approximation of cost-to-go functions of multistage stochastic problems with independent random variables. This framework encompasses most variants of the Stochastic Dual Dynamic Programming algorithm. Leveraging a Lipschitz assumption on the expected cost-to-go functions, we provide a new convergence and … Read more