We take a new approach to investigate synchronization in networks of coupled oscillators. We show that the coupled oscillator system when restricted to a proper region is a distributed partial primal-dual gradient algorithm for solving a well-defined convex optimization problem and its dual. We characterize conditions for synchronization solution of the KKT system of the optimization problem, b...

Transmit waveform design is one of the most important problems in active sensing and communication systems. This problem, due to the complexity and non-convexity, has been always the main topic of many papers for the decades. However, still an optimal solution which guarantees a global minimum for this multi-variable optimization problem is not found. In this paper, we propose an attracting met...

Deployment is a critical issue affecting the quality of service of camera networks. The deployment aims at adopting the least number of cameras to cover the whole scene, which may have obstacles to occlude the line of sight, with expected observation quality. This is generally formulated as a non-convex optimization problem, which is hard to solve in polynomial time. In this paper, we propose a...

We present an accelerated gradient method for non-convex optimization problems with Lipschitz continuous first and second derivatives. The method requires time O( −7/4 log(1/ )) to find an -stationary point, meaning a point x such that ‖∇f(x)‖ ≤ . The method improves upon the O( −2) complexity of gradient descent and provides the additional second-order guarantee that ∇f(x) −O( )I for the compu...

We develop primal-dual algorithms for distributed training of linear models in the Spark framework. We present the ProxCoCoA+ method which represents a generalization of the CoCoA+ algorithm and extends it to the case of general strongly convex regularizers. A primal-dual convergence rate analysis is provided along with an experimental evaluation of the algorithm on the problem of elastic net r...

The high level purpose of this paper is to describe some recent advances in the field of Mathematics called Real Algebraic Geometry, and discuss some of its applications to complexity theory, and non-convex optimization. In particular, one of the questions underlying the entire development, is the crucial question: What makes an optimization problem difficult or easy? Along the way, we try to p...

We consider the fundamental problem in non-convex optimization of efficiently reaching a stationary point. In contrast to the convex case, in the long history of this basic problem, the only known theoretical results on first-order non-convex optimization remain to be full gradient descent that converges in O(1/ε) iterations for smooth objectives, and stochastic gradient descent that converges ...

This survey contains recent developments for computing verified results of convex constrained optimization problems, with emphasis on applications. Especially, we consider the computation of verified error bounds for non-smooth convex conic optimization in the framework of functional analysis, for linear programming, and for semidefinite programming. A discussion of important problem transforma...