Joint uplink and downlink scheduling and UAV trajectory design in the presence of multiple unfriendly jammers and eavesdroppers

In this paper, we investigate secure uplink and downlink communications between an unmanned aerial vehicle (UAV) and multiple user equipments (UEs) in the presence of multiple ground-based eavesdroppers (EVs) and unfriendly jammers. In order to guarantee the secure uplink and downlink transmissions, we consider a novel secure transmission scheme, which involves a power splitting based downlink transmission and scheduling of uplink and downlink transmission. Explicitly, we aim to maximize the average secrecy rate (ASR) by optimizing the UAV trajectory, the transmit power of the UAV and UEs, and scheduling of uplink and downlink transmission. Although the formulated problem is nonconvex, we propose an efficient solution by jointly applying the techniques of block coordinate descent and successive convex approximation. Numerical results show that the proposed scheme achieves a better ASR than the benchmark schemes.     

Survey of Modified Classical Bifurcations Due to Deformation-Sensitive Loading with Nonlinear,Nonsmooth and Nonconvex Effects

Some loading devices show certain deformation-sensitive but conservative characteristics leading to changes of the classical postbifurcation behavior of structures. To compare the effect of dead and configuration-dependent loading devices, classical bifurcation models are considered. Fully nonlinear behavior is considered, in sense of material, geometry and loading. Originally or approximately nonsmooth functions are included with nonconvexity in the material or loading behaviour. Global equilibrium paths are modified by deformation-sensitive loading devices, nonsmoothness and nonconvexity.     

Proximal Point Algorithm On Riemannian Manifolds

Abstract

In this paper we consider the minimization problem with constraints. We will show that if the set of constraints is a Riemannian manifold of nonpositive sectional curvature, and the objective function is convex in this manifold, then the proximal point method in Euclidean space is naturally extended to solve that class of problems. We will prove that the sequence generated by our method is well defined and converge to a minimizer point. In particular we show how tools of Riemannian geometry, more specifically the convex analysis in Riemannian manifolds, can be used to solve nonconvex constrained problem in Euclidean, space.     

Necessary and sufficient condition for local minima of a class of nonconvex quadratic programs

The author (1992, 1993) earlier studied the equivalence of a class of 0–1 quadratic programs and their relaxed problems. Thus, a class of combinatorial optimization problems can be solved by solving a class of nonconvex quadratic programs. In this paper, a necessary and sufficient condition for local minima of this class of nonconvex quadratic programs is given; this will be the foundation for study of algorithms.Research supported by Huo Yingdong Educational Foundation '93.     

Optimal control of a fine structure

An optimal control problem for a multivalued system governed by a nonconvex variational problem, involving a regularization parameter >0, is proposed and studied. The solution to the variational problem exhibits typically rapid oscillations (a so-called fine structure) corresponding to a multiphase state of the material. We want to control only this fine structure. Existence of an optimal control is proved. Its convergence with 0 is studied by means of an optimal control problem for a relaxed variational problem involving (suitably generalized) Young measures. The uniqueness of the solution to the relaxed variational problem, which is nontrivial but is very important in the context of optimal control, is studied in special cases. A finite-element approximation is proposed.The second author gratefully acknowledges support for this research by the Alexander von Humboldt Foundation during his stay at the Institute for Mathematics of the University of Augsburg.     

Nonconvex energy functions,related eigenvalue hemivariational inequalities on the sphere and applications

The present paper deals with a new type of eigenvalue problems arising in problems involving nonconvex nonsmooth energy functions. They lead to the search of critical points (e.g. local minima) for nonconvex nonsmooth potential functions which in turn give rise to hemivariational inequalities. For this type of variational expressions the eigenvalue problem is studied here concerning the existence and multiplicity of solutions by applying a critical point theory appropriate for nonsmooth nonconvex functionals.     

On the Existence and Stability of Solutions for a System of Elliptic Equations

We show, using a dual variational method developed in the paper, the existence of solutions for a system of Dirichlet problems for partial differential equations involving p– and q–Laplacian. The stability of solutions and the existence of positive solutions is considered in the superlinear case.      

Quadratic Majorization for Nonconvex Loss with Applications to the Boosting Algorithm

Classical robust statistical methods dealing with noisy data are often based on modifications of convex loss functions. In recent years, nonconvex loss-based robust methods have been increasingly popular. A nonconvex loss can provide robust estimation for data contaminated with outliers. The significant challenge is that a nonconvex loss can be numerically difficult to optimize. This article proposes quadratic majorization algorithm for nonconvex (QManc) loss. The QManc can decompose a nonconvex loss into a sequence of simpler optimization problems. Subsequently, the QManc is applied to a powerful machine learning algorithm: quadratic majorization boosting algorithm (QMBA). We develop QMBA for robust classification (binary and multi-category) and regression. In high-dimensional cancer genetics data and simulations, the QMBA is comparable with convex loss-based boosting algorithms for clean data, and outperforms the latter for data contaminated with outliers. The QMBA is also superior to boosting when directly implemented to optimize nonconvex loss functions. Supplementary material for this article is available online.     

Separation theorems for nonconvex sets and application in optimization

The aim of this paper is to present separation theorems for two disjoint closed sets, without convexity condition. First, a separation theorem for a given closed cone and a point outside from this cone, is proved and then it is used to prove a separation theorem for two disjoint sets. Illustrative examples are provided to highlight the important aspects of these theorems. An application to optimization is also presented to prove optimality condition for a nonconvex optimization problem.     

A Parallel Algorithm for Large-Scale Nonconvex Penalized Quantile Regression

Penalized quantile regression (PQR) provides a useful tool for analyzing high-dimensional data with heterogeneity. However, its computation is challenging due to the nonsmoothness and (sometimes) the nonconvexity of the objective function. An iterative coordinate descent algorithm (QICD) was recently proposed to solve PQR with nonconvex penalty. The QICD significantly improves the computational speed but requires a double-loop. In this article, we propose an alternative algorithm based on the alternating direction method of multiplier (ADMM). By writing the PQR into a special ADMM form, we can solve the iterations exactly without using coordinate descent. This results in a new single-loop algorithm, which we refer to as the QPADM algorithm. The QPADM demonstrates favorable performance in both computational speed and statistical accuracy, particularly when the sample size n and/or the number of features p are large. Supplementary material for this article is available online.