An Accelerated Proximal Algorithm for the Difference of Convex Programming

In this paper, we propose an accelerated proximal point algorithm for the difference of convex (DC) optimization problem by combining the extrapolation technique with the proximal difference of convex algorithm. By making full use of the special structure of DC decomposition and the information of stepsize, we prove that the proposed algorithm converges at rate of O 1 / k 2 under milder conditions. The given numerical experiments show the superiority of the proposed algorithm to some existing algorithms.

Synthesizing Invariant Barrier Certificates via Difference-of-Convex Programming

A barrier certificate often serves as an inductive invariant that isolates an unsafe region from the reachable set of states, and hence is widely used in proving safety of hybrid systems possibly over the infinite time horizon. We present a novel condition on barrier certificates, termed the invariant barrier-certificate condition, that witnesses unbounded-time safety of differential dynamical systems. The proposed condition is by far the least conservative one on barrier certificates, and can be shown as the weakest possible one to attain inductive invariance. We show that discharging the invariant barrier-certificate condition—thereby synthesizing invariant barrier certificates—can be encoded as solving an optimization problem subject to bilinear matrix inequalities (BMIs). We further propose a synthesis algorithm based on difference-of-convex programming, which approaches a local optimum of the BMI problem via solving a series of convex optimization problems. This algorithm is incorporated in a branch-and-bound framework that searches for the global optimum in a divide-and-conquer fashion. We present a weak completeness result of our method, in the sense that a barrier certificate is guaranteed to be found (under some mild assumptions) whenever there exists an inductive invariant (in the form of a given template) that suffices to certify safety of the system. Experimental results on benchmark examples demonstrate the effectiveness and efficiency of our approach.

Robust Power Optimization for Downlink Cloud Radio Access Networks with Physical Layer Security

Since the cloud radio access network (C-RAN) transmits information signals by jointly transmission, the multiple copies of information signals might be eavesdropped on. Therefore, this paper studies the resource allocation algorithm for secure energy optimization in a downlink C-RAN, via jointly designing base station (BS) mode, beamforming and artificial noise (AN) given imperfect channel state information (CSI) of information receivers (IRs) and eavesdrop receivers (ERs). The considered resource allocation design problem is formulated as a nonlinear programming problem of power minimization under the quality of service (QoS) for each IR, the power constraint for each BS, and the physical layer security (PLS) constraints for each ER. To solve this non-trivial problem, we first adopt smooth ℓ 0 -norm approximation and propose a general iterative difference of convex (IDC) algorithm with provable convergence for a difference of convex programming problem. Then, a three-stage algorithm is proposed to solve the original problem, which firstly apply the iterative difference of convex programming with semi-definite relaxation (SDR) technique to provide a roughly (approximately) sparse solution, and then improve the sparsity of the solutions using a deflation based post processing method. The effectiveness of the proposed algorithm is validated with extensive simulations for power minimization in secure downlink C-RANs.