On the Embed and Project Algorithm for the Graph Bandwidth Problem

The graph bandwidth problem, where one looks for a labeling of graph vertices that gives the minimum difference between the labels over all edges, is a classical NP-hard problem that has drawn a lot of attention in recent decades. In this paper, we focus on the so-called Embed and Project Algorithm (EPA) introduced by Blum et al. in 2000,which in the main part has to solve a semidefinite programming relaxation with exponentially many linear constraints. We present several theoretical properties of this special semidefinite programming problem (SDP) and a cutting-plane-like algorithm to solve it, which works very efficiently in combination with interior-point methods or with the bundle method. Extensive numerical results demonstrate that this algorithm, which has only been studied theoretically so far, in practice gives very good labeling for graphs with n≤1000.

Fast Multi-label Learning

Embedding approaches have become one of the most pervasive techniques for multi-label classification. However, the training process of embedding methods usually involves a complex quadratic or semidefinite programming problem, or the model may even involve an NP-hard problem. Thus, such methods are prohibitive on large-scale applications. More importantly, much of the literature has already shown that the binary relevance (BR) method is usually good enough for some applications. Unfortunately, BR runs slowly due to its linear dependence on the size of the input data. The goal of this paper is to provide a simple method, yet with provable guarantees, which can achieve competitive performance without a complex training process. To achieve our goal, we provide a simple stochastic sketch strategy for multi-label classification and present theoretical results from both algorithmic and statistical learning perspectives. Our comprehensive empirical studies corroborate our theoretical findings and demonstrate the superiority of the proposed methods.

Sum-Rate Optimization Constrained by Consumed Power for Multi- Antenna Non-Regenerative Relay Network

Background & Objective: This paper considers a multi-pair wireless network, which communicates peer-to-peer using some multi-antenna amplify-and-forward relays. Maximizing the throughput supposing that the total relay nodes’ power consumption is constrained, is the main objective of this investigation. We prove that finding the beamforming matrix is not a convex problem. Methods: Therefore, by using a semidefinite relaxation technique we find a semidefinite programming problem. Moreover, we propose a novel algorithm for maximizing the total signal to the total leakage ratio. Numerical analyses show the effectiveness of the proposed algorithm which offers higher throughput compared to the existing total leakage minimization algorithm, with much less complexity. Results and Conclusion: Furthermore, the effect of different parameters such as, the number of relays, the number of antennas in each relay, the number of transmitter/receiver pairs and uplink and downlink channel gains are investigated.

Reconfigurable Intelligent Surface assisted Two–Way Communications: Performance Analysis and Optimization

In this paper, we investigate the two-way communication between two users assisted by a re-configurable intelligent surface (RIS). The scheme that two users communicate simultaneously in the same time slot over Rayleigh fading channels is considered. The channels between the two users and RIS can either be reciprocal or non-reciprocal. For reciprocal channels, we determine the optimal phases at the RIS to maximize the signal-to-interference-plus-noise ratio (SINR). We then derive exact closed-form expressions for the outage probability and spectral efficiency for single-element RIS. By capitalizing the insights obtained from the single-element analysis, we introduce a gamma approximation to model the product of Rayleigh random variables which is useful for the evaluation of the performance metrics in multiple-element RIS. Asymptotic analysis shows that the outage decreases at $\left(\log(\rho)/\rho\right)^L$ rate where $L$ is the number of elements, whereas the spectral efficiency increases at $\log(\rho)$ rate at large average SINR $\rho$. For non-reciprocal channels, the minimum user SINR is targeted to be maximized. For single-element RIS, closed-form solutions are derived whereas for multiple-element RIS the problem turns out to be non-convex. The latter is relaxed to be a semidefinite programming problem, whose optimal solution is achievable and serves as a sub-optimal solution.

Reconfigurable Intelligent Surface assisted Two–Way Communications: Performance Analysis and Optimization

In this paper, we investigate the two-way communication between two users assisted by a re-configurable intelligent surface (RIS). The scheme that two users communicate simultaneously in the same time slot over Rayleigh fading channels is considered. The channels between the two users and RIS can either be reciprocal or non-reciprocal. For reciprocal channels, we determine the optimal phases at the RIS to maximize the signal-to-interference-plus-noise ratio (SINR). We then derive exact closed-form expressions for the outage probability and spectral efficiency for single-element RIS. By capitalizing the insights obtained from the single-element analysis, we introduce a gamma approximation to model the product of Rayleigh random variables which is useful for the evaluation of the performance metrics in multiple-element RIS. Asymptotic analysis shows that the outage decreases at $\left(\log(\rho)/\rho\right)^L$ rate where $L$ is the number of elements, whereas the spectral efficiency increases at $\log(\rho)$ rate at large average SINR $\rho$. For non-reciprocal channels, the minimum user SINR is targeted to be maximized. For single-element RIS, closed-form solutions are derived whereas for multiple-element RIS the problem turns out to be non-convex. The latter is relaxed to be a semidefinite programming problem, whose optimal solution is achievable and serves as a sub-optimal solution.

Linear Matrix Inequalities in Stability Problems: Retrospective and Theoretical Aspects

Some aspects of the development of the theory of linear matrix inequalities are considered. A number of results obtained at the initial stage of the development of this theory, both in the development of numerical methods and in obtaining analytical conditions for their solvability, are highlighted. The main attention is focused on the system of linear matrix inequalities arising in solving the absolute stabi lity problem. E. S. Pyatnitskiy and his followers showed that the solvability of this system is a criterion for the existence of a quadratic Lyapunov function and a sufficient condition for absolute stability. The prerequisites leading to this result are considered here. The use of the considered system of inequalities for studying the stability of hybrid systems described by differential inclusions and switching systems is shown. An analysis is given of citing some works of Pyatnitskiy’s school on the theory of stability and the theory of systems of linear matrix inequalities, from which the relevance of the results of these works at the present time follows.In developing numerical methods, it was first shown in the work of Pyatnitskiy and Skorodinskiy that the solvability problem for a system of linear matrix inequalities reduces to a convex programming problem. An interesting gradient algorithm for finding solutions to such a system is also presented. In analyzing analytical conditions of solvability, an unsolvability criterion for the system of our interest obtained by Kamenetskiy and Pyatnitskiy is noted. In modern terms, this result can be considered as a description of an admissible set in the dual semidefinite programming problem. A similar result is given in the famous book by S. Boyd et al. The paper shows that the result of Boyd et al. is a simple corollary of the unsolvability criterion. Here the unsolvability criterion is generalized and refined.

Prestress Design of Tensegrity Structures Using Semidefinite Programming

Finding appropriate prestresses which can stabilize the system is a key step in the design of tensegrity structures. A semidefinite programming- (SDP-) based approach is developed in this paper to determine appropriate prestresses for tensegrity structures. Three different stability criteria of tensegrity structures are considered in the proposed approach. Besides, the unilateral property of members and the evenness of internal forces are taken into account. The stiffness of the whole system can also be optimized by maximizing the minimum eigenvalue of the tangent stiffness matrix. Deterministic algorithms are used to solve the semidefinite programming problem in polynomial time. The applicability of the proposed approach is verified by three typical examples. Compared to previous stochastic-based approaches, the global optimality of the solution of the proposed approach is theoretically guaranteed and the solution is exactly reproducible.

An SQP-type Method with Superlinear Convergence for Nonlinear Semidefinite Programming

A sequentially semidefinite programming method is proposed for solving nonlinear semidefinite programming problem (NLSDP). Inspired by the sequentially quadratic programming (SQP) method, the algorithm generates a search direction by solving a quadratic semidefinite programming subproblem at each iteration. The [Formula: see text] exact penalty function and a line search strategy are used to determine whether the trial step can be accepted or not. Under mild assumptions, the proposed algorithm is globally convergent. In order to avoid the Maratos effect, we present a modified SQP-type algorithm with the second-order correction step and prove that the fast local superlinear convergence can be obtained under the strict complementarity and the second-order sufficient condition with the sigma term. Finally, some numerical experiments are given to show the effectiveness of the algorithm.