SDP-Based Bounds for the Quadratic Cycle Cover Problem via Cutting-Plane Augmented Lagrangian Methods and Reinforcement Learning

We study the quadratic cycle cover problem (QCCP), which aims to find a node-disjoint cycle cover in a directed graph with minimum interaction cost between successive arcs. We derive several semidefinite programming (SDP) relaxations and use facial reduction to make these strictly feasible. We investigate a nontrivial relationship between the transformation matrix used in the reduction and the structure of the graph, which is exploited in an efficient algorithm that constructs this matrix for any instance of the problem. To solve our relaxations, we propose an algorithm that incorporates an augmented Lagrangian method into a cutting-plane framework by utilizing Dykstra’s projection algorithm. Our algorithm is suitable for solving SDP relaxations with a large number of cutting-planes. Computational results show that our SDP bounds and efficient cutting-plane algorithm outperform other QCCP bounding approaches from the literature. Finally, we provide several SDP-based upper bounding techniques, among which is a sequential Q-learning method that exploits a solution of our SDP relaxation within a reinforcement learning environment. Summary of Contribution: The quadratic cycle cover problem (QCCP) is the problem of finding a set of node-disjoint cycles covering all the nodes in a graph such that the total interaction cost between successive arcs is minimized. The QCCP has applications in many fields, among which are robotics, transportation, energy distribution networks, and automatic inspection. Besides this, the problem has a high theoretical relevance because of its close connection to the quadratic traveling salesman problem (QTSP). The QTSP has several applications, for example, in bioinformatics, and is considered to be among the most difficult combinatorial optimization problems nowadays. After removing the subtour elimination constraints, the QTSP boils down to the QCCP. Hence, an in-depth study of the QCCP also contributes to the construction of strong bounds for the QTSP. In this paper, we study the application of semidefinite programming (SDP) to obtain strong bounds for the QCCP. Our strongest SDP relaxation is very hard to solve by any SDP solver because of the large number of involved cutting-planes. Because of that, we propose a new approach in which an augmented Lagrangian method is incorporated into a cutting-plane framework by utilizing Dykstra’s projection algorithm. We emphasize an efficient implementation of the method and perform an extensive computational study. This study shows that our method is able to handle a large number of cuts and that the resulting bounds are currently the best QCCP bounds in the literature. We also introduce several upper bounding techniques, among which is a distributed reinforcement learning algorithm that exploits our SDP relaxations.

Efficient Neural Network Verification via Layer-based Semidefinite Relaxations and Linear Cuts

We introduce an efficient and tight layer-based semidefinite relaxation for verifying local robustness of neural networks. The improved tightness is the result of the combination between semidefinite relaxations and linear cuts. We obtain a computationally efficient method by decomposing the semidefinite formulation into layerwise constraints. By leveraging on chordal graph decompositions, we show that the formulation here presented is provably tighter than current approaches. Experiments on a set of benchmark networks show that the approach here proposed enables the verification of more instances compared to other relaxation methods. The results also demonstrate that the SDP relaxation here proposed is one order of magnitude faster than previous SDP methods.

Secrecy Wireless-Powered Sensor Networks for Internet of Things

This paper investigates a secure wireless-powered sensor network (WPSN) with the aid of a cooperative jammer (CJ). A power station (PS) wirelessly charges for a user equipment (UE) and the CJ to securely transmit information to an access point (AP) in the presence of multiple eavesdroppers. Also, the CJ are deployed, which can introduce more interference to degrade the performance of the malicious eavesdroppers. In order to improve the secure performance, we formulate an optimization problem for maximizing the secrecy rate at the AP to jointly design the secure beamformer and the energy time allocation. Since the formulated problem is not convex, we first propose a global optimal solution which employs the semidefinite programming (SDP) relaxation. Also, the tightness of the SDP relaxed solution is evaluated. In addition, we investigate a worst-case scenario, where the energy time allocation is achieved in a closed form. Finally, numerical results are presented to confirm effectiveness of the proposed scheme in comparison to the benchmark scheme.

Improving Lagrange Dual Bounds for Quadratic Extremal Problems

Introduction. Due to the fact that quadratic extremal problems are generally NP-hard, various convex relaxations to find bounds for their global extrema are used, namely, Lagrangian relaxation, SDP-relaxation, SOCP-relaxation, LP-relaxation, and others. This article investigates a dual bound that results from the Lagrangian relaxation of all constraints of quadratic extremal problem. The main issue when using this approach for solving quadratic extremal problems is the quality of the obtained bounds (the magnitude of the duality gap) and the possibility to improve them. While for quadratic convex optimization problems such bounds are exact, in other cases this issue is rather complicated. In non-convex cases, to improve the dual bounds (to reduce the duality gap) the techniques, based on ambiguity of the problem formulation, can be used. The most common of these techniques is an extension of the original quadratic formulation of the problem by introducing the so-called functionally superfluous constraints (additional constraints that result from available constraints). The ways to construct such constraints can be general in nature or they can use specific features of the concrete problems. The purpose of the article is to propose methods for improving the Lagrange dual bounds for quadratic extremal problems by using technique of functionally superfluous constraints; to present examples of constructing such constraints. Results. The general concept of using functionally superfluous constraints for improving the Lagrange dual bounds for quadratic extremal problems is considered. Methods of constructing such constraints are presented. In particular, the method proposed by N.Z. Shor for constructing functionally superfluous constraints for quadratic problems of general form is presented in generalized and schematized forms. Also it is pointed out that other special techniques, which employ the features of specific problems for constructing functionally superfluous constraints, can be used. Conclusions. In order to improve dual bounds for quadratic extremal problems, one can use various families of functionally superfluous constraints, both of general and specific type. In some cases, their application can improve bounds or even provide an opportunity to obtain exact values of global extrema.

Robust Beamforming Design for SWIPT-Based Multi-Radio Wireless Mesh Network with Cooperative Jamming

Wireless mesh networks (WMNs) can provide flexible wireless connections in a smart city, internet of things (IoT), and device-to-device (D2D) communications. The performance of WMNs can be greatly enhanced by adopting a multi-radio technique, which enables a node to communicate with more nodes simultaneously. However, multi-radio WMNs face two main challenges, namely, energy consumption and physical layer secrecy. In this paper, both simultaneous wireless information and power transfer (SWIPT) and cooperative jamming technologies were adopted to overcome these two problems. We designed the SWIPT and cooperative jamming scheme, minimizing the total transmission power by properly selecting beamforming vectors of the WMN nodes and jammer to satisfy the individual signal-to-interference-plus-noise ratio (SINR) and energy harvesting (EH) constrains. Especially, we considered the channel estimate error caused by the imperfect channel state information. The SINR of eavesdropper (Eve) was suppressed to protect the secrecy of WMN nodes. Due to the fractional form, the problem was proved to be non-convex. We developed a tractable algorithm by transforming it into a convex one, utilizing semi-definite programming (SDP) relaxation and S-procedure methods. The simulation results validated the effectiveness of the proposed algorithm compared with the non-robust design.

A New Spatial Branch and Bound Algorithm for Quadratic Program with One Quadratic Constraint and Linear Constraints

This paper proposes a novel second-order cone programming (SOCP) relaxation for a quadratic program with one quadratic constraint and several linear constraints (QCQP) that arises in various real-life fields. This new SOCP relaxation fully exploits the simultaneous matrix diagonalization technique which has become an attractive tool in the area of quadratic programming in the literature. We first demonstrate that the new SOCP relaxation is as tight as the semidefinite programming (SDP) relaxation for the QCQP when the objective matrix and constraint matrix are simultaneously diagonalizable. We further derive a spatial branch-and-bound algorithm based on the new SOCP relaxation in order to obtain the global optimal solution. Extensive numerical experiments are conducted between the new SOCP relaxation-based branch-and-bound algorithm and the SDP relaxation-based branch-and-bound algorithm. The computational results illustrate that the new SOCP relaxation achieves a good balance between the bound quality and computational efficiency and thus leads to a high-efficiency global algorithm.