Successive Convex Relaxation Research Articles

Sensing Fairness-Based Energy Efficiency Optimization for UAV Enabled Integrated Sensing and Communication

Integrated sensing and communication (ISAC) is developing rapidly due to the advantages of bandwidth saving, hardware cost reduction and energy saving. This paper proposes a unmanned aerial vehicle (UAV) enabled ISAC (UAV-ISAC) system, where the UAV will sense ground users and forward sensed information to base station (BS). Radar mutual information (MI) is introduced to measure the UAV sensing performance. On the basis of sensing fairness for each user, a multi-objective resource optimization problem of maximizing both energy efficiency (EE) and minimum radar MI is studied by jointly optimizing user scheduling, transmit power and UAV trajectory. To solve the non-convex optimization problem, we decompose it into three sub-problems: user scheduling optimization, transmit power optimization, and UAV trajectory optimization. Each subproblem can be solved by successive convex approximation (SCA), fractional programming and relaxation technique. By iteratively optimizing the three sub-problems, we can obtain the suboptimal solution to original optimization problem. Simulation results show that the proposed optimization scheme can maximize the EE of UAV while ensuring sensing fairness for all users.

Joint Optimization of Trajectory and Discrete Reflection Coefficients for UAV-Aided Backscatter Communication System with NOMA

Backscatter communication is a promising technology for the Internet of Things (IoT) systems with low-energy consumption, in which the data transmission of the backscatter devices relies on reflecting the incident signal. However, limited by the low power characteristic of the reflected signal from backscatter devices, achieving efficient data collection for the widely distributed backscatter devices is a thorny problem. Considering that unmanned aerial vehicles (UAVs) have flexible deployment capability, employing UAVs in a backscatter communication network can achieve feasible data collection for backscatter devices. In this paper, we consider a UAV-aided backscatter system and introduce Non-orthogonal multiple access (NOMA) to enable the UAV to collect signals from multiple backscatter devices simultaneously. We formulate an optimization problem to maximize the communication throughput of the considered system by jointly designing the backscatter device matching, the trajectory of the UAV, and the reflection coefficients of the backscatter devices, which is a non-convex optimization problem and challenging to solve. Hence, we decouple the original problem into three sub-problems and propose an efficient iterative algorithm based on Block Coordinate Descent (BCD) to solve them. In detail, a game-based matching algorithm is designed to ensure the transmission needs of remote backscatter devices. The UAV trajectory and reflection coefficients of backscatter devices are optimized through the Successive Convex Approximation (SCA) algorithm and relaxation algorithm. By iterative optimization of the sub-problems, the original problem is solved. The simulation results show that the proposed scheme can obtain a significant throughput gain compared to benchmark schemes.

Optimal Transmission Strategy and Time Allocation for RIS-Enhanced Partially WPSNs

Wireless powered sensor networks (WPSNs) have evolved as a promising paradigm for energy-efficient communications. Recently, the proliferation of reconfigurable intelligent surface (RIS) has further been envisioned as a cost-effective solution for improving wireless power transfer (WPT) efficiency. In this paper, from the practical perspective of balancing the network sustainability and reliability, we consider a RIS-enhanced partially WPSN that composed of wireless-powered energy receivers (ERs) and battery-powered information receivers (IRs). Assuming the partially WPSN operates in time division multiple access (TDMA) mode, the joint optimization of covariance matrices, downlink/uplink (DL/UL) time allocation and RIS reflecting coefficients are investigated under the minimum DL rate constraint among all IRs for maximizing the achievable UL sum rate. Specifically, the single-IR single-ER (SISE) case is first studied based on the assumption of separate DL/UL RIS reflecting coefficients, in which an alternating optimization algorithm is proposed with semi-closed-form optimal solutions. In order to reduce the hardware overhead and signal processing complexity, we also investigate the case of identical DL/UL RIS reflecting coefficients, in which an iterative optimization algorithm is developed to tackle the coupled DL/UL transmissions. Then, we extend our work to the multiple-IRs multiple-ERs (MIME) case, where both the optimization problems corresponding to separate and identical DL/UL RIS reflecting schemes become more challenging to solve. To circumvent this intractability, we propose a successive convex relaxation (SCA) based alternating optimization algorithm and a low-complexity two-step algorithm. Finally, numerical results demonstrate the superior UL sum rate performance of our proposed algorithms over the adopted benchmarks.

Reconfigurable Intelligent Surface Empowered Downlink Non-Orthogonal Multiple Access

Power-domain non-orthogonal multiple access (NOMA) has become a promising technology to exploit the new dimension of the power domain to enhance the spectral efficiency of wireless networks. However, most existing NOMA schemes rely on the strong assumption that users’ channel gains are quite different, which may be invalid in practice. To unleash the potential of power-domain NOMA, we propose a reconfigurable intelligent surface (RIS)-empowered NOMA scheme to introduce desirable channel gain differences among the users by adjusting the phase shifts at the RIS. Our goal is to minimize the total transmit power by jointly optimizing the beamforming vectors at the base station, the phase-shift matrix at the RIS, and user ordering. To address challenge due to the highly coupled optimization variables, we present an alternating optimization framework to decompose the non-convex bi-quadratically constrained quadratic problem under a specific user ordering into two rank-one constrained matrices optimization problems via matrix lifting. To accurately detect the feasibility of the non-convex rank-one constraints and improve performance by avoiding early stopping in the alternating optimization procedure, we equivalently represent the rank-one constraint as the difference between nuclear norm and spectral norm. A difference-of-convex (DC) algorithm is further developed to solve the resulting DC programs via successive convex relaxation, followed by establishing the convergence of the proposed DC-based alternating optimization method. We further propose an efficient user ordering scheme with closed-form expressions, considering both the channel conditions and users’ target data rates. Simulation results validate the ability of an RIS in enlarging the channel-gain difference when the users’ original channel conditions are similar and the superiority of the proposed DC-based alternating optimization method in reducing the total transmit power.

Sparse Array Beamforming Design for Wideband Signal Models

We develop sparse array receive beamformer design methods achieving maximum signal-to-interference plus noise ratio (MaxSINR) for wideband sources and jammers. Both tapped delay line (TDL) filtering and the DFT realizations to wideband array processing are considered. The array sparsity stems from the limited number of available RF transmission chains that switch between the sensors, thereby configuring different arrays at different times. The sparse array configuration design problem is formulated as a quadratically constraint quadratic program (QCQP) and solved by using SDR (semidefinite relaxation). A computationally viable approach through SCA (successive convex relaxation) is also pursued. In order to realize an implementable design, in presence of missing autocorrelation lags, we propose parameter-free block Toeplitz matrix completion to estimate the received data correlation matrix across the entire array aperture. It is shown that the optimum wideband sparse array effectively utilizes the array aperture and provides considerable performance improvement over suboptimal array topologies.

Atomic Scheduling of Appliance Energy Consumption in Residential Smart Grids

Most of the current formulations of the optimal scheduling of appliance energy consumption use the vectors of appliances’ scheduled energy consumption over equally divided time slots of a day as optimization variables, which does not take into account the atomicity of certain appliances’ operations, i.e., the non-interruptibility of appliances’ operations and the non-throttleability of the energy consumption patterns specific to their operations. In this paper, we provide a new formulation of atomic scheduling of energy consumption based on the optimal routing framework; the flow configurations of users over multiple paths between the common source and destination nodes of a ring network are used as optimization variables, which indicate the starting times of scheduled energy consumption, and optimal scheduling problems are now formulated in terms of the user flow configurations. Because the atomic optimal scheduling results in a Boolean-convex problem for a convex objective function, we propose a successive convex relaxation technique for efficient calculation of an approximate solution, where we iteratively drop fractional-valued elements and apply convex relaxation to the resulting problem until we find a feasible suboptimal solution. Numerical results for the cost and peak-to-average ratio minimization problems demonstrate that the successive convex relaxation technique can provide solutions close to and often identical to global optimal solutions.

Approximate solution algorithm for multi-parametric non-convex programming problems with polyhedral constraints

In this paper, we developed a novel algorithmic approach for thesolution of multi-parametric non-convex programming problems withcontinuous decision variables. The basic idea of the proposedapproach is based on successive convex relaxation of each non-convexterms and sensitivity analysis theory. The proposed algorithm isimplemented using MATLAB software package and numericalexamples are presented to illustrate the effectiveness andapplicability of the proposed method on multi-parametric non-convexprogramming problems with polyhedral constraints.

SOLVING LARGE SCALE OPTIMIZATION PROBLEMS VIA GRID AND CLUSTER COMPUTING(<Special Issue>Network Design, Control and Optimization)

Solving large scale optimization problems requires a huge amount of com- putational power. The size of optimization problems that can be solved on a few CPUs has been limited due to a lack of computational power. Grid and cluster computing has received much attention as a powerful and inexpensive way of solving large scale opti- mization problems that an existing single CPU cannot process. The aim of this paper is to show that grid and cluster computing provides tremendous power to optimization methods. The methods that this article picks up are a successive convex relaxation method for quadratic optimization problems, a polyhedral homotopy method for poly- nomial systems of equations and a primal-dual interior-point method for semidefinite programs. Their parallel implementations on grids and clusters together with numerical results are reported. The article also mentions a grid portal system for optimization problems briefly.

Solving Nonconvex Problems of Multibody Dynamics with Joints, Contact, and Small Friction by Successive Convex Relaxation

Time-stepping methods using impulse-velocity approaches are guaranteed to have a solution for any friction coefficient, but they may have nonconvex solution sets. We present an example of a configuration with a nonconvex solution set for any nonzero value of the friction coefficient. We construct an iterative algorithm that solves convex subproblems and that is guaranteed, for sufficiently small friction coefficients, to retrieve, at a linear convergence rate, the velocity solution of the nonconvex linear complementarity problem whenever the frictionless configuration can be disassembled. In addition, we show that one step of the iterative algorithm provides an excellent approximation to the velocity solution of the original, possibly nonconvex, problem if the product between the friction coefficient and the slip velocity is small.

Parallel Implementation of Successive Convex Relaxation Methods for Quadratic Optimization Problems

As computing resources continue to improve, global solutions for larger size quadrically constrained optimization problems become more achievable. In this paper, we focus on larger size problems and get accurate bounds for optimal values of such problems with the successive use of SDP relaxations on a parallel computing system called Ninf (Network-based Information Library for high performance computing).

Some Fundamental Properties of Successive Convex Relaxation Methods on LCP and Related Problems

General successive convex relaxation methods (SRCMs) can be used to compute the convex hull of any compact set, in an Euclidean space, described by a system of quadratic inequalities and a compact convex set. Linear complementarity problems (LCPs) make an interesting and rich class of structured nonconvex optimization problems. In this paper, we study a few of the specialized lift-and-project methods and some of the possible ways of applying the general SCRMs to LCPs and related problems.

Complexity Analysis of Successive Convex Relaxation Methods for Nonconvex Sets

This paper discusses computational complexity of conceptual successive convex relaxation methods proposed by Kojima and Tunçel for approximating a convex relaxation of a compact subset F = {x ∈ C0 : p(x) ≤ 0 (∀p(·) ∈ 𝒫F)} of the n-dimensional Euclidean space Rn. Here, C0 denotes a nonempty compact convex subset of Rn, and 𝒫F a set of finitely or infinitely many quadratic functions. We evaluate the number of iterations which the successive convex relaxation methods require to attain a convex relaxation of F with a given accuracy ε, in terms of ε, the diameter of C0, the diameter of F, and some other quantities characterizing the Lipschitz continuity, the nonlinearity, and the nonconvexity of the set 𝒫F of quadratic functions.

Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization

Based on the authors’ previous work which established theoretical foundations of two, conceptual, successive convex relaxation methods, i.e., the SSDP (Successive Semidefinite Programming) Relaxation Method and the SSILP (Successive Semi-Infinite Linear Programming) Relaxation Method, this paper proposes their implementable variants for general quadratic optimization problems. These problems have a linear objective function c T x to be maximized over a nonconvex compact feasible region F described by a finite number of quadratic inequalities. We introduce two new techniques, “discretization” and “localization,” into the SSDP and SSILP Relaxation Methods. The discretization technique makes it possible to approximate an infinite number of semi-infinite SDPs (or semi-infinite LPs) which appeared at each iteration of the original methods by a finite number of standard SDPs (or standard LPs) with a finite number of linear inequality constraints. We establish:¶•Given any open convex set U containing F, there is an implementable discretization of the SSDP (or SSILP) Relaxation Method which generates a compact convex set C such that F⊆C⊆U in a finite number of iterations.¶The localization technique is for the cases where we are only interested in upper bounds on the optimal objective value (for a fixed objective function vector c) but not in a global approximation of the convex hull of F. This technique allows us to generate a convex relaxation of F that is accurate only in certain directions in a neighborhood of the objective direction c. This cuts off redundant work to make the convex relaxation accurate in unnecessary directions. We establish:¶•Given any positive number e, there is an implementable localization-discretization of the SSDP (or SSILP) Relaxation Method which generates an upper bound of the objective value within e of its maximum in a finite number of iterations.