Semidefinite Programming Problem Research Articles

A Semidefinite Relaxation Approach for Mobile Target Localization Based on TOA and Doppler Frequency Shift Measurements

In this paper, we discuss mobile source localization using both time of arrival and Doppler frequency shift (TOA-DFS) measurements, where the source moves at a nonuniform velocity. To obtain the position of a mobile source, we first formulate the weighted least squares (WLS) problem by ignoring the second-order noise terms. Due to the non-convexity, we apply the convex relaxation technique to transform the problem into a semi-definite programming problem. However, ignoring the second-order noise terms is only reasonable in the case of small noise levels. In view of this, we then directly establish the maximum likelihood (ML) estimator based on the measurements model without ignoring the second-order noise terms. Since the ML estimator is a non-convex problem, we also propose implementable semidefinite relaxation (SDR) technique to tackle it. Finally, the Cramér-Rao lower bound (CRLB) analysis and results verify that the proposed methods based on the TOA-DFS measurements can significantly enhance localization accuracy.

Energy-Efficient Beamforming and Resource Optimization for AmBSC-Assisted Cooperative NOMA IoT Networks

In this manuscript, we present an energy-efficient alternating optimization framework based on the multi-antenna ambient backscatter communication (AmBSC) assisted cooperative non-orthogonal multiple access (NOMA) for next-generation (NG) internet-of-things (IoT) enabled communication networks. Specifically, the energy-efficiency maximization is achieved for the considered AmBSC-enabled multi-cluster cooperative IoT NOMA system by optimizing the active-beamforming vector and power-allocation coefficients (PAC) of IoT NOMA users at the transmitter, as well as passive-beamforming vector at the multi-antenna assisted backscatter node. Usually, increasing the number of IoT NOMA users in each cluster results in inter-cluster interference (ICI) (among different clusters) and intra-cluster interference (among IoT NOMA users). To combat the impact of ICI, we exploit a zero-forcing (ZF) based active-beamforming, as well as an efficient clustering technique at the source node. Further, the effect of intra-cluster interference is mitigated by exploiting an efficient power-allocation policy that determines the PAC of IoT NOMA users under the quality-of-service (QoS), cooperation, SIC decoding, and power-budget constraints. Moreover, the considered non-convex passive-beamforming problem is transformed into a standard semi-definite programming (SDP) problem by exploiting the successive-convex approximation (SCA) approximation, as well as the difference of convex (DC) programming, where Rank-1 solution of passive-beamforming is obtained based on the penalty-based method. Furthermore, the numerical analysis of simulation results demonstrates that the proposed energy-efficiency maximization algorithm exhibits an efficient performance by achieving convergence within only a few iterations.

Attitude control for a full-scale flexible electric solar wind sail spacecraft on heliocentric and displaced non-Keplerian orbits

The electric solar wind sail (E-sail) spacecraft is a fuel-free propulsion system with promising potential in deep space explorations and solar activity monitoring missions. It is challenging to conduct attitude control for a full-scale flexible E-sail spacecraft due to its complex dynamical characteristics, including the spatial multi-scale, high-speed spinning, and large deformations. In this work, a double-loop attitude control strategy for the flexible E-sail spacecraft with high accuracy and efficiency is proposed. The E-sail spacecraft is firstly modeled by the referenced nodal coordinate formulation, where the rotations are described by the Lie group approach. A state-feedback controller developed from the Lyapunov stability theory is prescribed as the inner loop. The gain is obtained by solving a semidefinite programming problem with linear matrix inequality constraints, which is globally optimal and numerically reliable. A multi-variable proportional-integral-derivative controller is then introduced as the outer loop for flexibly enhancing the control performance. Several missions of the E-sail spacecraft, including attitude maneuvers on heliocentric and displaced non-Keplerian orbits, are studied to show that the desired attitude adjustments are accurately and efficiently achieved without exciting high-frequency oscillations. The double-loop control strategy has great prospects in designing and implementing orbital transformation and long-term stable Sun-orbiting missions.

Safety verification for Regime-Switching Jump Diffusions via barrier certificates

It is well known that for a stochastic hybrid system, if its failure probability does not exceed a given safety threshold, the safety of the system can be guaranteed. Thus, in this paper, we concern with the upper bound of failure probability for failure analysis of a class of nonautonomous stochastic hybrid systems, denoted as Regime-Switching Jump Diffusions (RSJDs). We start with the definition of RSJDs, which contain not only continuous flows described by stochastic differential equations, but also instantaneous behavior described by Markovian switching and instantaneous behavior described by Lévy jump. Then we decompose the failure probability of RSJDs in [t0,+∞) into two segments: one is defined over [T,+∞), and the other is defined over [t0,T]. For failure probability over [T,+∞), by utilizing multiple vectorial barrier certificates, an asymptotically decreasing bound of failure probability with respect to T is established, where a general nonnegative matrix is used instead of a special nonnegative matrix defined by the exponent of essentially nonnegative matrix for a broader applicability. For failure probability over [t0,T], a generalized c-martingale condition is adopted to obtain a T-dependent failure probability bound, in which two non-negative scalar functions are utilized to relax the conservativeness of infinitesimal generators. Finally, for rational RSJDs, we transform the decomposed failure analysis problems into two semi-definite programming (SDP) problems, and then solve them via sum of squares programming. The applicabilities and effectiveness of our computable decomposition methodology are illustrated through three examples.

STABILIZATION OF DISCRETE-TIME SYSTEMS WITH STATE-DELAYS AND SATURATING CONTROL INPUTS

Context. The presence of time delays occurs in many complex dynamical systems, particularly in the areas of modern communication and information technologies, such as the problem of stabilizing networked control systems and high-speed communication networks. In many cases, time-delays lead to a decrease in the efficiency of such systems and even to the loss of stability. In the last decade, many interesting solutions using the Lyapunov-Krasovskii functional have been proposed for stability analysis and synthesis of a stabilizing regulator for discrete-time dynamic systems with unknown but bounded state-delays. The presence of nonlinear constraints on the amplitude of controls such as saturation further complicates this problem and requires the development of new approaches and methods.
 Objective. The purpose of this study is to develop a procedure for calculating the control gain matrix of state feedback that ensures the asymptotic stability of the analyzed system, as well as a procedure for calculating the maximum permissible value of the state-delay under which the stability of the closed-loop system can be ensured for a given set of admissible initial conditions.
 Method. The paper uses the method of descriptor transformation of the model of a closed-loop system and extends the invariant ellipsoids method to systems with unknown but bounded state-delays. The application of the Lyapunov-Krasovskii functional and the technique of linear matrix inequalities made it possible to reduce the problem of calculating the control gain matrix to the problem of semi-definite programming, which can be solved numerically. An iterative algorithm for solving the bilinear matrix inequality is proposed for calculating the maximum permissible value of the time-delay.
 Results. The results of numerical modeling confirm the effectiveness of the proposed approach in the problems of stabilizing discrete-time systems under the conditions of state-delays and nonlinear constraints on controls, which allows to recommend the proposed method for practical use for the problem of stability analysis and synthesis of stabilizing regulator, as well as for calculating the maximum permissible value of time-delay.
 Conclusions. An approach is proposed that allows extending the invariant ellipsoids method to discrete-time dynamic systems with unknown but bounded state-delays for solving the problem of system stabilization using static state feedback based on the application of the Lyapunov-Krasovskii functional. The results of numerical modeling confirm the effectiveness of the proposed approach in the presence of the saturation type nonlinear constraints on the control signals.

Securing Multi-User Uplink Communications Against Mobile Aerial Eavesdropper via Sensing

In this paper, we propose a secure multi-user uplink communicaion scheme against a mobile aerial eavesdropper (AE) in Integrated Sensing and Communication (ISAC) systems, where the ISAC base station transmits radar signals to track and jam the AE. For tracking mobile AE, we adopt an extended Kalman filter technique to predict AE's motion state. Based on the tracking information we formulate an optimization problem to jointly design radar signal and receiver beamformer for improving the secrecy performance. To deal with the non-convex constraints and the coupled variables, we propose an alternating optimization based algorithm via iteratively solving a Fractional Programming (FP) problem and a Semidefinite Program (SDP) problem. Numerical results show that the proposed algorithm can achieve higher secrecy capacity and better fairness, compared to the state-of-the-art techniques.

On the regularization and optimization in quantum detector tomography

Quantum detector tomography (QDT) is a fundamental technique for calibrating quantum devices and performing quantum engineering tasks. In this paper, we utilize regularization to improve the QDT accuracy whenever the probe states are informationally complete or informationally incomplete. In the informationally complete scenario, without regularization, we optimize the resource (probe state) distribution by converting it to a semidefinite programming problem. Then in both the informationally complete and informationally incomplete scenarios, we discuss different regularization forms and prove the mean squared error scales as O(1/N) or tends to a constant with N state copies under the static assumption. We also characterize the ideal best regularization for the identifiable parameters, accounting for both the informationally complete and informationally incomplete scenarios. Numerical examples demonstrate the effectiveness of different regularization forms and a quantum optical experiment test shows that a suitable regularization form can reach a reduced mean squared error.

Optimal linear array orientation design for 3D direct position determination via semi-Definite relaxation

This paper presents an optimization strategy for array orientations in a three-dimensional (3D) direct position determination (DPD) system. Specifically, we consider a scenario in which uniform linear arrays (ULAs) are used to locate emitters, and we seek to optimize the array orientations in terms of localization accuracy. The E-optimality criterion, which minimizes the spectral norm of the Cramér-Rao lower bound (CRLB), is exploited to formulate this problem. As the objective function is non-convex with bilinear structures, we leverage semi-definite relaxation (SDR) to transform it into a convex semi-definite programming (SDP) problem by substituting the bilinear terms with a matrix variable. In addition, we tackle the array orientation configuration problem in the context of multi-emitter scenarios and develop an SDR solution for large-scale antenna array systems. Simulation results validate that the ULAs with array orientations designed by the proposed SDR-based method have near-optimal localization performance.

RIS-Assisted Ambient Backscatter Communication for SAGIN IoT

The space, air and ground integrated network (SAGIN) will greatly promote the development of the Internet of Things (IoT). Green IoT will be an important part of SAGIN. Ambient backscatter communication (AmBC) is a potential solution for green SAGIN IoT. To improve the achievable sum rate (ASR) of the AmBC system, we propose a reconfigurable intelligent surface (RIS) assisted AmBC system. In the single-BD AmBC scenario, we firstly give the phase shifts that maximize the gain of the reflection link of the AmBC system, and then give the optimal reflection coefficient. The proposed scheme does not need to solve the convex semidefinite program (SDP) problem, and has the characteristics of low computational complexity. In the multi-BD AmBC scenario, we first propose a multi-BD phase shifts initialization strategy to ensure the stability of the proposed scheme. Then, we give the optimal reflection coefficient and phase shifts based on iterative method. Simulations show that the RIS-assisted AmBC scheme is superior to the non-RIS-assisted AmBC scheme.

Handover-Aware Downlink Beamforming Design for LEO Multibeam Satellite Communications

Because of their capability of providing seamless connectivity, low-Earth-orbit (LEO) satellites will play a vital role in integrated satellite–terrestrial paradigms. However, high-speed satellite motion brings frequent handovers, during which redundant transmission and even loss of data easily occur. In this letter, we propose a handover-aware downlink beamforming design. To mitigate the impact of inter-satellite handover, we reduce the blocklength of coverage-edge users, aiming at completing transmission within the remaining visible time (RVT). Hence, the capacity of these users should be evaluated in the finite-block length (FBL) regime. After obtaining a surrogate of the FBL capacity function with a low-complexity method, we formulate the beamforming optimization model as a semidefinite programming (SDP) problem and solve it with CVX. The simulation results show that the proposed scheme can reach a 99.35% transmission success rate before inter-satellite handover.

Designing Robust Networks of Coupled Phase Oscillators With Applications to the High Voltage Electric Grid

We propose a mathematical framework for designing robust networks of coupled phase-oscillators by leveraging a vulnerability measure proposed by Tyloo et al. that quantifies the impact of a small perturbation at an individual phase-oscillator's natural frequency to the system's global synchronized frequencies. Given a complex network topology with specific governing dynamics, the proposed framework finds an optimal allocation of edge weights that minimizes such vulnerability measure(s) at the node(s) for which we expect perturbations to occur by solving a tractable semi-definite programming problem. We specify the mathematical model to high voltage electric grids where each node corresponds to a voltage phase angle associated with a bus and edges correspond to transmission lines. Edge weights are determined by the susceptance values along the transmission lines. In this application, frequency synchronization is increasingly challenged by the integration of renewable energy, yet is imperative to the grid's health and functionality. Our framework helps to alleviate this challenge by optimizing the placement of renewable generation and the susceptance values along the transmission lines.

A new scheme for approximating the weakly efficient solution set of vector rational optimization problems

In this paper, we provide a new scheme for approximating the weakly efficient solution set for a class of vector optimization problems with rational objectives over a feasible set defined by finitely many polynomial inequalities. More precisely, we present a procedure to obtain a sequence of explicit approximations of the weakly efficient solution set of the problem in question. Each approximation is the intersection of the sublevel set of a single polynomial and the feasible set. To this end, we make use of the achievement function associated with the considered problem and construct polynomial approximations of it over the feasible set from above. Remarkably, the construction can be converted to semidefinite programming problems. Several nontrivial examples are designed to illustrate the proposed new scheme.

Bearing-Based Relative Localization for Robotic Swarm With Partially Mutual Observations

Mutual localization provides a consensus of reference frame as an essential basis for cooperation in multi-robot systems. Previous works have developed certifiable and robust solvers for relative transformation estimation between each pair of robots. However, recovering relative poses for robotic swarm with partially mutual observations is still an unexploited problem. In this letter, we present a complete algorithm for it with optimality, scalability and robustness. Firstly, we fuse all odometry and bearing measurements in a unified minimization problem among the Stiefel manifold. Furthermore, we relax the original non-convex problem into a semi-definite programming (SDP) problem with a strict tightness guarantee. Then, to hold the exactness in noised cases, we add a convex (linear) rank cost and apply a convex iteration algorithm. We compare our approach with local optimization methods on extensive simulations with different robot amounts under various noise levels to show our global optimality and scalability advantage. Finally, we conduct real-world experiments to show the practicality and robustness.

Robust Weighted-Average Continuous-Time Consensus With Communication Time Delay.

Achieving consensus behavior robust to time delay in multiagent systems has attracted much attention. This work is concerned with optimizing the convergence rate of the consensus algorithm in such systems with time delays. Previous approaches optimize either the robustness to time delay or the convergence rate separately, while imposing a limit on the other. Eigenratio optimization is another method, which does not necessarily result in a unique set of weights. Here, the problem is treated in its general form as a multiobjective optimization problem. It is shown that the corresponding Pareto frontier depends solely on the optimal condition number of the Laplacian, and it includes the optimal answer of previously adopted approaches as special cases. A notion of optimal consensusability is then defined, which allows a particular point on the Pareto Frontier with special properties to be identified. The resulting optimization problem is shown to be convex, as is solved by reformulating it as a standard semidefinite programming problem. The optimal weights for individual topologies, clique lifted graphs, and different types of subgraphs are provided, where for the latter, the optimal weights have shown to be independent of the rest of topology. Through numerical simulations, the tradeoff between robustness and convergence rate is demonstrated.

Orientation estimation of cryo-EM images using projected gradient descent method

Orientation estimation is an important task in three-dimensional cryo-EM image reconstruction. By applying the common line method, the orientation estimation task can be formulated as a least squares (LS) problem or a least un-squared deviation (LUD) problem with orthogonality constraint. However, the non-convexity of the orthogonality constraint introduces numerical difficulties to the orientation estimation. The conventional approach is to reformulate the orthogonality constrained minimization problem into a semi-definite programming problem using convex relaxation strategies. In this paper, we consider a direct way to solve the constrained minimization problem without relaxation. We focus on the weighted LS problem because the LUD problem can be reformulated into a sequence of weighted LS problems using the iteratively re-weighted LS approach. As a classical approach, the projected gradient descent (PGD) method has been successfully applied to solve the convex constrained minimization problem. We apply the PGD method to the minimization problem with orthogonality constraint and show that the constraint set is a generalized prox-regular set, and it satisfies the norm compatibility condition. We also demonstrate that the objective function of the minimization problem satisfies the restricted strong convexity and the restricted strong smoothness over a constraint set. Therefore, the sequence generated by the PGD method converges when the initial conditions are satisfied. Experimental results show that the PGD method significantly outperforms the semi-definite relaxation methods from a computation standpoint, and the mean square error is almost the same or smaller.

An information extraction method based on improved mixed text density web pages

AbstractTo improve the effect of web page information extraction, this paper proposes an improved information extraction method of mixed text density grids. Under various power constraints of the relay node itself, this paper proposes a design scheme of joint beamforming and artificial noise based on safety and rate maximisation. Furthermore, with the help of semidefinite relaxation techniques and first‐order approximations, it can be approximated as a semidefinite programming problem that is easy to solve. In addition, this paper uses an iterative algorithm based on a continuous convex approximation to process data to improve the accuracy of web page data extraction. The experimental results show that the information extraction method based on improved mixed text density webpages proposed in this paper has a good information extraction effect.

Computation of parameter dependent robust invariant sets for LPV models with guaranteed performance

This paper presents an iterative algorithm to compute a Robust Control Invariant (RCI) set, along with an invariance-inducing control law, for Linear Parameter-Varying (LPV) systems. As real-time measurements of the scheduling parameters are typically available, we allow the RCI set description and the invariance-inducing controller to be scheduling parameter dependent. Thus, the considered formulation leads to parameter-dependent conditions for the set invariance, which are replaced by sufficient Linear Matrix Inequalities (LMIs) via Polya’s relaxation. These LMI conditions are then combined with a novel volume maximization approach in a Semidefinite Programming (SDP) problem, which aims at computing the desirably large RCI set. Besides ensuring invariance, it is also possible to guarantee performance within the RCI set by imposing a chosen quadratic performance level as an additional constraint in the SDP problem. Using numerical examples, we show that the presented iterative algorithm can generate RCI sets for large parameter variations where commonly used robust approaches fail.

Consensus Decision Modeling with Distributionally Robust Chance Constraint in Uncertain Environments

Group decision-making (GDM) in an ambiguous environment has consistently become a research focus in the decision science field during the past decade. Existing minimum cost consensus models either control the total budget in a deterministic context or focus on improving the utility of decision makers. In this study, a novel consensus model with a distributionally robust chance constraint (DRO-MCCM) is explored. First, two distributionally robust chance constraints consensus models are developed based on the varied utility preferences of decision-makers and taking into consideration the uncertainty of the unit adjustment cost. Next, construct conditional value-at-risk (CVaR) to approximate the cost chance constraint, simulate the viewpoint of decision makers with ambiguous preferences such as utility function and Gaussian distribution, and convert the model into a feasible semidefinite programming problem using dual theory and the moment method. Finally, the supply chain management scenario involving new product prices employs these models. Comparison and sensitivity analyses demonstrates the model’s superiority and effectiveness.

Logarithmic Barrier Method Via Minorant Function for Linear Semidefinite Programming

Abstract We propose in this study, a new logarithmic barrier approach to solve linear semidefinite programming problem. We are interested in computation of the direction by Newton’s method and of the displacement step using minorant functions instead of line search methods in order to reduce the computation cost. Our new approach is even more beneficial than classical line search methods. This purpose is confirmed by some numerical simulations showing the e˙ectiveness of the algorithm developed in this work, which are presented in the last section of this paper.

Motion Parameter Estimation for Mobile Sources Using Semidefinite Programming

In this paper, we address the motion parameter estimation problem using time difference of arrival (TDOA) measurements, where a mobile source starts from an initial position with constant velocity. The problem is formulated as three non-convex constrained weighted least squares (CWLS) problems. Owing to their non-convex nature, local convergence may occur when solving them by an iterative algorithm, implying good estimates are not guaranteed. We propose to solve these CWLS problems by applying semidefinite relaxation (SDR) to generate three different semidefinite programming (SDP) problems, namely, the motion unconstrained semidefinite programming (MU-SDP), motion constrained semidefinite programming (MC-SDP), and motion parameter direct semidefinite programming (MPD-SDP). The MU-SDP, MC-SDP, and MPD-SDP methods are then extended to the scenario of unknown propagation speed (PS), which is common in underwater and underground localizations. Specifically, for this scenario, the two-step MU-SDP, two-step MC-SDP, and two-step MPD-SDP methods are designed to keep the relaxed SDP problems tight by introducing the penalty function. Simulation results show that MU-SDP performs worse than MC-SDP and MPD-SDP owing to the ignorance of the motion equation, and MC-SDP and MPD-SDP are able to reach the Cramér-Rao lower bound (CRLB) accuracy.