Semidefinite Programming Problem Research Articles

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.

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.

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.

We introduce a semi-supervised space adjustment framework in this paper. In the introduced framework, the dataset contains two subsets: (a) training data subset (space-one data (SOD)) and (b) testing data subset (space-two data (STD)). Our semi-supervised space adjustment framework learns under three assumptions: (I) it is assumed that all data points in the SOD are labeled, and only a minority of the data points in the STD are labeled (we call the labeled space-two data as LSTD), (II) the size of LSTD is very small comparing to the size of SOD, and (III) it is also assumed that the data of SOD and the data of STD have different distributions. We denote the unlabeled space-two data by ULSTD, which is equal to STD - LSTD. The aim is to map the training data, i.e., the data from the training labeled data subset and those from LSTD (note that all labeled data are considered to be training data, i.e., SOD ∪ LSTD) into a shared space (ShS). The mapped SOD, ULSTD, and LSTD into ShS are named MSOD, MULSTD, and MLSTD, respectively. The proposed method does the mentioned mapping in such a way that structures of the data points in SOD and MSOD, in STD and MSTD, in ULSTD and MULSTD, and in LSTD and MLSTD are the same. In the proposed method, the mapping is proposed to be done by a principal component analysis transformation on kernelized data. In the proposed method, it is tried to find a mapping that (a) can maintain the neighbors of data points after the mapping and (b) can take advantage of the class labels that are known in STD during transformation. After that, we represent and formulate the problem of finding the optimal mapping into a non-linear objective function. To solve it, we transform it into a semidefinite programming (SDP) problem. We solve the optimization problem with an SDP solver. The examinations indicate the superiority of the learners trained in the data mapped by the proposed approach to the learners trained in the data mapped by the state of the art methods.

Optimal curing resource allocation for epidemic spreading processes

Optimal energy constrained deception attacks in cyber–physical systems with multiple channels: A fusion attack approach

In this paper, we consider a semidefinite programming (SDP) relaxation of the quadratic knapsack problem. After applying low-rank factorization, we get a nonconvex problem, whose feasible region is an algebraic variety with certain good geometric properties, which we analyze. We derive a rank condition under which these two formulations are equivalent. This rank condition is much weaker than the classical rank condition if the coefficient matrix has certain special structures. We also prove that, under an appropriate rank condition, the nonconvex problem has no spurious local minima without assuming linearly independent constraint qualification. We design a feasible method that can escape from nonoptimal nonregular points. Numerical experiments are conducted to verify the high efficiency and robustness of our algorithm as compared with other solvers. In particular, our algorithm is able to solve a one-million-dimensional sparse SDP problem accurately in about 20 minutes on a modest computer. Funding: The research of K.-C. Toh is supported by the Ministry of Education, Singapore, under its Academic Research Fund Tier 3 grant call [Grant MOE-2019-T3-1-010].

Traditional wireless data aggregation (WDA) technology based on the principle of separated communication and computation is difficult to achieve large‐scale access under the limited spectrum resources, especially in scenarios with strict constraints on time latency. As an outstanding fast WDA technology, over‐the‐air computation (AirComp) can reduce transmit time while improving spectrum efficiency. Most edge devices in wireless networks are battery‐powered. Therefore, optimizing the transmit power of devices could prolong the life cycle of nodes and save the system power consumption. In this research, we aim to minimize the device transmit power subject to aggregation error constraint. Additionally, to improve the harsh wireless transmission environment, we use reconfigurable intelligent surface (RIS) to assist AirComp. To solve the presented nonconvex problem, we present a two‐step solution method. Specifically, we introduce matrix lifting technology to transform the original problems into semidefinite programming problems (SDP) in the first step and then propose an alternate difference‐of‐convex (DC) framework to solve the SDP subproblems. The numerical results show that RIS‐assisted communication can greatly save system power and reduce aggregation error. And the proposed alternate DC method is superior to the alternate semidefinite relaxation (SDR) method.

We can only extract the differential time delay (DTD) measurements between the direct and reflected paths in multistatic localization when there is no synchronization in time between the transmitter and receiver and among the receivers. This paper first addresses the problem of multistatic localization of a fixed object when the transmitter position is not available by using the DTD measurements. We propose a two-step optimization method for jointly estimating the object and transmitter positions. In the first step, we formulate a non-convex constrained weighted least squares (CWLS) problem by transforming the DTD measurement model and introducing nuisance variables. Such a non-convex CWLS problem is then relaxed to a tractable convex semidefinite programming (SDP) problem by applying semidefinite relaxation. In the second step, the error coming from relaxation and approximation in the SDP solution is reduced iteratively through solving a generalized trust region subproblem (GTRS) in each iteration. If the receivers are synchronized such that the time difference of arrival (TDOA) measurements can be acquired in addition to DTD, we formulate a different CWLS problem by utilizing both DTD and TDOA measurements, which is solved by convex relaxation as well. The relaxed SDP problem can achieve the optimal solution of the CWLS problem, and further refinement is no longer needed. We conduct the mean square error (MSE) analysis to validate that both proposed methods are able to achieve the Cramer-Rao Lower Bound (CRLB) performance under small Gaussian noise, which is also validated by simulations.

Wireless acoustic sensor network (WASN) has shown a superiority over conventional microphone arrays in many aspects. There exists an important tradeoff between the performance and power consumption, as usually the sensors are power driven with a limited amount of battery resource. Given a prescribed performance bound, in literature sensor selection (SS) and rate allocation (RA) methods can be leveraged to optimize the energy efficiency. In this work, we propose a joint rate allocation and sensor selection (RASS) approach to simultaneously optimize the sensor subset and rate distribution, which is formulated by minimizing the total transmission power in terms of selection and bit-rate variables and constraining the residual noise power. It can be shown that under a set of linear constraints on beamforming, the linearly-constrained minimum variance (LCMV) beamformer is the optimal noise reduction filter. Based on this, the RASS reduces to a mixed semi-definite and bilinear programming problem, which is then solved using a two-step algorithm. As the selection and bit-rate unknowns are bilinear, we first consider to optimize their product, resulting in an upper bound of RASS. Then, we use McCormick envelopes to relax the bilinear constraint, resulting in a linear program. The final selection and bit-rate solutions are obtained by posterior randomized rounding. It can be shown that SS and RA are special cases of the proposed RASS. Numerical results using simulated WASNs validate the power efficiency of the proposed method as well as the robustness against dynamic factors.

In this article, we propose a joint strategy of power and bandwidth allocation (JSPBA) for multiple maneuvering target tracking (MMTT) in the multi-input multi-output (MIMO) radar with collocated antennas. The basis of our strategy is to optimally allocate the transmitted resources of power and effective bandwidth by the prior information in the closed-loop system of cognitive tracking. On account of the predicted conditional Cramér-Rao lower bound (PC-CRLB) offering a more accurate and time-sensitive lower bound than the standard posterior CRLB (PCRLB), the PC-CRLB of the range, Doppler frequency, and direction-of-arrival (DOA) is derived, normalized and adopted as the optimization criterion. Moreover, in order to solve the nonconvex problem, the initial nonconvex problem is converted into a series of convex problems, which are further formulated as the standard semi-definite programming (SDP) problems and then be solved, by introducing the convex relaxation technique and the two-step solution technique. Simulations confirm the superiority of the proposed JSPBA algorithm, in terms of the overall tracking accuracy in the MMTT scenario.

Micro-phasor measurement units (µPMUs) with a micro-second resolution and milli-degree accuracy capability are expected to play an important role in improving the state estimation accuracy in the distribution network with increasing penetration of distributed generations. Therefore, this paper investigates the problem of how to place a limited number of µPMUs to improve the state estimation accuracy. Combined with pseudo-measurements and supervisory control and data acquisition (SCADA) measurements, an optimal µPMU placement model is proposed based on a two-step state estimation method. The E-optimal experimental criterion is utilized to measure the state estimation accuracy. The nonlinear optimization problem is transformed into a mixed-integer semidefinite programming (MISDP) problem, whose optimal solution can be obtained by using the improved Benders decomposition method. Simulations on several systems are carried out to evaluate the effective performance of the proposed model.

Computing Optimal Upper Bounds on the H2-norm of ODE-PDE Systems using Linear Partial Inequalities

In this paper, the multistatic localization problem with unknown propagation speed is investigated using differential delays and Doppler shifts between the signals from direct and indirect paths. A series of pseudo-linear equations are formulated via the transformation of measurement models. A weighted least squares (WLS) formulation is then proposed after ignoring the second-order error terms, which can be rewritten as a non-convex optimization problem with the relationships among variables included as constraints. To deal with the non-convexity of the problem, semidefinite relaxation is applied, resulting in a convex semidefinite program (SDP). Several reasonable second-order cone constraints constructed via basic inequality and Cauchy-Schwarz inequality are added to tighten the relaxed SDP problem. By preserving the second-order error terms in equations, the bias of the estimate from the WLS formulation is also derived and then subtracted to nearly eliminate the bias and reach a bias-reduced solution. Simulation results show that the mean square error (MSE) of the proposed method approaches the Cramer-Rao lower bound (CRLB) and the bias is reduced significantly.

This work investigates the robust beamforming for a multi-antenna internet of things (IoT) system using wireless information and power transmission (WIPT), given that imperfect channel state information (CSI) assumption is accounted.In particular, we investigate the problem of maximizing the worst-case energy harvested, taking into account the quality of service (QoS) constraint of user rate. The proposed problem is naturally a nonconvex problem, which is hard to tackle directly. On one hand, we rely on a classical method, that is semidefinite programming problem (SDP), to handle this by transforming the original nonconvex optimization problem with infinite number of constraints to a relaxed convex one. On the other hand, we propose another algorithm using Symbiotic Organisms Search (SOS) approach that can effciently solve the formulated problem. In the end, numerical results are provided to verify the effectiveness of the SDP-based algorithm in comparison with that of the SOS-based algorithm.

This article develops a robust source localization method using time delay and Doppler shift measurements, where the sensor motion effect accompanied by sensor location errors cannot be ignored. We begin by transforming the time delay and Doppler shift measurement models into a series of nonlinear equations that take sensor location errors into account, and then construct a constrained weighted least squares (CWLS) problem based on these equations. Because of the nonconvex nature of the problem, we relax it into a semidefinite programming (SDP) problem via convex relaxation and further propose a scheme to eliminate the influence of the additional estimation bias caused by the approximation applied in the transformation of measurement models. To perform the bias reduction, we derive the theoretical expression of the solution bias and then subtract it to obtain a bias-reduced solution. We also derive the closed-form expression of the hybrid Cramer–Rao lower bound (HCRLB) as the performance benchmark. Theoretical analysis and simulation results demonstrate that the mean-square error (MSE) performance of the proposed method can achieve the HCRLB accuracy and the bias can be significantly reduced with bias reduction.

The paper proposes a control design approach for the three-phase grid-side converter of a power amplifier inverter for Power-Hardware-In-the-Loop application. The main challenge for the controller design is to achieve a trade-off between the tight DC link voltage regulation and grid-side power quality objectives due to the established topology of the power amplifier and the associated ripple of the DC link voltage. To this end, a full-state feedback controller with a gain scheduling strategy is adopted to ensure fast control of the DC link voltage, while minimizing the negative impact of DC link voltage ripple on the AC output current. We formulate the problem of gain selection as a two-stage semidefinite programming problem, which allows using efficient convex program solvers. We also describe a bumpless transfer strategy to minimize any transients associated with control output discontinuities following gain transition. The paper demonstrates comprehensive validation of the designed controller, including offline simulations, Control-Hardware-In-the-Loop tests with an embedded controller, as well as experimental validation on a 2-kV 167-kVA module.

Fast tuning of transmission power flow routers for transient stability constrained optimal power flow under renewable uncertainties

The task of inducing, via continuous static state-feedback control, an asymptotically stable heteroclinic orbit in a nonlinear control system is considered in this paper. The main motivation comes from the problem of ensuring convergence to a so-called point-to-point maneuver in an underactuated mechanical system. Namely, to a smooth curve in its state–control space which is consistent with the system dynamics and connects two (linearly) stabilizable equilibrium points. The proposed method uses a particular parameterization, together with a state projection onto the maneuver as to combine two linearization techniques for this purpose: the Jacobian linearization at the equilibria on the boundaries and a transverse linearization along the orbit. This allows for the computation of stabilizing control gains offline by solving a semidefinite programming problem. The resulting nonlinear controller, which simultaneously asymptotically stabilizes both the orbit and the final equilibrium, is time-invariant, locally Lipschitz continuous, requires no switching, and has a familiar feedforward plus feedback–like structure. The method is also complemented by synchronization function–based arguments for planning such maneuvers for mechanical systems with one degree of underactuation. Numerical simulations of the non-prehensile manipulation task of a ball rolling between two points upon the “butterfly” robot demonstrates the efficacy of the synthesis.

Dynamical systems coupled in the form of a complex networks have been exploited for modeling numerous real-world distributed systems. Due to the decentralized nature of dynamical networks, and inaccessibility or high control cost, in some applications, it is desired to employ pinning control techniques. A natural question that comes to mind is how one can select the proper pinned nodes and their feedback gains, (i.e., the performance metrics) to optimize the convergence rate to the homogeneous stationary state (i.e., the performance index) while the total pinning cost is kept below a given limit. This optimization problem has been converted into a semidefinite programming problem, and the optimal feedback gains have been derived analytically. Based on these results, for a given set of pinned nodes, an algorithm for determining the optimal feedback gains has been developed. Furthermore, it is shown that the edges between the pinned nodes do not impact the optimal results, and adding an edge between free nodes or between free and pinned nodes modifies the optimal convergence rate in nondecreasing order. Based on the derived analytical results, several interesting properties have been discovered. For symmetric networks, the nodes within a vertex orbit have the same optimal feedback gains, which are independent of the edges within each vertex orbit. For a number of topologies, closed-form formulas are provided for the optimal results. For a network of Chua systems, the impact of cut number is illustrated by comparing the optimal results for different topologies.