Semidefinite Programming Problem Research Articles

An SDP relaxation in the complex domain for the efficient coordination of BESS and DGs in single-phase distribution grids while considering reactive power capabilities

This research proposes an efficient energy management system design for medium-voltage distribution networks from a perspective of convex optimization. The exact nonlinear programming problem that represents the operation of distributed energy resources (DERs), such as batteries and photovoltaic sources, is approximated as an equivalent semi-definite programming (SDP) problem. This SDP approximation is achieved in the complex domain using Hermitian matrices, which are the equivalent of positive semi-definite matrices in the real number domain. Two test feeders, one with 27 nodes and the other with 33 nodes, are modified to match the characteristics of Colombian rural and urban networks. The 27-bus grid is adapted to accommodate the average demand and solar generation conditions of the municipality of Capurganá, while those of the city of Medellín are used for the 33-bus grid. One of the main contributions of this research is that it considers a DER operation with variable power factors, which allows for significant improvements compared to the traditional use of a unitary power factor scheme. Numerical comparisons are conducted using metaheuristic optimizers, including parallel versions of the particle swarm optimizer, the vortex search algorithm, and the continuous version of the genetic algorithm. These comparisons demonstrate the effectiveness of the proposed SDP approximation. All simulations are carried out in the MATLAB software (version 2022b), using the disciplined convex optimizer (CVX) and the SDPT3 and SEDUMI solvers.

Enhanced Moving Source Localization with Time and Frequency Difference of Arrival: Motion-Assisted Method for Sub-Dimensional Sensor Networks

Localizing a moving source by Time Difference of Arrival (TDOA) and Frequency Difference of Arrival (FDOA) commonly requires at least N+1 sensors in N-dimensional space to obtain more than N pairs of TDOAs and FDOAs, thereby establishing more than 2N equations to solve for 2N unknowns. However, if there are insufficient sensors, the localization problem will become underdetermined, leading to non-unique solutions or inaccuracies in the minimum norm solution. This paper proposes a localization method using TDOAs and FDOAs while incorporating the motion model. The motion between the source and sensors increases the equivalent length of the baseline, thereby improving observability even when using the minimum number of sensors. The problem is formulated as a Maximum Likelihood Estimation (MLE) and solved through Gauss–Newton (GN) iteration. Since GN requires an initialization close to the true value, the MLE is transformed into a semidefinite programming problem using Semidefinite Relaxation (SDR) technology, while SDR results in a suboptimal estimate, it is sufficient as an initialization to guarantee the convergence of GN iteration. The proposed method is analytically shown to reach the Cramér–Rao Lower Bound (CRLB) accuracy under mild noise conditions. Simulation results confirm that it achieves CRLB-level performance when the number of sensors is lower than N+1, thereby corroborating the theoretical analysis.

A Predictor-Corrector Algorithm for Semidefinite Programming that Uses the Factor Width Cone

AbstractWe propose an interior point method (IPM) for solving semidefinite programming problems (SDPs). The standard interior point algorithms used to solve SDPs work in the space of positive semidefinite matrices. Contrary to that the proposed algorithm works in the cone of matrices of constant factor width. We prove global convergence and provide a complexity analysis. Our work is inspired by a series of papers by Ahmadi, Dash, Majumdar and Hall, and builds upon a recent preprint by Roig-Solvas and Sznaier [arXiv:2202.12374, 2022].

A powered descent trajectory planning method with quantitative consideration of safe distance to obstacle

High scientific value areas on celestial bodies such as the Moon and Mars are often located in hazardous terrains. To achieve safe landing exploration, a novel planning method is proposed, which can ensure that the planned trajectory maintains a user-specified distance from obstacles, thus reducing potential collision risk induced by factors such as the body size and model uncertainties. Firstly, the basic model of the trajectory optimization problem and its convexification version is given. The obstacles are modeled as polynomial functions, based on which the “if-then” obstacle avoidance logic explicitly considering the safe distance, is described as a sum of squares constraint. This constraint formulation applies to any obstacle described by a finite number of polynomials, independent of the specific expression of the polynomials (reflecting the shape of the obstacle). Subsequently, the convexification process for the obstacle avoidance constraint is given. Finally, the sequential sum of squares programming problem for the obstacle avoidance trajectory is established, which boils down to a series of semidefinite programming problems. Simulation results show that the closest distance between the planned trajectory and obstacles strictly satisfies the specified distance constraint, and the trajectory could avoid non-convex obstacles. With the promising convergence properties of underlying convex optimization algorithms, advanced autonomous obstacle avoidance guidance schemes are expected to be formed based on the proposed trajectory planning method.

On parametric semidefinite programming with unknown boundaries

In this paper, we study parametric semidefinite programs (SDPs) where the solution space of both the primal and dual problems change simultaneously. Given a bounded set, we aim to find the a priori unknown maximal permissible perturbation set within it where the semidefinite program problem has a unique optimum and is analytic with respect to the parameters. Our approach reformulates the parametric SDP as a system of partial differential equations (PDEs) where this maximal analytical permissible set (MAPS) is the set on which the system of PDEs is well-posed. A sweeping Euler scheme is developed to approximate this a priori unknown perturbation set. We prove local and global error bounds for this second-order sweeping Euler scheme and demonstrate the method in comparison to existing SDP solvers and its performance on several two-parameter and three-parameter SDPs for which the MAPS can be visualized.

Characterizations of ε-Approximate Solutions for Robust Convex Semidefinite Programming Problems

Characterizations of ε-Approximate Solutions for Robust Convex Semidefinite Programming Problems

Enhanced angle estimation in MIMO radar: Combine RD-MUSIC and SDP optimization

Multiple Input Multiple Output (MIMO) radars are increasingly employed. However, with a growing array of antenna components, the dimension of the sampling covariance matrix and the computational complexity of traditional Direction of Arrival (DOA) and Direction of Departure (DOD) estimation algorithms increase exponentially. In order to accurately and effectively identify and locate signal sources and to optimize the computational complexity, this paper presents an innovative approach for detecting target numbers based on a Semi-Definite Programming (SDP) problem, which generates efficient solutions even for higher dimensions. In addition, the SDP’s robustness is developed through shrinkage adaptation using Large Covariance Matrices (LCMs), which are crucial for angle estimation. Following the effective detection of a target, the tapering estimation is applied to the LCMs under the computationally efficient Reduced-Dimension Multiple Signal Classification (RD-MUSIC) algorithm, which is more accurate and less computationally intensive than 2D search algorithms. Simulation outcomes demonstrate that the efficacy of our proposed model achieves high success rates in target detection and enhanced precision in DOA and DOD estimations.

A distributionally robust chance-constrained kernel-free quadratic surface support vector machine

This paper studies the problem of constructing a robust nonlinear classifier when the data set involves uncertainty and only the first- and second-order moments are known a priori. A distributionally robust chance-constrained kernel-free quadratic surface support vector machine (SVM) model is proposed using the moment information of the uncertain data. The proposed model is reformulated as a semidefinite programming problem and a second-order cone programming problem for efficient computations. A geometric interpretation of the proposed model is also provided. For commonly used data without prescribed uncertainty, a cluster-based data-driven approach is introduced to retrieve the hidden moment information that enables the proposed model for robust classification. Extensive computational experiments using synthetic and public benchmark data sets with or without uncertainty involved support the superior performance of the proposed model over other state-of-the-art SVM models, particularly when the data sets are massive and/or imbalanced.

Efficient 2D DOA Estimation via Decoupled Projected Atomic Norm Minimization

This paper presents an efficient two-dimensional (2D) direction of arrival (DOA) estimation method, termed as decoupled projected atomic norm minimization (D-PANM), to solve the angle-ambiguity problem. It first introduces a novel atomic metric via projecting the original atom set onto a smoothing space, based on which we formulate an equivalent semi-definite programming (SDP) problem. Then, two relatively low-complexity decoupled Toeplitz matrices can be obtained to estimate the DOAs. We further exploit the structural information hidden in the newly constructed data to avoid pair matching for the azimuth and elevation angles when the number of sensors is odd, and then propose a fast and feasible decoupled alternating projections (D-AP) algorithm, reducing computational complexity to a great extent. Numerical simulations are performed to demonstrate that the proposed algorithm is no longer restricted by angle ambiguity scenarios, but instead provides a more stable estimation performance, even when multiple signals share the same angles in both azimuth and elevation dimensions. Additionally, it greatly improves the resolution, with control of the computation load compared with the existing atomic norm minimization (ANM) algorithm.

Physical Layer Security of the MIMO-NOMA Systems under Near-Field Scenario

In this paper, we propose a secure transmission framework for near-field MIMO-NOMA systems. This architecture integrates beamforming mechanisms for both transmission and reception, allowing the base station to send encrypted information to authorized users, effectively countering eavesdropping attempts in a near-field environment. To optimize the secrecy communication capability in the near field, a two-phase alternating optimization algorithm is introduced. In the first phase, the semidefinite relaxation (SDR) method is used to relax constraints in the problem and convert it into a semidefinite programming (SDP) problem. In the second phase, the successive convex approximation (SCA) algorithm is employed to transform the original non-convex problem into a convex optimization problem, obtaining a locally optimal solution through multiple iterations. Simulation results validate that the proposed near-field communication strategy exhibits superior secrecy communication capabilities under various parameter settings compared to far-field communication strategies.

Hydrazine Concentration Estimation in the Secondary Circuit of a Nuclear Power Plant: a IQC LPV approach

This paper addresses the design of a new linear parameter varying estimator, to estimate the hydrazine molar concentration in the secondary circuit of a nuclear power plant. The method uses dynamic shaping filters of arbitrary order, to formulate the estimation performance objectives. The theory is based on L2-gain synthesis, based on uncertainty characterization via static integral quadratic constraint multipliers. It is shown that the problem can be formulated as a semi definite programming problem, that can be solved efficiently using numerical solvers. The proposed approach is applied to real data, coming from a nuclear power plant located in France. The obtained results demonstrates the efficiency of the proposed approach.

Verification of Parametric Properties of Linear Discrete-time Stochastic Systems

We study the problem of verifying parametric properties of linear discrete-time stochastic systems (linear DTSSs) over bounded and unbounded time horizons. A linear DTSS captures random processes, where the one-step transition relation between the current random vector X and the next-step random vector X’ is linear and is given by X’ = AX + W, where A is an n x n matrix and W is a random noise vector. We assume that the initial and noise random vectors are multivariate normal, each of which is expressed as a pair of two parameters, namely, mean and covariance. We are interested in verifying whether the system always satisfies a given parametric property specified as a subset of the parameters’ space over bounded or unbounded time horizons. For bounded time steps, we reduce the problem to the satisfiability of a semidefinite programming problem. For the unbounded time steps, we propose a novel abstraction procedure to reduce the verification problem to that of a finite graph, wherein, the nodes of the graph correspond to the regions of a partition of the random vector space, in contrast to existing works that partition the state-space. More precisely, we partition the parameter space of normal random vectors, namely, the space of means and covariance matrices, and apply semi-definite programming to compute the edges. We show that our abstraction procedure is sound. Our experiments with varying number of vehicle dynamics and different dimensions of random models show that our methods are reasonably scalable.

A Customized ADMM Approach for Large-Scale Nonconvex Semidefinite Programming

We investigate a class of challenging general semidefinite programming problems with extra nonconvex constraints such as matrix rank constraints. This problem has extensive applications, including combinatorial graph problems, such as MAX-CUT and community detection, reformulated as quadratic objectives over nonconvex constraints. A customized approach based on the alternating direction method of multipliers (ADMM) is proposed to solve the general large-scale nonconvex semidefinite programming efficiently. We propose two reformulations: one using vector variables and constraints, and the other further reformulating the Burer–Monteiro form. Both formulations admit simple subproblems and can lead to significant improvement in scalability. Despite the nonconvex constraint, we prove that the ADMM iterates converge to a stationary point in both formulations, under mild assumptions. Additionally, recent work suggests that in this matrix form, when the matrix factors are wide enough, the local optimum with high probability is also the global optimum. To demonstrate the scalability of our algorithm, we include results for MAX-CUT, community detection, and image segmentation.

Tapping into Permutation Symmetry for Improved Detection of k-Symmetric Extensions.

Symmetric extensions are essential in quantum mechanics, providing a lens through which to investigate the correlations of entangled quantum systems and to address challenges like the quantum marginal problem. Though semi-definite programming (SDP) is a recognized method for handling symmetric extensions, it struggles with computational constraints, especially due to the large real parameters in generalized qudit systems. In this study, we introduce an approach that adeptly leverages permutation symmetry. By fine-tuning the SDP problem for detecting k-symmetric extensions, our method markedly diminishes the searching space dimensionality and trims the number of parameters essential for positive-definiteness tests. This leads to an algorithmic enhancement, reducing the complexity from O(d2k) to O(kd2) in the qudit k-symmetric extension scenario. Additionally, our approach streamlines the process of verifying the positive definiteness of the results. These advancements pave the way for deeper insights into quantum correlations, highlighting potential avenues for refined research and innovations in quantum information theory.

Exploiting NOMA and RIS in Integrated Sensing and Communication

A novel integrated sensing and communication (ISAC) system is proposed, where a dual-functional base station is utilized to transmit the superimposed non-orthogonal multiple access (NOMA) communication signal for serving communication users and sensing targets simultaneously. Furthermore, a new reconfigurable intelligent surface (RIS)-aided-sensing structure, where a dedicated RIS is deployed to provide virtual line-of-sight (LoS) links for radar targets, is also proposed to address the significant path loss or blockage of LoS links for the sensing task. Based on this setup, the beampattern gain at the RIS for the radar target is derived and adopted as a sensing metric. The objective of this paper is to maximize the minimum beampattern gain by jointly optimizing active beamforming at the base station (BS), power allocation coefficients among NOMA users and passive beamforming at the RIS. To tackle the non-convexity of the formulated optimization problem, the beampattern gain and constraints are first transformed into more tractable forms. Then, an iterative block coordinate descent (IBCD) algorithm is proposed by employing successive convex approximation (SCA), Schur complement, semidefinite relaxation (SDR) and sequential rank-one constraint relaxation (SRCR) methods. To reduce the complexity of the proposed IBCD algorithm, a low-complexity iterative alternating optimization (IAO) algorithm is proposed. Particularly, the active beamforming is optimized by solving a semidefinite programming (SDP) problem and the closed-form solutions of the power allocation coefficients are derived. Numerical results show that: i) the proposed RIS-NOMA-ISAC system always outperforms the RIS-ISAC system without NOMA in beampattern gain and illumination power; ii) the low-complexity IAO algorithm achieves a comparable performance to that achieved by the IBCD algorithm. iii) high beampattern gain can be achieved by the proposed joint optimization algorithms in underloaded and overloaded communication scenarios.

Feynman integrals from positivity constraints

We explore inequality constraints as a new tool for numerically evaluating Feynman integrals. A convergent Feynman integral is non-negative if the integrand is non-negative in either loop momentum space or Feynman parameter space. Applying various identities, all such integrals can be reduced to linear sums of a small set of master integrals, leading to infinitely many linear constraints on the values of the master integrals. The constraints can be solved as a semidefinite programming problem in mathematical optimization, producing rigorous two-sided bounds for the integrals which are observed to converge rapidly as more constraints are included, enabling high-precision determination of the integrals. Positivity constraints can also be formulated for the ϵ expansion terms in dimensional regularization and reveal hidden consistency relations between terms at different orders in ϵ. We introduce the main methods using one-loop bubble integrals, then present a nontrivial example of three-loop banana integrals with unequal masses, where 11 top-level master integrals are evaluated to high precision.

Energy-Efficient Beamforming and Resource Optimization for STAR-IRS Enabled Hybrid-NOMA 6G Communications

In this manuscript, we propose an energy-efficient optimization framework for a multi-cluster simultaneous transmitting and reflecting intelligent reflecting surfaces (STAR-IRS) enabled time-division multiple-access (TDMA) based hybrid-NOMA system to realize the future sixth-generation (6G) wireless communication systems. Specifically, the energy-efficiency maximization is achieved by optimizing the successive-interference cancellation (SIC) decoding order, time-allocation, and active-beamforming vectors at the transmitter, as well as transmission and reflection coefficients at the STAR-IRS under quality-of-service (QoS), conservation of energy, time-allocation, phase-shifts, and SIC-decoding constraints. Moreover, the proposed alternating optimization algorithm tackles the considered highly non-convex optimization problem in four steps. In first step, for computing the SIC-decoding order of NOMA users, an efficient optimization technique is proposed which maximizes the sum of combined channel gains by optimizing the transmission and reflection beamforming vectors of the considered STAR-IRS assisted hybrid-NOMA system. Further, in second step, an optimal time-allocation for each cluster in transmission and reflection region is computed for given SIC-decoding order. With decoding order and time-allocation in hand, active-beamforming vectors are computed by exploiting the sequential-convex approximation (SCA) and second-order-conic programming (SOCP) in third step. Finally, in the fourth step, the transmission and reflection coefficients of STAR-IRS are computed by transforming the non-convex optimization problem into a semi-definite programming (SDP) problem. Th numerical simulation results demonstrate that the proposed optimization framework exhibits an efficient energy efficiency performance and converges within a few iterations.

Spatial Channel Covariance Estimation and Two-Timescale Beamforming for IRS-Assisted Millimeter Wave Systems

We consider the problem of spatial channel covariance matrix (CCM) estimation for intelligent reflecting surface (IRS)-assisted millimeter wave (mmWave) communication systems. Spatial CCM is essential for two-timescale beamforming in IRS-assisted systems; however, estimating the spatial CCM is challenging due to the passive nature of reflecting elements and the large size of the CCM resulting from massive reflecting elements of the IRS. In this paper, we propose a CCM estimation method by exploiting the low-rankness as well as the positive semi-definite (PSD) 3-level Toeplitz structure of the CCM. Estimation of the CCM is formulated as a semidefinite programming (SDP) problem and an alternating direction method of multipliers (ADMM) algorithm is developed. Our analysis shows that the proposed method is theoretically guaranteed to attain a reliable CCM estimate with a sample complexity much smaller than the dimension of the CCM. Thus the proposed method can help achieve a significant training overhead reduction. Simulation results are presented to illustrate the effectiveness of our proposed method and the performance of two-timescale beamforming scheme based on the estimated CCM.

Integration of NOMA With Reflecting Intelligent Surfaces: A Multi-Cell Optimization With SIC Decoding Errors

Reflecting intelligent surfaces (RIS) has gained significant attention due to its high energy and spectral efficiency in next-generation wireless networks. By using low-cost passive reflecting elements, RIS can smartly reconfigure the signal propagation to extend the wireless communication coverage. On the other hand, non-orthogonal multiple access (NOMA) has been proven as a key air interface technique for supporting massive connections over limited resources. Utilizing the superposition coding and successive interference cancellation (SIC) techniques, NOMA can multiplex multiple users over the same spectrum and time resources by allocating different power levels. This paper proposes a new optimization scheme in a multi-cell RIS-NOMA network to enhance the spectral efficiency under SIC decoding errors. In particular, the power budget of the base station and the transmit power of NOMA users while the passive beamforming of RIS is simultaneously optimized in each cell. Due to objective function and quality of service constraints, the joint problem is formulated as non-convex, which is very complex and challenging to obtain the optimal global solution. To reduce the complexity and make the problem tractable, we first decouple the original problem into two sub-problems for power allocation and passive beamforming. Then, the efficient solution of each sub-problem is obtained in two-steps. In the first-step of For power allocation sub-problem, we transform it to a convex problem by the inner approximation method and then solve it through a standard convex optimization solver in the second-step. Accordingly, in the first-step of passive beamforming, it is transformed into a standard semi-definite programming problem by successive convex approximation and different of convex programming methods. Then, penalty based method is used to achieve a Rank-1 solution for passive beamforming in second-step. Numerical results demonstrate the benefits of the proposed optimization scheme in the multi-cell RIS-NOMA network.

Integrated sensing and communication waveform design in the Internet of Vehicles

We design an integrated sensing and communication (ISAC) transmission waveform with total transmission power, waveform similarity, and peak-to-average power ratio (PAPR) constraints in the Internet of Vehicles (IoV). The integrated waveform can complete radar target detection while communicating with downlink cellular users. The design of IoV with ISAC can not only improve the spectrum efficiency, but also further promote the integration of communication and sensing equipment. In this paper, the IoV with ISAC is formally modeled from the communication view and sensing view respectively. On this basis, taking the joint optimization of multi-user interference (MUI) total energy and Cramér Rao Bound (CRB) as the objective function, the transmission waveforms are designed with total transmission power, waveform similarity, and PAPR constraints respectively. To obtain the global optimal solution, we adopt a low complexity algorithm to solve the optimization problem under the total transmit power constraint and the optimization problem based on waveform similarity and PAPR constraints is further transformed into a nonconvex quadratically constrained quadratic programming (QCQP) problem, which is solved by using the semidefinite relaxation (SDR) technique. By ranking one approximation of the constrained optimization problem, it can be transformed into the solution of semidefinite programming (SDP) problems. The numerical results show that the designed ISAC transmission waveform can achieve an efficient tradeoff between sensing and communication performance, and each KPI is superior to one of the only communication and only sensing waveform design schemes. And it can improve the real-time information interaction ability of the IoV.