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

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.

We 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

On parametric semidefinite programming with unknown boundaries

In this article, we explore the characterizations of ε-approximate solutions for convex semidef- inite programming problems that involve uncertain data. It first reviews essential findings regarding the op- timality condition and duality of robust convex semidefinite programming problems. Then, we establish the optimality and duality conditions concerning the problem by assuming specific constraint qualifications. The study investigates ε-Kuhn-Tucker vectors and their relationships with the optimal solutions, maximizers of the corresponding Lagrangian dual problem, saddle points of the Lagrangian, and Kuhn-Tucker vectors. Finally, the article establishes the characterization of ε-approximate solution sets for the problem by studying the connection among three sets: the set of Lagrange multipliers corresponding to ε-approximate solutions, the set of ε-Kuhn- Tucker vectors, and the set of approximate solutions for their Lagrangian dual problems. The characterization is illustrated with several examples.

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

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

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.

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

Verification of Parametric Properties of Linear Discrete-time Stochastic Systems

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.

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.

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.

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.

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.

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.

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