Semidefinite Programming Problem Research Articles

Multi-fingered grasping force optimization based on generalized penalty-function concepts

This paper presents an efficient multi-fingered grasping force optimization (GFO) method based on generalized penalty-function concepts. In view of the fact that the mainstream multi-fingered GFO method often treats the second-order cone programming (SOCP) problem as a semi-definite programming (SDP) problem, whose computational complexity is high, we hereby use the barrier function to construct the regularized optimization problem. The trade-off representation of different dimension objective functions is given, and the penalty factor is introduced to form the augmented optimization objective function. For specific operational tasks, by adjusting the penalty factor, a more compact, stable or slack, flexible grasping scheme could be obtained. Monte Carlo simulation is used to determine the probability of successful grasping when variability is introduced, and the robustness of the proposed method in the change of contact position and the friction coefficient between hand and object is verified. Experimental results and dynamic simulation are given, which show that the proposed algorithm outperforms the mainstream SDP method in execution time and iteration number, and the obtained force distribution has both continuity and distribution. Operational flexibility is instructive for practical applications.

Refining the partition for multifold conic optimization problems

ABSTRACT In this paper, we give a unified treatment of two different definitions of complementarity partition of multifold conic programs introduced independently in Bonnans and Ramírez [Perturbation analysis of second-order cone programming problems, Math Program. 2005;104(2–30):205–227] for conic optimization problems, and in Peña and Roshchina [A complementarity partition theorem for multifold conic systems, Math Program. 2013;142(1–2):579–589] for homogeneous feasibility problems. We show that both can be treated within the same unified geometric framework and extend the latter notion to optimization problems. We also show that the two partitions do not coincide, and their intersection gives a seven-set index partition. Finally, we demonstrate that the partitions are preserved under the application of nonsingular linear transformations, and in particular, that a standard conversion of a second-order cone program into a semidefinite programming problem preserves the partitions.

Exploiting low-rank structure in semidefinite programming by approximate operator splitting

In contrast to many other convex optimization classes, state-of-the-art semidefinite programming solvers are still unable to efficiently solve large-scale instances. This work aims to reduce this scalability gap by proposing a novel proximal algorithm for solving general semidefinite programming problems. The key characteristic of the proposed algorithm is to be able to exploit the low-rank property inherent to several semidefinite programming problems. Exploiting the low-rank structure provides a substantial speedup and allows the operator splitting method to efficiently scale to larger instances. As opposed to other low-rank based methods, the proposed algorithm has convergence guarantees for general semidefinite programming problems. Additionally, an open-source semidefinite programming solver called ProxSDP is made available and its implementation details are discussed. Case studies are presented in order to evaluate the performance of the proposed methodology.

Sparse PSD approximation of the PSD cone

While semidefinite programming (SDP) problems are polynomially solvable in theory, it is often difficult to solve large SDP instances in practice. One technique to address this issue is to relax the global positive-semidefiniteness (PSD) constraint and only enforce PSD-ness on smaller $$k\times k$$ principal submatrices—we call this the sparse SDP relaxation. Surprisingly, it has been observed empirically that in some cases this approach appears to produce bounds that are close to the optimal objective function value of the original SDP. In this paper, we formally attempt to compare the strength of the sparse SDP relaxation vis-a-vis the original SDP from a theoretical perspective. In order to simplify the question, we arrive at a data independent version of it, where we compare the sizes of SDP cone and the $$k$$ -PSD closure, which is the cone of matrices where PSD-ness is enforced on all $$k\times k$$ principal submatrices. In particular, we investigate the question of how far a matrix of unit Frobenius norm in the $$k$$ -PSD closure can be from the SDP cone. We provide two incomparable upper bounds on this farthest distance as a function of k and n. We also provide matching lower bounds, which show that the upper bounds are tight within a constant in different regimes of k and n. Other than linear algebra techniques, we extensively use probabilistic methods to arrive at these bounds. One of the lower bounds is obtained by observing a connection between matrices in the $$k$$ -PSD closure and matrices satisfying the restricted isometry property.

Optimal sensor attacks in cyber-physical systems with Round-Robin protocol

This paper investigates the stealthy attack problem in cyber-physical systems (CPSs) under the scheduling effects of the Round-Robin protocol (RRP), where the transmission order of the sensor nodes is modeled as a periodic function and only one node is allowed to access the shared communication channel at each time instant. The adoption of the RRP can efficiently avoid data collisions, while it makes the existing innovation-based stealthy attack strategies invalid. Thus, the aim is to design an attack strategy under the protocol-induced behaviors to maximize the attack effect and remain strictly stealthy simultaneously. Different from the existing results, where only single-sensor application scenarios were considered, multi-sensor ones with limited bandwidth are investigated in this work. A new attack model is proposed to overcome the protocol-introduced effects, which utilizes only part of the historical and the current innovations at each time instant. Based on the attack model, the stealthy condition is obtained and converted into the form of linear matrix inequality (LMI). Then, the error covariance under the attack and the RRP is derived to quantify the attack effect. Subsequently, the optimal attack strategy is obtained by solving a semi-definite programming (SDP) problem. Finally, simulations are performed to demonstrate the results.

Numerical upper bounds to the ultimate load bearing capacity of three‐dimensional reinforced concrete structures

ABSTRACTThis contribution is addressing the ultimate limit state design of massive three‐dimensional reinforced concrete structures based on a finite‐element implementation of yield design theory. The strength properties of plain concrete are modeled either by means of a tension cutoff Mohr Coulomb or a Rankine condition, while the contribution of the reinforcing bars is taken into account by means of a homogenization method. This homogenization method can either represent regions of uniformly distributed steel rebars smeared into the concrete domain, but it can also be extended to model single rebars diluted into a larger region, thereby simplifying mesh generation and mesh size requirements in this region. The present paper is mainly focused on the implementation of the upper bound kinematic approach formulated as a convex minimization problem. The retained strength condition for the plain concrete and homogenized reinforced regions are both amenable to a formulation involving positive semidefinite constraints. The resulting semidefinite programming problems can, therefore, be solved using state‐of‐the‐art dedicated solvers. The whole computational procedure is applied to some illustrative examples, where the implementation of both static and kinematic methods produces a relatively accurate bracketing of the exact failure load for this kind of structures.

Beamforming Design for Multiuser Transmission Through Reconfigurable Intelligent Surface

This article investigates the problem of resource allocation for multiuser communication networks with a reconfigurable intelligent surface (RIS)-assisted wireless transmitter. In this network, the sum transmit power of the network is minimized by controlling the phase beamforming of the RIS and transmit power of the base station. This problem is posed as a joint optimization problem of transmit power and RIS control, whose goal is to minimize the sum transmit power under signal-to-interference-plus-noise ratio (SINR) constraints of the users. To solve this problem, a dual method is proposed, where the dual problem is obtained as a semidefinite programming problem. After solving the dual problem, the phase beamforming of the RIS is obtained in the closed form, while the optimal transmit power is obtained by using the standard interference function. Simulation results show that the proposed scheme can reduce up to 94% and 27% sum transmit power compared to the maximum ratio transmission (MRT) beamforming and zero-forcing (ZF) beamforming techniques, respectively.

Gridless One-Bit Direction-of-Arrival Estimation Via Atomic Norm Denoising

In this letter, a gridless one-bit direction-of-arrival (DOA) estimation approach with one-snapshot is proposed to be robust against the off-grid errors and sign inconsistency because of the one-bit measurements corrupted by additive noise. Different with the existing off-grid DOA estimators, an atomic norm minimization is considered to mitigate the grid mismatch with atoms instead of pre-divided discretized dictionary. Then, the sign inconsistency of one-bit measurements is solved by introducing a linear loss function. The resulting optimization problem is convex and can be equivalent to a semidefinite programming (SDP) problem, which, however, is computational demanding. Therefore, alternating direction multiplier method (ADMM) is employed to speed up the implementation. To avoid the spectrum searching, an effective dual polynomial method is developed with closed-form solution for DOA estimation. Meanwhile, the proposed method does not require $a~prior$ information of the number of targets. Simulation results demonstrate the superiority and effectiveness of the proposed method.

Addressing the Conditional and Correlated Wind Power Forecast Errors in Unit Commitment by Distributionally Robust Optimization

In this paper, a study of the day-ahead unit commitment problem with stochastic wind power generation is presented, which considers conditional, and correlated wind power forecast errors through a distributionally robust optimization approach. Firstly, to capture the characteristics of random wind power forecast errors, the least absolute shrinkage, and selection operator (Lasso) is utilized to develop a robust conditional error estimator, while an unbiased estimator is used to obtain the covariance matrix. The conditional error, and the covariance matrix are then used to construct an enhanced ambiguity set. Secondly, we develop an equivalent mixed integer semidefinite programming (MISDP) formulation of the two-stage distributionally robust unit commitment model with a polyhedral support of random variables. Further, to efficiently solve this problem, a novel cutting plane algorithm that makes use of the extremal distributions identified from the second-stage semidefinite programming (SDP) problems is introduced. Finally, numerical case studies show the advantage of the proposed model in capturing the spatiotemporal correlation in wind power generation, as well as the economic efficiency, and robustness of dispatch decisions.

Sum-Rate Optimization Constrained by Consumed Power for Multi- Antenna Non-Regenerative Relay Network

Background & Objective: This paper considers a multi-pair wireless network, which communicates peer-to-peer using some multi-antenna amplify-and-forward relays. Maximizing the throughput supposing that the total relay nodes’ power consumption is constrained, is the main objective of this investigation. We prove that finding the beamforming matrix is not a convex problem. Methods: Therefore, by using a semidefinite relaxation technique we find a semidefinite programming problem. Moreover, we propose a novel algorithm for maximizing the total signal to the total leakage ratio. Numerical analyses show the effectiveness of the proposed algorithm which offers higher throughput compared to the existing total leakage minimization algorithm, with much less complexity. Results and Conclusion: Furthermore, the effect of different parameters such as, the number of relays, the number of antennas in each relay, the number of transmitter/receiver pairs and uplink and downlink channel gains are investigated.

Secure transmission for wireless collaborative systems based on second‐order channel statistics

AbstractIn this article, we investigate the maximum secrecy rates (SRs) and the optimal beamforming schemes for a collaborative wireless network under uncertain channel levels of available second‐order statistics. Both decode‐and‐forward (DF) and amplify‐and‐forward (AF) strategies under the aggregate and individual power constraints are considered. We have derived the maximum SRs and the optimal beamforming vectors under aforementioned conditions and an efficient cooperation paradigm to provide secrecy in a multirelay environment. A closed‐form maximum SR in the DF scheme under aggregate power constraint is presented. Next, the upper and lower bounds for the maximum SR under aggregate power constraint in the AF scheme is derived. The optimization problem in the AF scheme can be converted into a convex semidefinite programming problem, which is efficiently solved by the bisection method. Moreover, the optimal beamforming weights are also obtained to maximize the SRs under both aggregate and individual relay power constraints. Numerical results are provided to demonstrate the effectiveness of proposed algorithms.

Source localisation using TDOA and FDOA measurements under unknown noise power knowledge

In this study, the authors develop a new algorithm for passive source localisation using time-difference-of-arrival (TDOA) and frequency-difference-of-arrival (FDOA) measurements. In this algorithm, the position and velocity of the target source are successively estimated by formulating two alone semidefinite programming problems, where the variances of the TDOA and FDOA measurement noises can be separated and neglected. Unlike the traditional methods, the authors’ approach does not require the prior knowledge of the TDOA/FDOA measurement noise power. Simulation results indicate that the proposed algorithm outperforms the existing methods under unknown noise power knowledge.

Moving Target Localization in Multiple-Input Multiple-Output Radar Systems Without the Prior Knowledge of Measurement Noise Powers

This letter investigates the problem of locating a moving target using bistatic range (BR) and bistatic range rate (BRR) measurements in multi-input and multi-output (MIMO) radar systems. The proposed estimator constructs two separate semidefinite programming (SDP) problems, in which both the BR and BRR measurement noise powers can be neglected. By utilizing the optimization toolbox, we efficiently solve the SDP problems and give the source position and velocity successively. Unlike the traditional techniques, the proposed method does not require the exact prior knowledge of the measurement noise powers. Simulation results validate the performance of the proposed method.

The Geometric Properties of a Class of Nonsymmetric Cones

Geometric methods are important for researching the differential properties of metric projectors, sensitivity analysis, and the augmented Lagrangian algorithm. Sun [3] researched the relationship among the strong second-order sufficient condition, constraint nondegeneracy, B-subdifferential nonsingularity of the KKT system, and the strong regularity of KKT points in investigating nonlinear semidefinite programming problems. Geometric properties of cones are necessary in studying second-order sufficient condition and constraint nondegeneracy. In this paper, we study the geometric properties of a class of nonsymmetric cones, which is widely applied in optimization problems subjected to the epigraph of vector k-norm functions and low-rank-matrix approximations. We compute the polar, the tangent cone, the linear space of the tangent cone, the critical cone, and the affine hull of this critical cone. This paper will support future research into the sensitivity and algorithms of related optimization problems.

Two Timescale Robust Energy-Efficient Precoding for Dual-Polarized MIMO Systems

In this work, we investigate in depth the two timescale robust system energy efficiency (EE) precoding design for the multiuser dual-polarized multiple-input multiple-output (MIMO) system. To achieve good performance, low feedback overhead as well as low implementation complexity, the dual-structured linear precoding scheme is adopted, which is based on the two timescale channel state information (CSI) and the dual-polarized antenna structure. The subgrouping technique, which is based on polarization, is also utilized to divide spatially grouped users into co-polarized subgroups to further reduce channel feedback overhead. The proposed robust system EE precoding design can achieve the maximization of the worst-case system EE, with the norm-bounded channel errors of all users. This robust EE optimization is naturally decomposed into two parts. In the first part, based on the polarized spatial correlation information, the block diagonalization is utilized to design the preprocessing matrix. In the second part, based on the relationship between the mean square error and the signal-to-interference-plus-noise ratio, the linear precoding matrix can be optimized by utilizing the sign-definiteness lemma and the fractional programming technique. Specifically, the corresponding nonconvex EE fractional optimization problem is converted to a series of semidefinite programming problems, which are solved by the convex optimization method efficiently. Simulation results indicate that the proposed two timescale based dual-structured precoding has many advantages on the robust system EE performance in the dual-polarized multiuser MIMO system.

Adaptive Target Separation Detection

This article considers target separation detection (TSD) in the presence of homogeneous Gaussian interference. The problem is formulated as a hypothesis test for two typical situations: 1) the use of a low range-resolution radar or fast-track update rate; 2) the use of a high-resolution radar or low-track update rate. At the design stage, TSD tests are devised according to the generalized likelihood ratio test criterion. The computation of each decision statistic requires the solution of a semidefinite programming problem obtained leveraging the relationship among linear matrix inequalities and nonnegative trigonometric polynomials. All the obtained decision rules ensure the bounded constant false alarm rate property. At the analysis stage, the performance of some benchmark detectors is given in terms of detection and false alarm probabilities. Finally, numerical examples are provided to show the performance of the proposed tests as compared with the benchmarks as well as the gain achievable over some counterparts devised to monitor a separation event.

A filter-type method for solving nonlinear semidefinite programming

Recently, there has been much interest in filter methods for solving linear and nonlinear problems. We describe extensions of such methods to nonlinear semidefinite programming problems. These extensions are based on linear subproblems that combine filter techniques and trust region methods. The results of convergence and the rate of convergence are given. Numerical examples on semidefinite programming problems are also reported.

Truss geometry and topology optimization with global stability constraints

In this paper, we introduce geometry optimization into an existing topology optimization workflow for truss structures with global stability constraints, assuming a linear buckling analysis. The design variables are the cross-sectional areas of the bars and the coordinates of the joints. This makes the optimization problem formulations highly nonlinear and yields nonconvex semidefinite programming problems, for which there are limited available numerical solvers compared with other classes of optimization problems. We present problem instances of truss geometry and topology optimization with global stability constraints solved using a standard primal-dual interior point implementation. During the solution process, both the cross-sectional areas of the bars and the coordinates of the joints are concurrently optimized. Additionally, we apply adaptive optimization techniques to allow the joints to navigate larger move limits and to improve the quality of the optimal designs.

A primal-dual interior point trust-region method for nonlinear semidefinite programming

In this paper, we propose a primal-dual interior point trust-region method for solving nonlinear semidefinite programming problems. The method consists of the outer iteration (SDPIP) that finds a Karush–Kuhn–Tucker (KKT) point and the inner iteration (SDPTR) that calculates an approximate barrier KKT point. Algorithm SDPTR combines a commutative class of Newton-like directions with the steepest descent type direction within the framework of the trust-region strategy. We present a trust-region method in primal-dual space and prove the global convergence property of the proposed method. Some numerical experiments are given. In addition, we also present second-order approximations to the primal-dual merit function, and a trust-region method in primal space in Appendix.

Constrained Uncertain System Stabilization with Enlargement of Invariant Sets

An enhanced method able to perform accurate stability of constrained uncertain systems is presented. The main objective of this method is to compute a sequence of feedback control laws which stabilizes the closed-loop system. The proposed approach is based on robust model predictive control (RMPC) and enhanced maximized sets algorithm (EMSA), which are applied to improve the performance of the closed-loop system and achieve less conservative results. In fact, the proposed approach is split into two parts. The first is a method of enhanced maximized ellipsoidal invariant sets (EMES) based on a semidefinite programming problem. The second is an enhanced maximized polyhedral set (EMPS) which consists of appending new vertices to their convex hull to minimize the distance between each new vertex and the polyhedral set vertices to ensure state constraints. Simulation results on two examples, an uncertain nonisothermal CSTR and an angular positioning system, demonstrate the effectiveness of the proposed methodology when compared to other works related to a similar subject. According to the performance evaluation, we recorded higher feedback gain provided by smallest maximized invariant sets compared to recently studied methods, which shows the best region of stability. Therefore, the proposed algorithm can achieve less conservative results.