Semidefinite Programming Research Articles

Cooperative Localization in Hybrid Active and Passive Wireless Sensor Networks With Unknown Tx Power

Hybrid active and passive wireless sensor networks (HWSNs) gain advantages in extending the network lifetime and reducing the overall cost. Because the passive sensors without battery harvest the energy from distributed active sensor signal beam, and only a few active sensors can maintain a large-scale network. Thus, how to track the passive sensor’s location is essential for network management. Since the active sensors are sparsely deployed, cooperative localization which employs passive sensors to locate themselves together is a promising solution. In this paper, we analyze the energy beam generated by the active sensors on the cooperative localization accuracy of the passive sensors. We consider the spatial-temporal cooperative localization based on the received signal strength (RSS) model with unknown Tx power information of each sensor due to the limited processing capabilities, circuit complexity and energy constraints. We formulate the Fisher information matrix (FIM) and the corresponding Cramér-Rao lower bound (CRLB) for the static full connected network and dynamic spatial-temporal recursive network. Accordingly, energy beamforming schemes are proposed to optimize localization accuracy and energy efficiency problems. For the optimal localization problem, we derive the closed-form solution of the optimal energy beamforming wave. For the optimal energy efficiency problem, we propose a semi-definite programming (SDP) solution to achieve optimal energy consumption with a self-calibration method, which can address the over-relax problem. Extensive simulation results indicate that our proposed beamforming schemes have high localization accuracy and lower power consumption compared with the existing power allocation-based schemes.

Moving Object Localization in Distributed MIMO With Clock and Frequency Offsets

This paper investigates the localization problem of the moving target in distributed multiple-input multiple-output (MIMO) systems, where the transmitters are subject to clock and frequency offsets. The position and velocity of the moving target are jointly determined by applying time delay (TD) and Doppler shift (DS) measurements. For this purpose, two efficient solution approaches are proposed to estimate the unknowns of the moving target, as well as the clock and frequency offsets. Firstly, the nonconvex localization problem is first relaxed into a convex semidefinite programming (SDP) form. Additionally, a two-stage closed-form solution (TSCFS) approach is proposed to determine the unknowns. The performance of the proposed methods is examined by the mean squared error (MSE) analysis. The simulated results show that the performance of the proposed solutions are able to approach the Cramér-rao lower bound (CRLB) accuracy at low noise levels. In contrast, the SDP solution performs better than the TSCFS in the presence of high noise levels or a small number of receivers.

The inflation hierarchy and the polarization hierarchy are complete for the quantum bilocal scenario

It is a fundamental but difficult problem to characterize the set of correlations that can be obtained by performing measurements on quantum mechanical systems. The problem is particularly challenging when the preparation procedure for quantum states is assumed to comply with a given causal structure. Recently, a first completeness result for this quantum causal compatibility problem has been given based on the so-called quantum inflation technique. However, completeness was achieved by imposing additional technical constraints, such as an upper bound on the Schmidt rank of the observables. Here, we show that these complications are unnecessary in the quantum bilocal scenario, a much-studied abstract model of entanglement swapping experiments. We prove that the quantum inflation hierarchy is complete for the bilocal scenario in the commuting observable model of locality. We also give a bilocal version of an observation by Tsirelson, namely, in finite dimensions, the commuting observable model and the tensor product model of locality coincide. These results answer questions recently posed by Renou and Xu [arXiv:2210.09065v2 (2022)]. Finally, we point out that our techniques can be interpreted more generally as giving rise to a semidefinite programming hierarchy that is complete for the problem of optimizing polynomial functions in the states of operator algebras defined by generators and relations. The completeness of this polarization hierarchy follows from a quantum de Finetti theorem for states on maximal C*-tensor products.

Computing the quantumguesswork: a quadratic assignment problem

The quantum guesswork quantifies the minimum number of queries needed to guess the state of a quantum ensemble if one is allowed to query only one state at a time. Previous approaches to the computation of the guesswork were based on standard semi-definite programming techniques and therefore lead to approximated results. In contrast, we show that computing the quantum guesswork of qubit ensembles with uniform probability distribution corresponds to solving a quadratic assignment problem and we provide an algorithm that, upon the input of any qubit ensemble over a discrete ring, after finitely many steps outputs the exact closed-form expression of its guesswork. While in general the complexity of our guesswork-computing algorithm is factorial in the number of states, our main result consists of showing a more-than-quadratic speedup for symmetric ensembles, a scenario corresponding to the three-dimensional analog of the maximization version of the turbine-balancing problem. To find such symmetries, we provide an algorithm that, upon the input of any point set over a discrete ring, after finitely many steps outputs its exact symmetries. The complexity of our symmetries-finding algorithm is polynomial in the number of points. As examples, we compute the guesswork of regular and quasi-regular sets of qubit states.

An equivalent nonlinear optimization model with triangular low-rank factorization for semidefinite programs

ABSTRACT In this paper, we propose a new nonlinear optimization model to solve semidefinite optimization problems (SDPs), providing some properties related to local optimal solutions. The proposed model is based on another nonlinear optimization model given by [S. Burer and R. Monteiro, A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization, Math. Program. Ser. B 95 (2003), pp. 329–357], but it has several nice properties not seen in the existing one. Firstly, the decision variable of the proposed model is a triangular low-rank matrix. Secondly, the existence of a strict local optimum of the proposed model is guaranteed under some conditions, whereas the existing model has no strict local optimum. In other words, it is difficult to construct solution methods equipped with fast convergence using the existing model. We also present some numerical results, showing that the use of the proposed model allows to deliver highly accurate solutions.

Doubly Constrained Waveform Optimization for Integrated Sensing and Communications

This paper investigates threshold-constrained joint waveform optimization for an integrated sensing and communication (ISAC) system. Unlike existing studies, we employ mutual information (MI) and sum rate (SR) as sensing and communication metrics, respectively, and optimize the waveform under constraints to both metrics simultaneously. This provides significant flexibility in meeting system performance. We formulate three different optimization problems that constrain the radar performance only, the communication performance only, and the ISAC performance, respectively. New techniques are developed to solve the original problems, which are NP-hard and cannot be directly solved by conventional semi-definite programming (SDP) techniques. Novel gradient descent methods are developed to solve the first two problems. For the third non-convex optimization problem, we transform it into a convex problem and solve it via convex toolboxes. We also disclose the connections between three optimizations using numerical results. Finally, simulation results are provided and validate the proposed optimization solutions.

Efficient Sensor Node Selection for Observability Gramian Optimization.

Optimization approaches that determine sensitive sensor nodes in a large-scale, linear time-invariant, and discrete-time dynamical system are examined under the assumption of independent and identically distributed measurement noise. This study offers two novel selection algorithms, namely an approximate convex relaxation method with the Newton method and a gradient greedy method, and confirms the performance of the selection methods, including a convex relaxation method with semidefinite programming (SDP) and a pure greedy optimization method proposed in the previous studies. The matrix determinant of the observability Gramian was employed for the evaluations of the sensor subsets, while its gradient and Hessian were derived for the proposed methods. In the demonstration using numerical and real-world examples, the proposed approximate greedy method showed superiority in the run time when the sensor numbers were roughly the same as the dimensions of the latent system. The relaxation method with SDP is confirmed to be the most reasonable approach for a system with randomly generated matrices of higher dimensions. However, the degradation of the optimization results was also confirmed in the case of real-world datasets, while the pure greedy selection obtained the most stable optimization results.

A Semidefinite Relaxation Based Branch-and-Bound Method for Tight Neural Network Verification

We introduce a novel method based on semidefinite program (SDP) for the tight and efficient verification of neural networks. The proposed SDP relaxation advances the present state of the art in SDP-based neural network verification by adding a set of linear constraints based on eigenvectors. We extend this novel SDP relaxation by combining it with a branch-and-bound method that can provably close the relaxation gap up to zero. We show formally that the proposed approach leads to a provably tighter solution than the present state of the art. We report experimental results showing that the proposed method outperforms baselines in terms of verified accuracy while retaining an acceptable computational overhead.

Concurrent Multi-Label Prediction in Event Streams

Streams of irregularly occurring events are commonly modeled as a marked temporal point process. Many real-world datasets such as e-commerce transactions and electronic health records often involve events where multiple event types co-occur, e.g. multiple items purchased or multiple diseases diagnosed simultaneously. In this paper, we tackle multi-label prediction in such a problem setting, and propose a novel Transformer-based Conditional Mixture of Bernoulli Network (TCMBN) that leverages neural density estimation to capture complex temporal dependence as well as probabilistic dependence between concurrent event types. We also propose potentially incorporating domain knowledge in the objective by regularizing the predicted probability. To represent probabilistic dependence of concurrent event types graphically, we design a two-step approach that first learns the mixture of Bernoulli network and then solves a least-squares semi-definite constrained program to numerically approximate the sparse precision matrix from a learned covariance matrix. This approach proves to be effective for event prediction while also providing an interpretable and possibly non-stationary structure for insights into event co-occurrence. We demonstrate the superior performance of our approach compared to existing baselines on multiple synthetic and real benchmarks.

Sum-of-squares certificates for Vizing's conjecture via determining Gröbner bases

The famous open Vizing conjecture claims that the domination number of the Cartesian product graph of two graphs G and H is at least the product of the domination numbers of G and H. Recently Gaar, Krenn, Margulies and Wiegele used the graph class G of all graphs with nG vertices and domination number kG and reformulated Vizing's conjecture as the problem that for all graph classes G and H the Vizing polynomial is sum-of-squares (SOS) modulo the Vizing ideal. By solving semidefinite programs (SDPs) and clever guessing they derived SOS-certificates for some values of kG, nG, kH, and nH.In this paper, we consider their approach for kG=kH=1. For this case we are able to derive the unique reduced Gröbner basis of the Vizing ideal. Based on this, we deduce the minimum degree (nG+nH−1)/2 of an SOS-certificate for Vizing's conjecture, which is the first result of this kind. Furthermore, we present a method to find certificates for graph classes G and H with nG+nH−1=d for general d, which is again based on solving SDPs, but does not depend on guessing and depends on much smaller SDPs. We implement our new method in SageMath and give new SOS-certificates for all graph classes G and H with kG=kH=1 and nG+nH≤15.

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.

A preconditioned iterative interior point approach to the conic bundle subproblem

AbstractThe conic bundle implementation of the spectral bundle method for large scale semidefinite programming solves in each iteration a semidefinite quadratic subproblem by an interior point approach. For larger cutting model sizes the limiting operation is collecting and factorizing a Schur complement of the primal-dual KKT system. We explore possibilities to improve on this by an iterative approach that exploits structural low rank properties. Two preconditioning approaches are proposed and analyzed. Both might be of interest for rank structured positive definite systems in general. The first employs projections onto random subspaces, the second projects onto a subspace that is chosen deterministically based on structural interior point properties. For both approaches theoretic bounds are derived for the associated condition number. In the instances tested the deterministic preconditioner provides surprisingly efficient control on the actual condition number. The results suggest that for large scale instances the iterative solver is usually the better choice if precision requirements are moderate or if the size of the Schur complemented system clearly exceeds the active dimension within the subspace giving rise to the cutting model of the bundle method.

Elastoplastic and limit analysis of reinforced concrete with an equilibrium-based finite element formulation

This paper introduces a 3D equilibrium-based finite element formulation for reinforced concrete. The stress unknowns in the proposed formulation strongly satisfy the equilibrium equations throughout the entire volume, with the exception of the mesh interfaces where the rebars intersect. The concrete stress tensor is interpolated using tetrahedral piece-wise linear elements with statically admissible discontinuities, while the rebars are considered as 1D elements embedded into the concrete, intersecting the triangular mesh. Thus, the global equilibrium is ensured by writing for each face of the mesh the traction continuity equation, while including the rebar stress contribution for the intersected triangular faces. The elastic equilibrium formulation is then written into an equivalent optimization problem, extended to the elastoplastic case by simply adding semi-definite matrix constraints on the concrete stress tensor, corresponding to a Rankine or a truncated Mohr–Coulomb criterion. As for the rebars, they are considered to obey a 1D perfectly elastoplastic behavior. The present formulation is also developed for limit analysis, to directly obtain an estimate of the lower bound of the collapse load. The resulting semi-definite programming optimization problems are solved using an in–house interior point algorithm.

Transmit Power Optimization and Precoding Design in Multiuser Satellite MIMO Downlink With SINR Constraints

In this paper, a multi-user (MU) satellite Multiple-Input Multiple-Output (MIMO) downlink including a multi-beam GEO satellite and multiple terrestrial users is considered. In order to meet the signal to interference plus noise ratio (SINR) requirements of different users while minimizing the satellite transmit power, the onboard precoding problem is investigated. Zero forcing (ZF) and regularized zero-forcing (RZF) precoding are first adopted and it is found that they can satisfy the SINR constraints by multiplying by a power allocation matrix. The required power allocation matrix is divided into two types: equal power factor schemes and unequal power factor schemes, and they are derived separately, where Karush-Kuhn-Tucker (KKT) conditions are used in RZF unequal power factor scheme and a heuristic algorithm is proposed to optimize the regularization factor in RZF schemes to further reduce the transmit power. Then, a semi-definite programming beamforming (SDPBF) is proposed to jointly optimize precoding and power allocation matrices. The closed-form expressions of the transmit power of the above five precoding schemes are derived and their performance is compared in the simulation part. The research results show that under the same conditions, the performance of RZF is better than that of ZF, the unequal power factor scheme is better than the equal power factor scheme, and SDPBF has the best performance. Besides, it is also found that in addition to changing the precoding method, activating more beams will further reduce the transmit power.

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.

Convex Optimization for Nonequilibrium Steady States on a Hybrid Quantum Processor.

Finding the transient and steady state properties of open quantum systems is a central problem in various fields of quantum technologies. Here, we present a quantum-assisted algorithm to determine the steady states of open system dynamics. By reformulating the problem of finding the fixed point of Lindblad dynamics as a feasibility semidefinite program, we bypass several well-known issues with variational quantum approaches to solving for steady states. We demonstrate that our hybrid approach allows us to estimate the steady states of higher dimensional open quantum systems and discuss how our method can find multiple steady states for systems with symmetries.

Semidefinite programming bounds for binary codes from a split Terwilliger algebra

We study the upper bounds for $A(n,d)$, the maximum size of codewords with length $n$ and Hamming distance at least $d$. Schrijver studied the Terwilliger algebra of the Hamming scheme and proposed a semidefinite program to bound $A(n, d)$. We derive more sophisticated matrix inequalities based on a split Terwilliger algebra to improve Schrijver's semidefinite programming bounds on $A(n, d)$. In particular, we improve the semidefinite programming bounds on $A(18,4)$ to $6551$.

PSWF-based decoupled atomic norm minimization for DOD and DOA estimation in MIMO radar with arbitrary linear arrays

Decoupled atomic norm minimization (D-ANM) is a computationally-efficient gridless two-dimensional parameter estimation method via dividing the two-level Toeplitz matrix into two one-level matrices, and can be applied to the one-snapshot direction-of-departure (DOD) and direction-of-arrival (DOA) estimation in bistatic multiple-input multiple-output (MIMO) radar with uniform linear arrays (ULAs). Nevertheless, in actual applications, the transmitting and receiving arrays are not always ULAs, which leads to the Toeplitz structure not satisfied and the regular D-ANM algorithm not available. In this work, the one-snapshot DOD and DOA estimation in bistatic MIMO radar with arbitrary linear arrays (ALAs) is investigated. The Prolate spheroidal wave functions (PSWFs) are utilized to express such a D-ANM problem followed by a semi-definite programming (SDP) solution. The PSWF-based method can extend the application of D-ANM from ULAs to ALAs, and avoid the grid partition of the conventional sparse representation methods. The numerical experiments are conducted to verify the proposed method.

On Best Low Rank Approximation of Positive Definite Tensors

Tensors, or multi-indexed arrays, play an important role in machine learning and signal processing. These higher-order generalizations of matrices allow for preservation of higher-order structure in data, and low rank decompositions of tensors allow for recovery of underlying information. In many cases, the underlying tensor of interest is positive (semi)definite, i.e., the homogeneous polynomial associated to the tensor has nonnegative evaluation on all inputs. An archetypal problem is that one has a noisy measurement of some low rank signal tensor of interest. This measurement itself does not have low rank, so one must compute a best low rank approximation of the measured tensor to recover component information. As it turns out, the set of tensors of rank less than or equal to is, in general, not closed when , and, as a consequence, best low rank approximations can fail to exist. In the case when a best low rank approximation does not exist, near optimal low rank approximations suffer numerical issues and cannot be used to reliably approximate underlying component information. As a consequence, existence guarantees for best low rank tensor approximations are of great practical and theoretical interest. This article develops deterministic guarantees for the existence of best low rank approximations of tensors which are positive semidefinite. In particular, we show that the set of low rank positive semidefinite tensors is relatively closed as a subset of the set of positive semidefinite tensors. We use this fact to give a deterministic bound which may be used to guarantee the existence of a best low rank approximation of a noisy low rank positive semidefinite tensor. Furthermore, we show that this condition guarantees uniqueness of the canonical polyadic decomposition for best low rank approximations. In addition, for order three tensors, we prove that our bound is sharp and that it can be computed using semidefinite programming. The main results of this article are illustrated through numerical experiments, which show that our bound is highly predictive of numerical issues when attempting to recover underlying component information from a noisy low rank positive semidefinite tensor.

Multi-output multilevel best linear unbiased estimators via semidefinite programming

Multifidelity forward uncertainty quantification (UQ) problems often involve multiple quantities of interest and heterogeneous models (e.g., different grids, equations, dimensions, physics, surrogate and reduced-order models). While computational efficiency is key in this context, multi-output strategies in multilevel/multifidelity methods are either sub-optimal or non-existent. In this paper we extend multilevel best linear unbiased estimators (MLBLUE) to multi-output forward UQ problems and we present new semidefinite programming formulations for their optimal setup. Not only do these formulations yield the optimal number of samples required, but also the optimal selection of low-fidelity models to use. While existing MLBLUE approaches are single-output only and require a non-trivial nonlinear optimization procedure, the new multi-output formulations can be solved reliably and efficiently. We demonstrate the efficacy of the new methods and formulations in practical UQ problems with model heterogeneity.