Positive Semidefinite Programming Research Articles

Lining up a positive semi-definite six-point bootstrap

In this work, we initiate a positive semi-definite numerical bootstrap program for multi-point correlators. Considering six-point functions of operators on a line, we reformulate the crossing symmetry equation for a pair of comb-channel expansions as a semi-definite programming problem. We provide two alternative formulations of this problem. At least one of them turns out to be amenable to numerical implementation. Through a combination of analytical and numerical techniques, we obtain rigorous bounds on CFT data in the triple-twist channel for several examples.

Two-Dimensional Super-Resolution Direction-of-Arrival Estimation for Arbitrary Planar Array Geometries

The two-dimensional grid-free compressed sensing method with planar microphone arrays is expected to achieve super-resolution direction-of-arrival (DOA) estimation for acoustic sources in the hemispherical space in front of array. The existing methods are not yet well compatible with arbitrary planar array geometries. In this paper, we aim to break this limitation and explore a novel method applicable to arbitrary planar array geometries. First, the covariance between source-induced signals at arbitrary two microphones is formulated as related to the two-dimensional inverse Fourier transform of the source strength. Then, a mathematical model with a source sparsity constraint is built in the continuous domain to denoise the covariances of measured signals, which can be solved by decoupled positive semidefinite programming. Finally, the DOAs are extracted from the solution of the mathematical model via a matrix pencil and pairing method. Simulation and experimental results demonstrate the effectiveness of the proposed method.

Energy Storage Optimization Method for Flexible Interconnected Low-voltage Distribution Network Based on Positive Semi-definite Programming

With the fast development of the electricity market, the number of small and medium-sized new energy generation in the urban low-voltage distribution networks is increasing. These “retail investors” hope to sell the extra electricity for financial gain. However, these renewable energy generation units have small capacity and obvious intermittent output. Thus, they can not be added to the operation of the power system alone. In this paper, the model is understood from the perspective of optimization theory, and the solution method is given. The optimization problem of the model is given, and the required gradient and Hessian functions are defined. By applying mathematical induction and the properties of the special matrix column, a decomposition formula of the matrix column is derived. Through using this inequality and the standard decomposition of a positive semi-definite matrix, a feasible scheme is formed. The eigenvalue relationship between the sum of elements of a positive semi-definite block matrix and its quasi-diagonal block is given under certain conditions, while the number of such blocks is not required. The standard form of the dual problem is drawn by the method of Lagrange multipliers, and the relationship between the dual solution and the solution of the original problem is studied. The stability of the system is further improved, which provides reference significance for improving the utilization rate of photovoltaic new energy, optimizing the charging and discharging of the energy storage system, and ensuring the safe and reliable operation of the distribution network.

Simultaneous Diagonalization via Congruence of Hermitian Matrices: Some Equivalent Conditions and a Numerical Solution

This paper aims at solving the Hermitian SDC problem, i.e., that of \textit{simultaneously diagonalizing via $*$-congruence} a collection of finitely many (not need pairwise commute) Hermitian matrices. Theoretically, we provide some equivalent conditions for that such a matrix collection can be simultaneously diagonalized via $^*$-congruence.% by a nonsingular matrix. Interestingly, one of such conditions leads to the existence of a positive definite solution to a semidefinite program (SDP). From practical point of view, we propose an algorithm for numerically solving such problem. The proposed algorithm is a combination of (1) a positive semidefinite program detecting whether the initial Hermitian matrices are simultaneously diagonalizable via $*$-congruence, and (2) a Jacobi-like algorithm for simultaneously diagonalizing via $*$-congruence the commuting normal matrices derived from the previous stage. Illustrating examples by hand/coding in \textsc{Matlab} are also presented.

Hyperbolic Relaxation of $k$-Locally Positive Semidefinite Matrices

A successful computational approach for solving large-scale positive semidefinite (PSD) programs is to enforce PSD-ness on only a collection of submatrices. For our study, we let $\mathcal{S}^{n,k}$ be the convex cone of $n\times n$ symmetric matrices where all $k\times k$ principal submatrices are PSD. We call a matrix in this $k$-locally PSD. In order to compare $\mathcal{S}^{n,k}$ to the cone of PSD matrices, we study eigenvalues of $k$-locally PSD matrices. The key insight in this paper is that there is a convex cone $H(e_k^n)$ so that if $X \in \mathcal{S}^{n,k}$, then the vector of eigenvalues of $X$ is contained in $H(e_k^n)$. The cone $H(e_k^n)$ is the hyperbolicity cone of the elementary symmetric polynomial $e_k^n$ (where $e_k^n(x) = \sum_{S \subseteq [n] : |S| = k} \prod_{i \in S} x_i$) with respect to the all ones vector. Using this insight, we are able to improve previously known upper bounds on the Frobenius distance between matrices in $\mathcal{S}^{n,k}$ and PSD matrices. We also study the quality of the convex relaxation $H(e^n_k)$. We first show that this relaxation is tight for the case of $k = n -1$, that is, for every vector in $H(e^n_{n -1})$ there exists a matrix in $\mathcal{S}^{n, n -1}$ whose eigenvalues are equal to the components of the vector. We then prove a structure theorem on nonsingular matrices in $\mathcal{S}^{n,k}$ all of whose $k\times k$ principal minors are zero, which we believe is of independent interest. This result shows shows that for $1< k < n -1$ “large parts” of the boundary of $H(e_k^n)$ do not intersect with the eigenvalues of matrices in $\mathcal{S}^{n,k}$.

When a system of real quadratic equations has a solution

We provide a sufficient condition for solvability of a system of real quadratic equations pi(x)=yi, i=1,…,m, where pi:Rn⟶R are quadratic forms. By solving a positive semidefinite program, one can reduce it to another system of the type qi(x)=αi, i=1,…,m, where qi:Rn⟶R are quadratic forms and αi=traceqi. We prove that the latter system has solution x∈Rn if for some (equivalently, for any) orthonormal basis A1,…,Am in the space spanned by the matrices of the forms qi, the operator norm of A12+…+Am2 does not exceed η/m for some absolute constant η>0. The condition can be checked in polynomial time and is satisfied, for example, for random qi provided m≤γn for an absolute constant γ>0. We prove a similar sufficient condition for a system of homogeneous quadratic equations to have a non-trivial solution. While the condition we obtain is of an algebraic nature, the proof relies on analytic tools including Fourier analysis and measure concentration.

Distributed State Estimation in Digital Distribution Networks Based on Proximal Atomic Coordination

With the emerging digitalization technologies represented by edge computing, distribution networks are gradually transforming into digital distribution networks (DDNs). The realization of edge computing drives the distributed operation of DDNs, where multiple areas exchange boundary information through edge computing devices. Benefitting from the data acquisition and computing capacity of edge computing devices, it is feasible to perform accurate and real-time state estimation on the edge side. Aiming at the state perception with edge computing devices in DDNs, this paper proposes a distributed state estimation (DSE) method based on the proximal atomic coordination (PAC) algorithm. Firstly, based on convex relaxation optimization, the state estimation model is converted into a positive semidefinite programming model to solve the nonconvexity caused by nonlinear measurements, which ensures the accuracy and convergence of state estimation. Then, a DSE method based on the PAC algorithm is proposed to exchange information of each area, which reduces the computation time and realizes the efficient state estimation on the edge side. The model and the effectiveness of the proposed method are numerically demonstrated on the modified PG&#x0026;E 69-node system and the test case from a practical pilot in Guangzhou, China.

Two-dimensional grid-free compressive beamforming with spherical microphone arrays

Compressive beamforming based on microphone array measurements and compressive sensing theory is an emerging and practical acoustic source identification technology. The grid-free compressive beamforming that treats the target source region as a continuum has attracted tremendous attention due to the advantage of circumventing the basis mismatch defect caused by the gridding. The grid-free methods have been developed for the linear and the planar microphone array measurements to identify sources in a local region. In contrast, for the spherical microphone array measurements suitable for 360° panoramic identification, the current research mainly focuses on grid-based methods. In this paper, we propose a two-dimensional grid-free compressive beamforming method with spherical microphone arrays. First of all, the relationship between the spherical harmonic function vector and the two-dimensional Fourier basis function vector is determined. Then, the mathematical model relating the measured sound pressures to the source distribution is built under a continuous setting. Next, a source sparsity constraint is imposed based on the total variation norm to solve the mathematical model, and its dual problem is deduced. Subsequently, the dual problem is solved by positive semidefinite programming. Finally, the DOAs of sources are estimated through polynomial rooting and utilizing the dual optimal variables, and the source strengths are quantified via least-square fitting. An experimental case is conducted to examine the practicability and superiority of the proposed method to the existing grid-based compressive beamforming method. Plenty of Monte Carlo simulations are carried out to disclose the influence of the source coherence, the angular distance between sources, the noise, the number of snapshots and the number of microphones. This study offers a new approach for the accurate and 360° panoramic identification of acoustic sources.

An Interior Point-Proximal Method of Multipliers for Linear Positive Semi-Definite Programming

In this paper we generalize the Interior Point-Proximal Method of Multipliers (IP-PMM) presented in Pougkakiotis and Gondzio (Comput Optim Appl 78:307–351, 2021. https://doi.org/10.1007/s10589-020-00240-9) for the solution of linear positive Semi-Definite Programming (SDP) problems, allowing inexactness in the solution of the associated Newton systems. In particular, we combine an infeasible Interior Point Method (IPM) with the Proximal Method of Multipliers (PMM) and interpret the algorithm (IP-PMM) as a primal-dual regularized IPM, suitable for solving SDP problems. We apply some iterations of an IPM to each sub-problem of the PMM until a satisfactory solution is found. We then update the PMM parameters, form a new IPM neighbourhood, and repeat this process. Given this framework, we prove polynomial complexity of the algorithm, under mild assumptions, and without requiring exact computations for the Newton directions. We furthermore provide a necessary condition for lack of strong duality, which can be used as a basis for constructing detection mechanisms for identifying pathological cases within IP-PMM.

Enhancement of direction-of-arrival estimation performance of spherical ESPRIT via atomic norm minimisation

Estimation of signal parameters via rotational invariance technique (ESPRIT) with spherical microphone arrays (abbreviated as spherical ESPRIT) has recently attracted considerable attention because it can panoramically estimate the direction-of-arrivals of acoustic sources and avoid the degradation in accuracy due to discretisation of the region of interest. However, the state-of-the-art spherical ESPRIT still fails to perform well at high frequencies and under the cases involving coherent sources, a small number of data snapshots, or low signal-to-noise ratios (SNRs). To overcome these limitations, an atomic norm minimisation (ANM) method is proposed, which can measure the source sparsity in a continuous domain. Hence, a novel covariance matrix can be obtained by solving the positive semidefinite programming that is equivalent to the ANM. It is not only independent of the orthogonality of the sampled spherical harmonics but can also decorrelate the sources and denoise the measured signals. The principle involves using the novel covariance matrix as opposed to the original covariance matrix as the spherical ESPRIT input. The simulations and experiments demonstrate that the cooperation with ANM aids the spherical ESPRIT in overcoming or mitigating the aforementioned limitations. Furthermore, the experiments demonstrate that the proposed method is effective in a typical reverberant environment.

Two-Dimensional Multiple-Snapshot Grid-Free Compressive Beamforming Using Alternating Direction Method of Multipliers

Compressive beamforming with planar microphone arrays is capable of estimating the two-dimensional direction-of-arrivals (DOAs) and quantifying the strengths of acoustic sources effectively. The multiple-snapshot grid-free method has recently been concerned due to the advantages that it can circumvent the basis mismatch conundrum of the conventional grid-based method and improve the performance of the single-snapshot grid-free method. The existing atomic norm minimization based strategy uses an off-the-peg interior point method (IPM) based solver to solve the positive semidefinite programming equivalent to the atomic norm minimization. We present an alternative algorithm based on alternating direction method of multipliers (ADMM) in this paper. Both simulations and experiments demonstrate that whether a standard uniform rectangular array or a non-uniform array constituted by a small number of microphones is employed, the two-dimensional multiple-snapshot grid-free compressive beamforming using our ADMM based algorithm can estimate the DOAs and quantify the strengths of acoustic sources well, and reaching the same or even better DOA estimation accuracy as the one using the IPM based solver, our ADMM based algorithm is distinctly faster.

Blind three dimensional deconvolution via convex optimization

In this paper we discuss recovering two signals from their convolution in 3 dimensions. One of the signals is assumed to lie in a known subspace and the other one is assumed to be sparse. Various applications such as super resolution, radar imaging, and direction of arrival estimation can be described in this framework. We introduce a method to estimate parameters of a signal in a low-dimensional subspace which is convolved with another signal comprised of some impulses in time domain. We transform the problem to a convex optimization in the form of a positive semi-definite program using lifting and the atomic norm. We demonstrate that unknown parameters can be recovered by lowpass observations. Numerical simulations show excellent performance of the proposed method.

Localisation algorithm based on weighted semi-definite programming

In order to improve the performance of reduced complexity positive semi-definite programming (RCSDP) algorithm based on time difference of arrival (TDOA), a weighted positive semi-definite programming (WSDP) scheme is proposed in this paper. Based on the squared distance differences between the target node to one anchor node and to the other anchor node, the location of the target node is described as the optimal solution of a non-convex optimisation problem. The semi-definite relaxation technique is used to transform the original non-convex problem into a weighted convex problem, which takes the measurement noise into consideration, and then the estimated location of the target node is obtained. The simulation results show that the localisation performance of WSDP algorithm is better than that of RCSDP algorithm, regardless of whether the target node is located inside or outside the area surrounded by anchor nodes.

A panoramic continuous compressive beamformer with cuboid microphone arrays

Compressive beamforming is a powerful approach for the direction-of-arrival (DOA) estimation and strength quantification of acoustic sources. The conventional grid-based discrete compressive beamformer suffers from the basis mismatch conundrum. Its result degrades under the situation that sources fall off the grid. The existing continuous compressive beamformer with linear or planar microphone arrays can circumvent the conundrum, but work well only for sources in a local region. Here we develop a panoramic continuous compressive beamformer with cuboid microphone arrays based on an atomic norm minimization (ANM) and a matrix pencil and paring method. To solve the positive semidefinite programming equivalent to the ANM efficiently, we formulate a solving algorithm based on the alternating direction method of multipliers. We also present an iterative reweighted ANM to enhance sparsity and resolution. The beamformer is capable of estimating the DOAs and quantifying the strengths of acoustic sources panoramically and accurately, whether a standard uniform or a sparse cuboid microphone array is utilized.

On Doubly Positive Semidefinite Programming Relaxations

On Doubly Positive Semidefinite Programming Relaxations

Two-dimensional grid-free compressive beamforming.

Compressive beamforming realizes the direction-of-arrival (DOA) estimation and strength quantification of acoustic sources by solving an underdetermined system of equations relating microphone pressures to a source distribution via compressive sensing. The conventional method assumes DOAs of sources to lie on a grid. Its performance degrades due to basis mismatch when the assumption is not satisfied. To overcome this limitation for the measurement with plane microphone arrays, a two-dimensional grid-free compressive beamforming is developed. First, a continuum based atomic norm minimization is defined to denoise the measured pressure and thus obtain the pressure from sources. Next, a positive semidefinite programming is formulated to approximate the atomic norm minimization. Subsequently, a reasonably fast algorithm based on alternating direction method of multipliers is presented to solve the positive semidefinite programming. Finally, the matrix enhancement and matrix pencil method is introduced to process the obtained pressure and reconstruct the source distribution. Both simulations and experiments demonstrate that under certain conditions, the grid-free compressive beamforming can provide high-resolution and low-contamination imaging, allowing accurate and fast estimation of two-dimensional DOAs and quantification of source strengths, even with non-uniform arrays and noisy measurements.

Regularly decomposable tensors and classical spin states

A spin-$j$ state can be represented by a symmetric tensor of order $N=2j$ and dimension $4$. Here, $j$ can be a positive integer, which corresponds to a boson; $j$ can also be a positive half-integer, which corresponds to a fermion. In this paper, we introduce regularly decomposable tensors and show that a spin-$j$ state is classical if and only if its representing tensor is a regularly decomposable tensor. In the even-order case, a regularly decomposable tensor is a completely decomposable tensor but not vice versa; a completely decomposable tensors is a sum-of-squares (SOS) tensor but not vice versa; an SOS tensor is a positive semi-definite (PSD) tensor but not vice versa. In the odd-order case, the first row tensor of a regularly decomposable tensor is regularly decomposable and its other row tensors are induced by the regular decomposition of its first row tensor. We also show that complete decomposability and regular decomposability are invariant under orthogonal transformations, and that the completely decomposable tensor cone and the regularly decomposable tensor cone are closed convex cones. Furthermore, in the even-order case, the completely decomposable tensor cone and the PSD tensor cone are dual to each other. The Hadamard product of two completely decomposable tensors is still a completely decomposable tensor. Since one may apply the positive semi-definite programming algorithm to detect whether a symmetric tensor is an SOS tensor or not, this gives a checkable necessary condition for classicality of a spin-$j$ state. Further research issues on regularly decomposable tensors are also raised.

Weighted Algebraic Connectivity Maximization for Optical Satellite Networks

In this paper, the topology configuration methods for heterogeneous optical satellite networks are investigated. Our objectives are to maximize weighted algebraic connectivity with respect to both network initialization and reconfiguration scenarios subject to onboard hardware constraints. The problems are not strictly convex and have been proven as NP-hard. In order to solve the problems in polynomial time, the original problems are relaxed to a convex optimization problem. Specifically, the relaxed problem is transformed to a positive semidefinite programming form, which can be solved exactly and more efficiently. Furthermore, based on the perturbation theory of matrices, we propose two greedy heuristic methods to deal with the initialization and reconfiguration case from the relaxed solutions, respectively. Simulation results show that the proposed algorithms are able to accomplish network initialization and reconfiguration correctly in the overwhelming majority of situations. The final sub-optimal solutions can also be obtained under low computational complexity.

Further relationship between second-order cone and positive semidefinite matrix cone

It is well known that second-order cone (SOC) programming can be regarded as a special case of positive semidefinite programming using the arrow matrix. This paper further studies the relationship between SOCs and positive semidefinite matrix cones. In particular, we explore the relationship to expressions regarding distance, projection, tangent cone, normal cone and the KKT system. Understanding these relationships will help us see the connection and difference between the SOC and its PSD reformulation more clearly.

On linear conic relaxation of discrete quadratic programs

A special reformulation-linearization technique based linear conic relaxation is proposed for discrete quadratic programming (DQP). We show that the proposed relaxation is tighter than the traditional positive semidefinite programming relaxation. More importantly, when the proposed relaxation problem has an optimal solution with rank one or two, optimal solutions to the original DQP problem can be explicitly generated. This rank-two property is further extended to binary quadratic optimization problems and linearly constrained DQP problems. Numerical results indicate that the proposed relaxation is capable of providing high quality and robust lower bounds for DQP.