Semidefinite Programming Algorithm Research Articles

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].

Efficient symbol level precoding design for full-duplex integrated sensing and communication systems

Full-duplex (FD) integrated sensing and communication (ISAC) system attracts much attention because adopting the FD mode can enhance the transmission efficiency and reduce the transmission time of the ISAC system. However, the inevitable self-interference (SI) in FD mode would result in severe performance degradation. This paper considers the FD ISAC system and investigates a symbol-level precoding (SLP) design, which minimizes the Cramér-Rao Bound of target angle estimation while considering the constant-envelope power restraint, SI suppression and the quality of service requirement. Due to the non-convex fractional problem and the unit-modulus constraint, the optimization problem is reformulated by implementing the Schur complement, where the optimal waveform is obtained via the semidefinite programming algorithm. The SLP design that optimizes each symbol vector for all users may suffer high computational complexity of matrix multiplication with high numbers of users. A grouped SLP strategy in which the communication users are divided into several groups on the basis of channel correlation is proposed to reduce the complexity. The proposed grouped SLP design only suppresses group-to-group interference and converts the intra-group interference into a constructive interference, which can reduce the dimension of each precoded signal significantly, resulting in a low computational complexity. Simulation results illustrate that the proposed SLP design can effectively mitigate SI, which can improve the accuracy of target detection. Moreover, the proposed grouped-SLP design can further reduce computational complexity without dramatic performance degradation.

IPRSDP: a primal-dual interior-point relaxation algorithm for semidefinite programming

IPRSDP: a primal-dual interior-point relaxation algorithm for semidefinite programming

Certifying the quantum Fisher information from a given set of mean values: a semidefinite programming approach

We introduce a semidefinite programming algorithm to find the minimal quantum Fisher information compatible with an arbitrary dataset of mean values. This certification task allows one to quantify the resource content of a quantum system for metrology applications without complete knowledge of the quantum state. We implement the algorithm to study quantum spin ensembles. We first focus on Dicke states, where our findings challenge and complement previous results in the literature. We then investigate states generated during the one-axis twisting dynamics, where in particular we find that the metrological power of the so-called multi-headed cat states can be certified using simple collective spin observables, such as fourth-order moments for small systems, and parity measurements for arbitrary system sizes.

An Experimental Evaluation of Semidefinite Programming and Spectral Algorithms for Max Cut

We experimentally evaluate the performance of several Max Cut approximation algorithms. In particular, we compare the results of the Goemans and Williamson algorithm using semidefinite programming with Trevisan’s algorithm using spectral partitioning. The former algorithm has a known .878 approximation guarantee whereas the latter has a .614 approximation guarantee. We investigate whether this gap in approximation guarantees is evident in practice or whether the spectral algorithm performs as well as the SDP. We also compare the performances to the standard greedy Max Cut algorithm which has a .5 approximation guarantee, two additional spectral algorithms, and a heuristic from Burer, Monteiro, and Zhang (BMZ). The algorithms are tested on Erdős–Renyi random graphs, complete graphs from TSPLIB, and real-world graphs from the Network Repository. We find, unsurprisingly, that the spectral algorithms provide a significant speed advantage over the SDP. In our experiments, the spectral algorithms and BMZ heuristic return cuts with values which are competitive with those of the SDP.

Source Localization Based on Hybrid AOA, TDOA, and RSS Measurements

In this paper, a constrained weighted least squares (CWLS) problem with two quadratic constraints for hybrid AOA, TDOA, and RSS localization is formulated. The Lagrange multiplier method cannot be efficiently used in this situation, so we provide four methods to solve the CWLS problem: 1) weighted least squares (WLS); 2) iterative constrained weighted least squares (ICWLS); 3) Lagrange multiplier method combined with Newton’s method (LMM+NM); 4) semidefinite programming (SDP). Furthermore, in order to obtain a good balance between accuracy and computational burden, we develop a fixed point iteration (FPI)-based algorithm that is derived from an approximate maximum likelihood estimator (MLE). Simulation results show that the accuracy of the proposed FPI method is better than the other four methods; besides, its computational burden is one in ten of the SDP algorithm.

Semidefinite programming algorithm for the quantum mechanical bootstrap.

We present a semidefinite programming algorithm to find eigenvalues of Schrödinger operators within the bootstrap approach to quantum mechanics. The bootstrap approach involves two ingredients: a nonlinear set of constraints on the variables (expectation values of operators in an energy eigenstate), plus positivity constraints (unitarity) that need to be satisfied. By fixing the energy we linearize all the constraints and show that the feasibility problem can be presented as an optimization problem for the variables that are not fixed by the constraints and one additional slack variable that measures the failure of positivity. To illustrate the method we are able to obtain high-precision, sharp bounds on eigenenergies for arbitrary confining polynomial potentials in one dimension.

Distributed Synchronous and Asynchronous Algorithms for Semidefinite Programming With Diagonal Constraints

This article develops distributed synchronous and asynchronous algorithms for the large-scale semidefinite programming with diagonal constraints, which has wide applications in combinatorial optimization, image processing, and community detection. The information of the semidefinite programming is allocated to multiple interconnected agents such that each agent aims to find a solution by communicating to its neighbors. Based on the low-rank property of solutions and the Burer–Monteiro factorization, we transform the original problem into a distributed optimization problem over unit spheres to reduce variable dimensions and ensure positive semidefiniteness without involving semidefinite projections, which are computationally expensive. For the distributed optimization problem, we propose distributed synchronous and asynchronous algorithms, both of which reduce computational burden and storage space compared with existing centralized algorithms. Specifically, the distributed synchronous algorithm almost surely escapes strict saddle points and converges to the set of optimal solutions to the optimization problem. In addition, the proposed distributed asynchronous algorithm allows communication delays and converges to critical points to the optimization problem under mild conditions. By applying the proposed algorithms to image segmentation, we illustrate the efficiency and convergence performance of the two proposed algorithms.

Semidefinite Programming Algorithms for 3-D AOA-Based Hybrid Localization

Semidefinite Programming Algorithms for 3-D AOA-Based Hybrid Localization

Community Detection With Known, Unknown, or Partially Known Auxiliary Latent Variables

Empirical observations suggest that in practice, community membership does not completely explain the dependency between the edges of an observation graph. The residual dependence of the graph edges are modeled in this paper, to first order, by auxiliary node latent variables that affect the statistics of the graph edges but <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">carry no information about the communities of interest.</i> We then study community detection in graphs obeying the stochastic block model and censored block model with auxiliary latent variables. We analyze the conditions for exact recovery when these auxiliary latent variables are unknown, representing unknown nuisance parameters or model mismatch. We also analyze exact recovery when these secondary latent variables have been either fully or partially revealed. Finally, we propose a semidefinite programming algorithm for recovering the desired labels when the secondary labels are either known or unknown. We show that exact recovery is possible by semidefinite programming down to the respective maximum likelihood exact recovery threshold.

Joint User Scheduling and Hybrid Beamforming Design for Massive MIMO LEO Satellite Multigroup Multicast Communication Systems.

In the satellite multigroup multicast communication systems based on the DVB-S2X standard, due to the limitation of the DVB-S2X frame structure, user scheduling and beamforming design have become the focus of academic research. In this work, we take the massive multi-input multi-output (MIMO) low earth orbit (LEO) satellite communication system adopting the DVB-S2X standard as the research scenario, and the LEO satellite adopts a uniform planar array (UPA) based on the fully connected hybrid structure. We focus on the coupling design of user scheduling and beamforming; meanwhile, the scheme design takes the influence of residual Doppler shift and phase disturbance on channel errors into account. Under the constraints of total transmission power and quality of service (QoS), we study the robust joint user scheduling and hybrid beamforming design aimed at maximizing the energy efficiency (EE). For this problem, we first adopt the hierarchical clustering algorithm to group users. Then, the semidefinite programming (SDP) algorithm and the concave convex process (CCCP) framework are applied to tackle the optimization of user scheduling and hybrid beamforming design. To handle the rank-one matrix constraint, the penalty iteration algorithm is proposed. To balance the performance and complexity of the algorithm, the user preselected step is added before joint design. Finally, to obtain the digital beamforming matrix and the analog beamforming matrix in a hybrid beamformer, the alternative optimization algorithm based on the majorization-minimization framework (MM-AltOpt) is proposed. Numerical simulation results show that the EE of the proposed joint user scheduling and beamforming design algorithm is higher than that of the traditional decoupling design algorithms.

Consensus of Positive Networked Systems on Directed Graphs.

This article addresses the distributed consensus problem for identical continuous-time positive linear systems with state-feedback control. Existing works of such a problem mainly focus on the case where the networked communication topologies are of either undirected and incomplete graphs or strongly connected directed graphs. On the other hand, in this work, the communication topologies of the networked system are described by directed graphs each containing a spanning tree, which is a more general and new scenario due to the interplay between the eigenvalues of the Laplacian matrix and the controller gains. Specifically, the problem involves complex eigenvalues, the Hurwitzness of complex matrices, and positivity constraints, which make analysis difficult in the Laplacian matrix. First, a necessary and sufficient condition for the consensus analysis of directed networked systems with positivity constraints is given, by using positive systems theory and graph theory. Unlike the general Riccati design methods that involve solving an algebraic Riccati equation (ARE), a condition represented by an algebraic Riccati inequality (ARI) is obtained for the existence of a solution. Subsequently, an equivalent condition, which corresponds to the consensus design condition, is derived, and a semidefinite programming algorithm is developed. It is shown that, when a protocol is solved by the algorithm for the networked system on a specific communication graph, there exists a set of graphs such that the positive consensus problem can be solved as well.

Chaos-Enhanced Adaptive Hybrid Butterfly Particle Swarm Optimization Algorithm for Passive Target Localization.

This paper considers the problem of finding the position of a passive target using noisy time difference of arrival (TDOA) measurements, obtained from multiple transmitters and a single receiver. The maximum likelihood (ML) estimator’s objective function is extremely nonlinear and non-convex, making it impossible to use traditional optimization techniques. In this regard, this paper proposes the chaos-enhanced adaptive hybrid butterfly particle swarm optimization algorithm, named CAHBPSO, as the hybridization of butterfly optimization (BOA) and particle swarm optimization (PSO) algorithms, to estimate passive target position. In the proposed algorithm, an adaptive strategy is employed to update the sensory fragrance of BOA algorithm, and chaos theory is incorporated into the inertia weight of PSO algorithm. Furthermore, an adaptive switch probability is employed to combine global and local search phases of BOA with the PSO algorithm. Additionally, the semidefinite programming is employed to convert the considered problem into a convex one. The statistical comparison on CEC2014 benchmark problems shows that the proposed algorithm provides a better performance compared to well-known algorithms. The CAHBPSO method surpasses the BOA, PSO and semidefinite programming (SDP) algorithms for a broad spectrum of noise, according to simulation findings, and achieves the Cramer–Rao lower bound (CRLB).

The Quantum Approximate Optimization Algorithm and the Sherrington-Kirkpatrick Model at Infinite Size

The Quantum Approximate Optimization Algorithm (QAOA) is a general-purpose algorithm for combinatorial optimization problems whose performance can only improve with the number of layers p. While QAOA holds promise as an algorithm that can be run on near-term quantum computers, its computational power has not been fully explored. In this work, we study the QAOA applied to the Sherrington-Kirkpatrick (SK) model, which can be understood as energy minimization of n spins with all-to-all random signed couplings. There is a recent classical algorithm by Montanari that, assuming a widely believed conjecture, can efficiently find an approximate solution for a typical instance of the SK model to within (1&amp;#x2212;&amp;#x03F5;) times the ground state energy. We hope to match its performance with the QAOA.Our main result is a novel technique that allows us to evaluate the typical-instance energy of the QAOA applied to the SK model. We produce a formula for the expected value of the energy, as a function of the 2p QAOA parameters, in the infinite size limit that can be evaluated on a computer with O(16p) complexity. We evaluate the formula up to p=12, and find that the QAOA at p=11 outperforms the standard semidefinite programming algorithm. Moreover, we show concentration: With probability tending to one as n&amp;#x2192;&amp;#x221E;, measurements of the QAOA will produce strings whose energies concentrate at our calculated value. As an algorithm running on a quantum computer, there is no need to search for optimal parameters on an instance-by-instance basis since we can determine them in advance. What we have here is a new framework for analyzing the QAOA, and our techniques can be of broad interest for evaluating its performance on more general problems where classical algorithms may fail.

Noisy intermediate-scale quantum algorithm for semidefinite programming

Semidefinite programs (SDPs) are convex optimization programs with vast applications in control theory, quantum information, combinatorial optimization and operational research. Noisy intermediate-scale quantum (NISQ) algorithms aim to make an efficient use of the current generation of quantum hardware. However, optimizing variational quantum algorithms is a challenge as it is an NP-hard problem that in general requires an exponential time to solve and can contain many far from optimal local minima. Here, we present a current term NISQ algorithm for solving SDPs. The classical optimization program of our NISQ solver is another SDP over a lower dimensional ansatz space. We harness the SDP based formulation of the Hamiltonian ground state problem to design a NISQ eigensolver. Unlike variational quantum eigensolvers, the classical optimization program of our eigensolver is convex, can be solved in polynomial time with the number of ansatz parameters and every local minimum is a global minimum. We find numeric evidence that NISQ SDP can improve the estimation of ground state energies in a scalable manner. Further, we efficiently solve constrained problems to calculate the excited states of Hamiltonians, find the lowest energy of symmetry constrained Hamiltonians and determine the optimal measurements for quantum state discrimination. We demonstrate the potential of our approach by finding the largest eigenvalue of up to $2^{1000}$ dimensional matrices and solving graph problems related to quantum contextuality. We also discuss NISQ algorithms for rank-constrained SDPs. Our work extends the application of NISQ computers onto one of the most successful algorithmic frameworks of the past few decades.

Improved results on distributed observer-based positive edge consensus

This paper revisits the observer-based positive edge consensus problem for nodal networks. So far, existing positive edge consensus of directed networks with less conservative connectivity conditions have to use the global topology information. On the other hand, instead of using global topology information, the positive consensus conditions using the bounds of the eigenvalues of the Laplacian matrix are conservative. To tackle these problems, less conservative bounds of the eigenvalues of the Laplacian matrix are presented. Based on a general distributed observer-based approach, the necessary and sufficient conditions of the edge consensus are derived. And then, with the improved bounds of the Laplacian eigenvalues, less conservative sufficient conditions without using global topology information are given. By solving the algebraic Riccati inequalities, semi-definite programming algorithms are developed to obtain the solutions. Finally, simulation results are also given to illustrate the given results.

Robust Energy-Efficient Hybrid Beamforming Design for Massive MIMO LEO Satellite Communication Systems

In low earth orbit (LEO) satellite communication systems, the limited energy supply capacity and the difficulty in obtaining channel state information (CSI) are the practical challenges. Motivated by this, we focus on the design of robust hybrid beamforming to maximize the energy efficiency of LEO satellite communication systems. Assuming that the LEO satellite transmitter adopts the massive multi-input multi-output (MIMO) technology, considering the CSI errors caused by propagation delay and Doppler shift, under the constraints of transmit power and quality of service (QoS), a robust energy-efficient hybrid beamforming scheme is proposed. Since that there are no explicit expressions for the ergodic user rate and the ergodic signal-to-interference-plus-noise ratio, the approximate values with closed-form expressions are adopted. Then, we invoke the semidefinite programming (SDP) algorithm to transform the nonconvex quadratic constrained quadratic programming (QCQP) problem equivalently, and an inner and outer nested iterative algorithm combining quadratic transformation fractional programming (QTFP) and concave convex process (CCCP) is utilized to transfer a nonconvex problem into a convex problem. Meanwhile, we adopt a penalty function algorithm to solve the rank-one constraint in semidefinite programming algorithm. Finally, we invoke the normal form distance minimization (NFDM) algorithm and the alternating optimization (AltOpt) algorithm to jointly solve the digital beamforming matrix and analog beamforming matrix in hybrid beamformer. Numerical results validate that our proposed robust approach significantly outperforms the conventional one.

Joint Transceiver Design for Radar-Communication Spectral Sharing Systems by Minimizing Effective Interference Power

This brief investigates a spectrum sharing architecture for a colocated MIMO radar and a MIMO communication system. We develop an alternative formulation for joint resource allocation, aiming at minimizing the effective power of radar interference under the power and similarity constraints. The communication system mutual information (MI) is also simultaneously concerned in order to guarantee the communication performance. The degrees of freedom for system design are both the radar transceiver and the communication transmitter. For the joint designed original non-convex problem, an alternating iterative algorithm with multi-subproblems is derived based on the majorization-minimization algorithm, minimum variance distortionless response (MVDR) and semidefinite programming (SDP) algorithm. The convergence of the algorithm is guaranteed. Finally, we assess the effectiveness of the proposed approach via numerical results.

A New Projection Contraction Algorithm for Semidefinite Programming

This paper studies a projective contraction algorithm for solving semidefinite programming problems. The interior point algorithm is a classic algorithm for solving semidefinite programming problems. It can be divided into feasible and infeasible algorithms according to whether the initial point is feasible or not. The feasible interior point algorithm needs to use a self-dual strategy to find strictly feasible points. Although the infeasible interior point algorithm does not require the initial point to be strictly feasible, the theoretical analysis of the algorithm is very complicated. Compared with the interior point algorithm, the projection contraction algorithm has no requirement on the initial point, and the algorithm analysis is much simpler. The optimal condition of the semidefinite programming problem can be rewritten as a projection equation, which is then solved by the projection contraction algorithm. We propose a new projection contraction algorithm based on different descent directions to solve the semidefinite programming problem. Further, under the framework of the projection contraction algorithm, an improved algorithm is proposed, that is, changing the descending direction by solving the linear equation system, thus avoiding the high-dimensional inversion operation so that the algorithm can solve the large-scale semidefinite programming problem. Numerical experiments show that the algorithm is feasible and the improved algorithm is competitive with the interior point algorithm.

Quantum Annealing versus Digital Computing

Quantum annealing is getting increasing attention in combinatorial optimization. The quantum processing unit by D-Wave is constructed to approximately solve Ising models on so-called Chimera graphs. Ising models are equivalent to quadratic unconstrained binary optimization (QUBO) problems and maximum cut problems on the associated graphs. We have tailored branch-and-cut as well as semidefinite programming algorithms for solving Ising models for Chimera graphs to provable optimality and use the strength of these approaches for comparing our solution values to those obtained on the current quantum annealing machine, D-Wave 2000Q. This allows for the assessment of the quality of solutions produced by the D-Wave hardware. In addition, we also evaluate the performance of a heuristic by Selby. It has been a matter of discussion in the literature how well the D-Wave hardware performs at its native task, and our experiments shed some more light on this issue. In particular, we examine how reliably the D-Wave computer can deliver true optimum solutions and present some surprising results.