Standard Semidefinite Programming Research Articles

On semidefinite descriptions for convex hulls of quadratic programs

Quadratically constrained quadratic programs (QCQPs) are a highly expressive class of nonconvex optimization problems. While QCQPs are NP-hard in general, they admit a natural convex relaxation via the standard semidefinite program (SDP) relaxation. In this paper we study when the convex hull of the epigraph of a QCQP coincides with the projected epigraph of the SDP relaxation. We present a sufficient condition for convex hull exactness and show that this condition is further necessary under an additional geometric assumption. The sufficient condition is based on geometric properties of Γ, the cone of convex Lagrange multipliers, and its relatives Γ1 and Γ∘.

Integration of NOMA With Reflecting Intelligent Surfaces: A Multi-Cell Optimization With SIC Decoding Errors

Reflecting intelligent surfaces (RIS) has gained significant attention due to its high energy and spectral efficiency in next-generation wireless networks. By using low-cost passive reflecting elements, RIS can smartly reconfigure the signal propagation to extend the wireless communication coverage. On the other hand, non-orthogonal multiple access (NOMA) has been proven as a key air interface technique for supporting massive connections over limited resources. Utilizing the superposition coding and successive interference cancellation (SIC) techniques, NOMA can multiplex multiple users over the same spectrum and time resources by allocating different power levels. This paper proposes a new optimization scheme in a multi-cell RIS-NOMA network to enhance the spectral efficiency under SIC decoding errors. In particular, the power budget of the base station and the transmit power of NOMA users while the passive beamforming of RIS is simultaneously optimized in each cell. Due to objective function and quality of service constraints, the joint problem is formulated as non-convex, which is very complex and challenging to obtain the optimal global solution. To reduce the complexity and make the problem tractable, we first decouple the original problem into two sub-problems for power allocation and passive beamforming. Then, the efficient solution of each sub-problem is obtained in two-steps. In the first-step of For power allocation sub-problem, we transform it to a convex problem by the inner approximation method and then solve it through a standard convex optimization solver in the second-step. Accordingly, in the first-step of passive beamforming, it is transformed into a standard semi-definite programming problem by successive convex approximation and different of convex programming methods. Then, penalty based method is used to achieve a Rank-1 solution for passive beamforming in second-step. Numerical results demonstrate the benefits of the proposed optimization scheme in the multi-cell RIS-NOMA network.

Energy-Efficient Beamforming and Resource Optimization for AmBSC-Assisted Cooperative NOMA IoT Networks

In this manuscript, we present an energy-efficient alternating optimization framework based on the multi-antenna ambient backscatter communication (AmBSC) assisted cooperative non-orthogonal multiple access (NOMA) for next-generation (NG) internet-of-things (IoT) enabled communication networks. Specifically, the energy-efficiency maximization is achieved for the considered AmBSC-enabled multi-cluster cooperative IoT NOMA system by optimizing the active-beamforming vector and power-allocation coefficients (PAC) of IoT NOMA users at the transmitter, as well as passive-beamforming vector at the multi-antenna assisted backscatter node. Usually, increasing the number of IoT NOMA users in each cluster results in inter-cluster interference (ICI) (among different clusters) and intra-cluster interference (among IoT NOMA users). To combat the impact of ICI, we exploit a zero-forcing (ZF) based active-beamforming, as well as an efficient clustering technique at the source node. Further, the effect of intra-cluster interference is mitigated by exploiting an efficient power-allocation policy that determines the PAC of IoT NOMA users under the quality-of-service (QoS), cooperation, SIC decoding, and power-budget constraints. Moreover, the considered non-convex passive-beamforming problem is transformed into a standard semi-definite programming (SDP) problem by exploiting the successive-convex approximation (SCA) approximation, as well as the difference of convex (DC) programming, where Rank-1 solution of passive-beamforming is obtained based on the penalty-based method. Furthermore, the numerical analysis of simulation results demonstrates that the proposed energy-efficiency maximization algorithm exhibits an efficient performance by achieving convergence within only a few iterations.

Generalized dissipativity dynamic output feedback control for coronary artery system

This work addresses the problem of synchronizing pathological changes in a patient with coronary artery obstruction to the corresponding nominal coronary artery system (CAS) model. The CAS model is characterized by nonlinear terms, so the synchronization problem is transformed into an equivalent time-varying delay T-S fuzzy framework using the sector of nonlinearity approach. New bilinear matrix inequalities (BMIs) conditions for robust stability analysis and dynamic output feedback gain synthesis are derived based on a less conservative condition for stability assessment of T-S fuzzy systems. A simple change of coordinate is used to solve the cross terms of the bilinearities, allowing the problem to be formulated in terms of linear matrix inequalities (LMIs) and solved using standard semi-definite programming. The effectiveness of the controller design approach is demonstrated through extensive simulations under diverse performance criterion that the controller can achieve.

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.

Robust Weighted-Average Continuous-Time Consensus With Communication Time Delay.

Achieving consensus behavior robust to time delay in multiagent systems has attracted much attention. This work is concerned with optimizing the convergence rate of the consensus algorithm in such systems with time delays. Previous approaches optimize either the robustness to time delay or the convergence rate separately, while imposing a limit on the other. Eigenratio optimization is another method, which does not necessarily result in a unique set of weights. Here, the problem is treated in its general form as a multiobjective optimization problem. It is shown that the corresponding Pareto frontier depends solely on the optimal condition number of the Laplacian, and it includes the optimal answer of previously adopted approaches as special cases. A notion of optimal consensusability is then defined, which allows a particular point on the Pareto Frontier with special properties to be identified. The resulting optimization problem is shown to be convex, as is solved by reformulating it as a standard semidefinite programming problem. The optimal weights for individual topologies, clique lifted graphs, and different types of subgraphs are provided, where for the latter, the optimal weights have shown to be independent of the rest of topology. Through numerical simulations, the tradeoff between robustness and convergence rate is demonstrated.

Optimal curing resource allocation for epidemic spreading processes

The networked Susceptible–Infected–Susceptible (SIS) model is one of the well-developed models for epidemic spreading processes in networked systems. For a network that has an unstable healthy state, and follows the networked SIS model, a problem of interest is to stabilize the healthy state by increasing the curing rates of individuals, while minimizing the total cost associated with the additional curing rates over the network. Here this minimum control budget problem is reformulated as a standard semidefinite programming problem and an iterative algorithm is developed for determining the optimal additional curing rates for each agent in the network. We address arbitrary undirected topologies, with both symmetric and asymmetric infection rates. For a number of topologies, solutions have been provided for the case of uniform curing and infection rates for all possible ranges of the effective infection rate. Moreover, we have investigated the case of discrete-time SIS models and have shown that the algorithm developed based on the continuous-time model can be extended to this case. Based on the daily COVID-19 infections over the USA, the curing and infection rates have been estimated for a simulation scenario. Using these simulations, the algorithm is shown to have lower complexity than the well-known SDPT3 tool, and to have orders of magnitude lower computational and memory requirements.

Finite‐frequency output feedback MPC for Markov jump systems

AbstractThis paper investigates a general design framework of dynamic output feedback model predictive control (DOFMPC) for Markov jump systems within both time and frequency domain. Such a design with guaranteed H∞ and quadratic performance is formulated by a standard semi‐definite programming (SDP), and it is achieved by employing a special congruence transformation. The SDP condition greatly reduces the computational effort by eliminating bilinear matrix inequalities or equation constraints reported in existing references. Specifically, the H∞ norm of the transfer function is optimized within three types of frequency ranges on account of generalized Kalman–Yakubovic–Popov (GKYP) lemma. The quadratic index is optimized online via SDP. Finally, we verify the feasibility and effectiveness of the proposed method from both theoretical and practical point of view.

Joint Strategy of Power and Bandwidth Allocation for Multiple Maneuvering Target Tracking in Cognitive MIMO Radar With Collocated Antennas

In this article, we propose a joint strategy of power and bandwidth allocation (JSPBA) for multiple maneuvering target tracking (MMTT) in the multi-input multi-output (MIMO) radar with collocated antennas. The basis of our strategy is to optimally allocate the transmitted resources of power and effective bandwidth by the prior information in the closed-loop system of cognitive tracking. On account of the predicted conditional Cramér-Rao lower bound (PC-CRLB) offering a more accurate and time-sensitive lower bound than the standard posterior CRLB (PCRLB), the PC-CRLB of the range, Doppler frequency, and direction-of-arrival (DOA) is derived, normalized and adopted as the optimization criterion. Moreover, in order to solve the nonconvex problem, the initial nonconvex problem is converted into a series of convex problems, which are further formulated as the standard semi-definite programming (SDP) problems and then be solved, by introducing the convex relaxation technique and the two-step solution technique. Simulations confirm the superiority of the proposed JSPBA algorithm, in terms of the overall tracking accuracy in the MMTT scenario.

Exact SDP relaxations for quadratic programs with bipartite graph structures

For nonconvex quadratically constrained quadratic programs (QCQPs), we first show that, under certain feasibility conditions, the standard semidefinite programming (SDP) relaxation is exact for QCQPs with bipartite graph structures. The exact optimal solutions are obtained by examining the dual SDP relaxation and the rank of the optimal solution of this dual SDP relaxation under strong duality. Our results generalize the previous results on QCQPs with sign-definite bipartite graph structures, QCQPs with forest structures, and QCQPs with nonpositive off-diagonal data elements. Second, we propose a conversion method from QCQPs with no particular structure to the ones with bipartite graph structures. As a result, we demonstrate that a wider class of QCQPs can be exactly solved by the SDP relaxation. Numerical instances are presented for illustration.

Efficient Scheduling in Space–Air–Ground-Integrated Localization Networks

High accuracy and seamless position information formulates the basis of many modern wireless applications, such as the Internet of Things (IoT) and intelligent transportation systems (ITSs). In this article, aiming at the ground user equipment (UE) those in the “blind spots,” where only limited navigation signals are provided, the temporary aerial-aided “anchors” such as the unmanned aerial vehicles (UAVs) are introduced as alternating solutions. We first give the general fundamental limits of the three-dimensional space–air–ground-integrated localization networks (SAGILNs) using both time and angle measurements. Unlike most existing investigations, we treat aerial nodes as “agents” whose positions are not known beforehand. We then try to formulate an efficient scheduling strategy, where proper network <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">behaviors</i> , including the resource optimization and UAV deployment, are provided. We find that the proposed scheduling problems could be formulated as standard semidefinite programming (SDP) problems and solved by off-the-shelf solvers. Numerical results are provided to validate our analysis. The proposed methods and analyses provide meaningful insights for performance benchmarks for the implementation of SAGILN.

State estimation of the time–space propagation of COVID-19 using a distributed parameter observer based on a SEIR-type model

The real-time prediction and estimation of the spread of diseases, such as COVID-19 is of paramount importance as evidenced by the recent pandemic. This work is concerned with the distributed parameter estimation of the time–space propagation of such diseases using a diffusion–reaction epidemiological model of the susceptible–exposed–infected–recovered (SEIR) type. State estimation is based on continuous measurements of the number of infections and deaths per unit of time and of the host spatial domain. The observer design method is based on positive definite matrices to parameterize a class of Lyapunov functionals, in order to stabilize the estimation error dynamics. Thus, the stability conditions can be expressed as a set of matrix inequality constraints which can be solved numerically using sum of squares (SOS) and standard semi-definite programming (SDP) tools. The observer performance is analyzed based on a simplified case study corresponding to the situation in France in March 2020 and shows promising results.

Observer design for multivariable transport-reaction systems based on spatially distributed measurements

This paper is concerned with the design of observers for a class of one-dimensional multi-state transport-reaction systems considering distributed in-domain measurements over the spatial domain. A design based on the Lyapunov method is proposed for the stabilization of the estimation error dynamics. The approach uses positive definite matrices to parameterize a class of Lyapunov functionals that are positive in the Lebesgue space of integrable square functions. Thus, the stability conditions can be expressed as a set of LMI constraints which can be solved numerically using sum of squares (SOS) and standard semi-definite programming (SDP) tools. In order to evaluate the proposed methodology, a state observer is designed to estimate the variables of a nonisothermal tubular reactor model. Numerical simulations are presented to demonstrate the potentials of the proposed observer.

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.

Exact SDP relaxations of quadratically constrained quadratic programs with forest structures

We study the exactness of the semidefinite programming (SDP) relaxation of quadratically constrained quadratic programs (QCQPs). With the aggregate sparsity matrix from the data matrices of a QCQP with n variables, the rank and positive semidefiniteness of the matrix are examined. We prove that if the rank of the aggregate sparsity matrix is not less than \(n-1\) and the matrix remains positive semidefinite after replacing some off-diagonal nonzero elements with zeros, then the standard SDP relaxation provides an exact optimal solution for the QCQP under feasibility assumptions. In particular, we demonstrate that QCQPs with forest-structured aggregate sparsity matrix, such as the tridiagonal or arrow-type matrix, satisfy the exactness condition on the rank. The exactness is attained by considering the feasibility of the dual SDP relaxation, the strong duality of SDPs, and a sequence of QCQPs with perturbed objective functions, under the assumption that the feasible region is compact. We generalize our result for a wider class of QCQPs by applying simultaneous tridiagonalization on the data matrices. Moreover, simultaneous tridiagonalization is applied to a matrix pencil so that QCQPs with two constraints can be solved exactly by the SDP relaxation.

On the tightness of SDP relaxations of QCQPs

Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study conditions under which the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show that the SDP relaxation is tight whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. We present similar sufficient conditions under which the projected epigraph of the SDP gives the convex hull of the epigraph in the original QCQP. Our results also imply new sufficient conditions for the tightness (as well as convex hull exactness) of a second order cone program relaxation of simultaneously diagonalizable QCQPs.

An Enhanced MAX-SINR Strategy With Interference Leakage Power Constraint in Multiuser Multiantenna SWIPT Systems

Interference alignment (IA) is one of the most promising techniques to eliminate interference in multiuser multiantenna systems. Simultaneous wireless information and power transfer (SWIPT) technology is applied to harvest energy from radio frequency signals in green communications, while interference signal is assumed to be utilized as energy supply source in SWIPT. In this paper, an enhanced maximum signal to interference plus noise ratio (MAX-SINR) strategy with interference leakage power constraint is proposed, which realizes a tradeoff between information transmission and interference leakage in multiuser multiantenna SWIPT systems. Optimization problem is formulated and a three-step optimization solution is proposed. Finally, the problem can be transformed to a standard semi-definite programming (SDP) to obtain optimal power splitting factor and system sum-rate (spectral efficiency). Simulation results are presented to show that system sum-rate performance is enhanced with the increasing of receiving antenna numbers. The tradeoff between sum-rate and the harvested energy can be realized via the proposed IA strategy.

Counting frequent patterns in large labeled graphs: a hypergraph-based approach.

In recent years, the popularity of graph databases has grown rapidly. This paper focuses on single-graph as an effective model to represent information and its related graph mining techniques. In frequent pattern mining in a single-graph setting, there are two main problems: support measure and search scheme. In this paper, we propose a novel framework for designing support measures that brings together existing minimum-image-based and overlap-graph-based support measures. Our framework is built on the concept of occurrence/instance hypergraphs. Based on such, we are able to design a series of new support measures: minimum instance (MI) measure, and minimum vertex cover (MVC) measure, that combine the advantages of existing measures. More importantly, we show that the existing minimum-image-based support measure is an upper bound of the MI measure, which is also linear-time computable and results in counts that are close to number of instances of a pattern. We show that not only most major existing support measures and new measures proposed in this paper can be mapped into the new framework, but also they occupy different locations of the frequency spectrum. By taking advantage of the new framework, we discover that MVC can be approximated to a constant factor (in terms of number of pattern nodes) in polynomial time. In contrast to common belief, we demonstrate that the state-of-the-art overlap-graph-based maximum independent set (MIS) measure also has constant approximation algorithms. We further show that using standard linear programming and semidefinite programming techniques, polynomial-time relaxations for both MVC and MIS measures can be developed and their counts stand between MVC and MIS. In addition, we point out that MVC, MIS, and their relaxations are bounded within constant factor. In summary, all major support measures are unified in the new hypergraph-based framework which helps reveal their bounding relations and hardness properties.

Implications of ANEC for SCFTs in four dimensions

We explore consequences of the Averaged Null Energy Condition (ANEC) for scaling dimensions ∆ of operators in four-dimensional mathcal{N} = 1 superconformal field theories. We show that in many cases the ANEC bounds are stronger than the corresponding unitarity bounds on ∆. We analyze in detail chiral operators in the left(frac{1}{2}j,0right) Lorentz representation and prove that the ANEC implies the lower bound Delta ge frac{3}{2}j , which is stronger than the corresponding unitarity bound for j > 1. We also derive ANEC bounds on left(frac{1}{2}j,0right) operators obeying other possible shortening conditions, as well as general left(frac{1}{2}j,0right) operators not obeying any shortening condition. In both cases we find that they are typically stronger than the corresponding unitarity bounds. Finally, we elucidate operator-dimension constraints that follow from our mathcal{N} = 1 results for multiplets of mathcal{N} = 2, 4 superconformal theories in four dimensions. By recasting the ANEC as a convex optimization problem and using standard semidefinite programming methods we are able to improve on previous analyses in the literature pertaining to the nonsupersymmetric case.

Variational reduced density matrix method in the doubly-occupied configuration interaction space using four-particle N-representability conditions: Application to the XXZ model of quantum magnetism.

This work deals with the variational determination of the two-particle reduced density matrix (2-RDM) and the energy corresponding to the ground state of N-particle systems within the doubly occupied configuration interaction (DOCI) space. Here, we impose for the first time up to four-particle N-representability constraint conditions in the variational determination of the 2-RDM matrix elements using the standard semidefinite programming algorithms. The energies and 2-RDMs obtained from this treatment and the corresponding computational costs are compared with those arisen from previously reported less restrictive variational methods [D. R. Alcoba et al., J. Chem. Phys. 149, 194105 (2018)] as well as with the exact DOCI values. We apply the different approximations to the one-dimensional XXZ model of quantum magnetism, which has a rich phase diagram with one critical phase and constitutes a stringent test for the method. The numerical results show the usefulness of our treatment to achieve a high degree of accuracy.