Semidefinite Programming Approach Research Articles

Low-Rank Semidefinite Programming for the MAX2SAT Problem

This paper proposes a new algorithm for solving MAX2SAT problems based on combining search methods with semidefinite programming approaches. Semidefinite programming techniques are well-known as a theoretical tool for approximating maximum satisfiability problems, but their application has traditionally been very limited by their speed and randomized nature. Our approach overcomes this difficult by using a recent approach to low-rank semidefinite programming, specialized to work in an incremental fashion suitable for use in an exact search algorithm. The method can be used both within complete or incomplete solver, and we demonstrate on a variety of problems from recent competitions. Our experiments show that the approach is faster (sometimes by orders of magnitude) than existing state-of-the-art complete and incomplete solvers, representing a substantial advance in search methods specialized for MAX2SAT problems.

Bus injection relaxation based OPF in multi-phase neutral equipped distribution networks embedding wye- and delta-connected loads and generators

A novel Optimal Power Flow (OPF) model, based on Semi-Definite Programming approach, for multi-phase active Distribution Networks (DNs) equipped with neutral conductor(s) and both wye- and delta-connected loads/Distributed Generators (DGs) is proposed in this paper. The coupled power injection across phase-neutral and phase-phase conductors is decoupled by modelling of a shunt load/DG through a correction current injection approach and, consequently, explicit power injections are obtained for each conductor of a node. Furthermore, the complete ZIP load representation is considered in the OPF model and two approximations, based on the first-order Taylor series and approximation of the first-order Taylor series, are introduced for the modelling of constant current component of a ZIP load in terms of an optimization variable. Simulations are carried out on several medium and low voltage active DNs under various parameters of the ZIP models for the minimization of either slack-bus power or network losses objective functions. It is successfully demonstrated that the proposed relaxation recovers the global optimal solution of the original OPF problem for both load types (wye and delta) under various combinations of ZIP parameters in the case of lightly or moderately unbalanced DNs, whereas in the case of highly unbalanced DNs, a rank-2 solution is obtained when constant current component dominates the other load components. Furthermore, the impact of unrealistic assumption of Kron reduction approach on the quality of the proposed relaxation for a single or multiple point grounded DNs is also discussed. From a computational point of view, the proposed methodology can be realized on a real-time basis for small DNs. However, for medium and large DNs, the incorporation of additional sparsity exploitation or distributed algorithm techniques are required for its real-time implementation.

A semidefinite programming approach for polynomial switched optimal control problems

SummaryThis paper considers solving optimal control of switched systems with polynomial data globally, where the number of switches and the switching signal are not preassigned. The problem is transformed into an embedded polynomial form in which a continuous variable controls the switching policy. Then, using occupation measures, the embedded optimal control problem is formulated as an infinite‐dimensional linear programming (LP) over the space of measures. A polynomial sum‐of‐squares strengthening corresponding to conic dual of the measure LP problem provides an approximating optimal feedback control for both classical control and the switching signal. Furthermore, the optimal switching signal is calculated without mode projection. Finally, three simulation experiments are included to confirm the theoretical results in the paper.

Sparse polynomial interpolation: sparse recovery, super-resolution, or Prony?

We show that the sparse polynomial interpolation problem reduces to a discrete super-resolution problem on the $n$-dimensional torus. Therefore the semidefinite programming approach initiated by Cand\`es \& Fernandez-Granda \cite{candes_towards_2014} in the univariate case can be applied. We extend their result to the multivariate case, i.e., we show that exact recovery is guaranteed provided that a geometric spacing condition on the supports holds and the number of evaluations are sufficiently many (but not many). It also turns out that the sparse recovery LP-formulation of $\ell_1$-norm minimization is also guaranteed to provide exact recovery {\it provided that} the evaluations are made in a certain manner and even though the Restricted Isometry Property for exact recovery is not satisfied. (A naive sparse recovery LP-approach does not offer such a guarantee.) Finally we also describe the algebraic Prony method for sparse interpolation, which also recovers the exact decomposition but from less point evaluations and with no geometric spacing condition. We provide two sets of numerical experiments, one in which the super-resolution technique and Prony's method seem to cope equally well with noise, and another in which the super-resolution technique seems to cope with noise better than Prony's method, at the cost of an extra computational burden (i.e. a semidefinite optimization).

A Semidefinite Programming Approach for Harmonic Balance Method

The harmonic balance method is broadly employed for analyzing and predicting the periodic steady-state solution. Most of the traditional methods in the literature do not guarantee global optimality. Due to its nonconvexity, it is a complex task to find a global solution to the harmonic balance problem in the frequency domain. A convex relaxation in the form of semidefinite programming has attracted attention since its introduction because it yields a global solution in most cases. This paper introduces a novel optimization-based approach to predict periodic solution by determining the Fourier series coefficients with high accuracy. Unlike the other commonly used methods, the proposed approach is completely independent of initial conditions. In our proposed method, the nonlinear constraints composed of time-dependent trigonometric functions are converted into nonlinear algebraic polynomial equations. Then, nonlinear unknowns are convexified through moment-sum of squares approach. However, computing a global solution costs a higher runtime. Our approach is validated through small examples which contain only polynomial nonlinearities. In all cases, the Mosek solver shows a better performance in comparison with SDPT3 and SeDuMi solvers. The proposed method shows high computational cost as a result of an increase in the positive semidefinite matrix size, which can depend on the number of harmonics and the degree of nonlinearity.

A Statistical Model for Motifs Detection

We consider a statistical model for the problem of finding subgraphs with specified topology in an otherwise random graph. This task plays an important role in the analysis of social and biological networks. In these types of networks, small subgraphs with a specific structure have important functional roles, and they are referred to as motifs. Within this model, one or multiple copies of a subgraph is added (planted) in an Erdős–Renyi random graph with $n$ vertices and edge probability $q_{0}$ . We ask whether the resulting graph can be distinguished reliably from a pure Erdős–Renyi random graph, and we present two types of result. First we investigate the question from a purely statistical perspective, and ask whether there is any test that can distinguish between the two graph models. We provide necessary and sufficient conditions that are essentially tight for small enough subgraphs. Next we study two polynomial-time algorithms for solving the same problem: a spectral algorithm and a semidefinite programming (SDP) relaxation. For the spectral algorithm, we establish sufficient conditions under which it distinguishes the two graph models with high probability. Under the same conditions the spectral algorithm indeed identifies the hidden subgraph. The spectral algorithm is substantially sub-optimal with respect to the optimal test. We show that a similar gap is present for the more sophisticated SDP approach.

Multistage network data envelopment analysis: Semidefinite programming approach

Additive efficiency aggregation is one of the important techniques measuring the relative efficiency of decision-making units under network data envelopment analysis (DEA). However, the modelling of additive network DEA is limited to parametric methods to approximate optimal solutions in previous literature. Under multistage network structure, if some outputs leave the system from a given stage while others become inputs to the next stage and some new inputs enter at any stage, the additive network models become extremely nonlinear and are impossible to be solved by linear program. The current paper proposes to solve general additive two-stage models by using semidefinite programming, which is known as effective as linear program. We then extend the methodology to general multistage network structures (including serial processes, parallel processes, multistage processes with non-immediate successor flows, and multistage processes with feedbacks). A numerical data set and the case of regional R&D processes in China are revisited by using the newly developed approach.

IoT Communications Network for Wireless Power Transfer System State Estimation and Stabilization

Wireless energy harvesting has emerged as an appealing solution to prolong the lifespan of energy-constrained wireless networks and been regarded as a main functional unit for almost all perpetual communications. This paper proposes a channel coding-based Internet of Things communication network, and state estimation as well as stabilization algorithms for wireless power transfer systems. After modeling the energy harvesting module like wireless energy transfer systems as a state-space framework, Web-enabled smart sensors are deployed to obtain measurements. The sensing information is transmitted to the nearby base station for collection and digital communication under the condition of packet losses. A recursive systematic convolutional code is used to add redundancy into the quantize system states. After digital modulation, the signal is transmitted to a control center for system state estimation and stabilization. The optimal gain of the state estimation scheme is determined after minimizing the mean squared errors. Afterward, a stabilization algorithm is designed considering packet losses in the communication networks using the semidefinite programming approach. Furthermore, the convergence of the developed estimation algorithm is proved. Numerical simulation results demonstrate, considering packet loss, sensor faults, disturbances, and delay into account, that the proposed algorithms can estimate and stabilize the system states within a short period of time.

A Fast Matrix Majorization-Projection Method for Penalized Stress Minimization With Box Constraints

Kruskal's stress minimization, though nonconvex and nonsmooth, has been a major computational model for dissimilarity data in multidimensional scaling. Semidefinite programming (SDP) relaxation (by dropping the rank constraint) would lead to a high number of SDP cone constraints. This has rendered the SDP approach computationally challenging even for problems of small size. In this paper, we reformulate the stress optimization as an Euclidean distance matrix (EDM) optimization with box constraints. A key element in our approach is the conditional positive-semidefinite cone with rank cut. Although nonconvex, this geometric object allows a fast computation of the projection onto it, and it naturally leads to a majorization-minimization algorithm with the minimization step having a closed-form solution. Moreover, we prove that our EDM optimization follows a continuously differentiable path, which greatly facilitated the analysis of the convergence to a stationary point. The superior performance of the proposed algorithm is demonstrated against some of the state-of-the-art solvers in the field of sensor network localization and molecular conformation.

Consensus-Based Smart Grid State Estimation Algorithm

The distribution power subsystems are usually interconnected to each other, so the design of the interconnected optimal filtering algorithm for distributed state estimation is a challenging task. Driven by this motivation, this paper proposes a novel consensus filter based dynamic state estimation algorithm with its convergence analysis for modern power systems. The novelty of the scheme is that the algorithm is designed based on the mean squared error and semidefinite programming approaches. Specifically, the optimal local gain is computed after minimizing the mean squared error between the true and estimated states. The consensus gain is determined by a convex optimization process with a given suboptimal local gain. Furthermore, the convergence of the proposed scheme is analyzed after stacking all the estimation error dynamics. The Laplacian operator is used to represent the interconnected filter structure as a compact error dynamic for deriving the convergence condition of the algorithm. The developed approach is verified by using the renewable microgrid. It shows that the distributed scheme being explored is effective as it takes only 0.00004 seconds to properly estimate the system states and does not need to transmit the remote sensing signals to the central estimator.

Robust Shortest Path Problem With Distributional Uncertainty

Routing service considering uncertainty is at the core of intelligent transportation systems and has attracted increasing attention. Existing stochastic shortest path models require the exact probability distributions of travel times and usually assume that they are independent. However, the distributions are often unavailable or inaccurate due to insufficient data, and correlation of travel times over different links has been observed. This paper presents a robust shortest path (RSP) model that only requires partial distribution information of travel times, including the support set, mean, variance, and correlation matrix. We introduce a concept of robust mean-excess travel time to hedge against the risk from both the uncertainty of the random travel times and the uncertainty in their distributions. To solve the RSP problem, an equivalent dual formulation is derived and used to design tight lower and upper bound approximation methods, which adopt the scenario approach and semi-definite programming approach, respectively. To solve large problems, we further propose an efficient primal approximation method, which only needs to solve two deterministic shortest path problems and a mean-standard deviation shortest path problem, and analyze its approximation performance. Experiments validate the tightness of the proposed bounds and demonstrate the impact of uncertainty on the relative benefit and cost of robust paths.

Incorporating Convexity in Bond Portfolio Immunization Using Multifactor Model: A Semidefinite Programming Approach

Bond portfolio immunization is a classical issue in finance. Since Macaulay gave the concept of duration in 1938, many scholars proposed different kinds of duration immunization models. In the literature of bond portfolio immunization using multifactor model, to the best of our knowledge, researchers only use the first-order immunization, which is usually called as duration immunization, and no one has considered second-order effects in immunization, which is well known as “convexity” in the case of single-factor model. In this paper, we introduce the second-order information associated with multifactor model into bond portfolio immunization and reformulate the corresponding problems as tractable semidefinite programs. Both simulation analysis and empirical study show that the second-order immunization strategies exhibit more accurate approximation to the value change of bonds and thus result in better immunization performance.

On semidefinite relaxations for the block model

The stochastic block model (SBM) is a popular tool for community detection in networks, but fitting it by maximum likelihood (MLE) involves a computationally infeasible optimization problem. We propose a new semidefinite programming (SDP) solution to the problem of fitting the SBM, derived as a relaxation of the MLE. We put ours and previously proposed SDPs in a unified framework, as relaxations of the MLE over various subclasses of the SBM, which also reveals a connection to the well-known problem of sparse PCA. Our main relaxation, which we call SDP-1, is tighter than other recently proposed SDP relaxations, and thus previously established theoretical guarantees carry over. However, we show that SDP-1 exactly recovers true communities over a wider class of SBMs than those covered by current results. In particular, the assumption of strong assortativity of the SBM, implicit in consistency conditions for previously proposed SDPs, can be relaxed to weak assortativity for our approach, thus significantly broadening the class of SBMs covered by the consistency results. We also show that strong assortativity is indeed a necessary condition for exact recovery for previously proposed SDP approaches and not an artifact of the proofs. Our analysis of SDPs is based on primal-dual witness constructions, which provides some insight into the nature of the solutions of various SDPs. In particular, we show how to combine features from SDP-1 and already available SDPs to achieve the most flexibility in terms of both assortativity and block-size constraints, as our relaxation has the tendency to produce communities of similar sizes. This tendency makes it the ideal tool for fitting network histograms, a method gaining popularity in the graphon estimation literature, as we illustrate on an example of a social networks of dolphins. We also provide empirical evidence that SDPs outperform spectral methods for fitting SBMs with a large number of blocks.

An efficient second-order cone programming approach for optimal selection in tree breeding

It is common in forest tree breeding that selection of populations must consider conservation of genetic diversity, while at the same time attempting to maximize response to selection. To optimize selection in these situations, the constraint on genetic diversity can be mathematically described with the numerator relationship matrix as a quadratic constraint. Pong-Wong and Woolliams formulated the optimal selection problem using semidefinite programming (SDP). Their SDP approach gave an accurate optimal value, but required rather long computation time. In this paper, we propose an second-order cone programming (SOCP) approach to reduce the heavy computation cost. First, we demonstrate that a simple SOCP formulation achieves the same numerical solution as the SDP approach. A simple SOCP formulation is, however, not much more efficient compared to the SDP approach, so we focused on the sparsity structure of the numerator relationship matrix, and we develop a more efficient SOCP formulation using Henderson’s algorithm. Numerical results show that the proposed formulation, which we call a compact SOCP, greatly reduced computation time. In a case study, an optimal selection problem that demanded 39,200 s under the SDP approach was solved in less than 2 s by the compact SOCP formulation. The proposed approach is now available as a part of the software package OPSEL.

LQG Control With Minimum Directed Information: Semidefinite Programming Approach

We consider a discrete-time Linear-Quadratic-Gaussian (LQG) control problem in which Massey's directed information from the observed output of the plant to the control input is minimized while required control performance is attainable. This problem arises in several different contexts, including joint encoder and controller design for data-rate minimization in networked control systems. We show that the optimal control law is a Linear-Gaussian randomized policy. We also identify the state space realization of the optimal policy, which can be synthesized by an efficient algorithm based on semidefinite programming. Our structural result indicates that the filter-controller separation principle from the LQG control theory, and the sensor-filter separation principle from the zero-delay rate-distortion theory for Gauss-Markov sources hold simultaneously in the considered problem. A connection to the data-rate theorem for mean-square stability by Nair and Evans is also established.

A Semidefinite Programming Approach for Doppler Frequency Shift Based Stationary Target Localization

A Semidefinite Programming Approach for Doppler Frequency Shift Based Stationary Target Localization

Chordal Relaxation Based ACOPF for Unbalanced Distribution Systems With DERs and Voltage Regulation Devices

In emerging distribution systems with a proliferation of distributed energy resources (DERs) and flexible demand assets, operation characters of the unbalance network and voltage regulation devices need to be accurately addressed for ensuring the secure and economic operation. This paper focuses on the modeling and solution approach of AC optimal power flow (ACOPF) for unbalanced distribution systems with DERs and voltage regulation transformers (VRT). The ACOPF problem is formulated as a chordal relaxation based semidefinite programming (SDP) model, and a tighter convexification model of VRTs is proposed for mitigating solution inexactness. Analytical conditions are presented and proved to determine whether global optimal solution to the original ACOPF problem can be retrieved from solutions of the chordal relaxation based SDP model. Numerical studies on modified IEEE 34-bus and 8500-node systems show that the proposed approach presents a better computational performance as compared to rank relaxation based SDP approaches and general nonlinear solvers.

Hierarchical network systems: An application to high-technology industry in China

In real world situations, there is a hierarchical structure exists in a specific organization and each component has its network process. However, such hierarchical network system has not been well studied in previous literature, and misleading results often are produced. The current paper discusses a data envelopment analysis (DEA) modelling technique for a network structure where a hierarchical system consists of components having two-stage series processes. An additive network DEA is proposed to evaluate the performance of this type of network structure. The overall and divisional efficiencies of the system and each component can be derived, and the relationship between system efficiency, divisional efficiency and the ones of components is discussed. The newly developed additive network DEA is nonlinear and cannot be converted into a linear program. A semidefinite programming (SDP) approach is developed for effectively solving this model and the global solution can be guaranteed. Another linear multiplicative network DEA also developed for this hierarchical system. The two newly developed models are illustrated with a case of the performance evaluation of high-technology industry in China.

Semidefinite programming approach for the quadratic assignment problem with a sparse graph

The matching problem between two adjacency matrices can be formulated as the NP-hard quadratic assignment problem (QAP). Previous work on semidefinite programming (SDP) relaxations to the QAP have produced solutions that are often tight in practice, but such SDPs typically scale badly, involving matrix variables of dimension $$n^2$$ where n is the number of nodes. To achieve a speed up, we propose a further relaxation of the SDP involving a number of positive semidefinite matrices of dimension $$\mathcal {O}(n)$$ no greater than the number of edges in one of the graphs. The relaxation can be further strengthened by considering cliques in the graph, instead of edges. The dual problem of this novel relaxation has a natural three-block structure that can be solved via a convergent Alternating Direction Method of Multipliers in a distributed manner, where the most expensive step per iteration is computing the eigendecomposition of matrices of dimension $$\mathcal {O}(n)$$ . The new SDP relaxation produces strong bounds on quadratic assignment problems where one of the graphs is sparse with reduced computational complexity and running times, and can be used in the context of nuclear magnetic resonance spectroscopy to tackle the assignment problem.

A semidefinite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test

SummaryFinding the maximum eigenvalue of a symmetric tensor is an important topic in tensor computation and numerical multilinear algebra. In this paper, we introduce a new class of structured tensors called W‐tensors, which not only extends the well‐studied nonnegative tensors by allowing negative entries but also covers several important tensors arising naturally from spectral hypergraph theory. We then show that finding the maximum H‐eigenvalue of an even‐order symmetric W‐tensor is equivalent to solving a structured semidefinite program and hence can be validated in polynomial time. This yields a highly efficient semidefinite program algorithm for computing the maximum H‐eigenvalue of W‐tensors and is based on a new structured sums‐of‐squares decomposition result for a nonnegative polynomial induced by W‐tensors. Numerical experiments illustrate that the proposed algorithm can successfully find the maximum H‐eigenvalue of W‐tensors with dimension up to 10,000, subject to machine precision. As applications, we provide a polynomial time algorithm for computing the maximum H‐eigenvalues of large‐size Laplacian tensors of hyperstars and hypertrees, where the algorithm can be up to 13 times faster than the state‐of‐the‐art numerical method introduced by Ng, Qi, and Zhou in 2009. Finally, we also show that the proposed algorithm can be used to test the copositivity of a multivariate form associated with symmetric extended Z‐tensors, whose order may be even or odd.