Semidefinite Programming Techniques Research Articles

Genuine multipartite Bell inequality for device-independent conference key agreement

In this work, we present a new class of genuine multipartite Bell inequalities, that is particularly designed for multipartite device-independent (DI) quantum key distribution (QKD), also called DI conference key agreement. We prove the classical bounds of this inequality, discuss how to maximally violate it and show its usefulness by calculating achievable conference key rates via the violation of this Bell inequality. To this end, semidefinite programming techniques based on [Nat. Commun. 2, 238 (2011)] are employed and extended to the multipartite scenario. Our Bell inequality represents a nontrivial multipartite generalization of the Clauser-Horne-Shimony-Holt inequality and is motivated by the extension of the bipartite Bell state to the n-partite Greenberger-Horne-Zeilinger state. For DIQKD, we suggest an honest implementation for any number of parties and study the effect of noise on achievable asymptotic conference key rates.

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.

Lower Bound Limit Analysis Using Power Cone Programming for Solving Stability Problems in Rock Mechanics for Generalized Hoek–Brown Criterion

This study introduces a methodology to solve plane strain stability problems in rock mechanics, following the generalized Hoek and Brown yield criterion, by employing the lower bound finite elements limit analysis in conjunction with the power cone programming. The efficacy of the proposed approach has been demonstrated by solving three different types of stability problems: (1) finding the bearing capacity of strip footings on rock media, (2) assessing the stability of finite rock slopes, and (3) the stability analysis of unlined rectangular tunnels in rock mass. In all the cases, the results obtained from the analysis have been compared thoroughly with that computed using (1) nonlinear programming, and (2) semi-definite programming technique. The present approach has been found to be computationally very robust and it generates very accurate solutions.

Time-of-arrival–based localization algorithm in mixed line-of-sight/non-line-of-sight environments

A novel time-of-arrival–based localization algorithm in mixed line-of-sight/non-line-of-sight environments is proposed. First, an optimization problem of target localization in the known distribution of line-of-sight and non-line-of-sight is established, and mixed semi-definite and second-order cone programming techniques are used to transform the original problem into a convex optimization problem which can be solved efficiently. Second, a worst-case robust least squares criterion is used to form an optimization problem of target localization in unknown distribution of line-of-sight and non-line-of-sight, where all links are treated as non-line-of-sight links. This problem is also solved using the similar techniques used in the known distribution of line-of-sight and non-line-of-sight case. Finally, computer simulation results show that the proposed algorithms have better performance in both the known distribution and the unknown distribution of line-of-sight and non-line-of-sight environments.

Lower bound limit analysis of unsupported vertical circular excavations in rocks using Hoek‐Brown failure criterion

SummaryThe stability of vertical unsupported circular excavations in rock media, obeying generalized Hoek‐Brown yield criterion, has been investigated by using the lower bound finite elements limit analysis. An axisymmetric analysis, composed of a planar domain with a mesh of three‐noded triangular elements, has been carried out. The optimization problem is dealt with by using the semidefinite programming technique avoiding the need of either smoothing the yield surface or making any assumption associated with the circumferential stress (σθ). A detailed parametric study has been executed, and the effects of different input material parameters, namely, geological strength index (GSI), yield parameter (mi), and the disturbance factor (D) on the results have been studied. For different height to radius ratios of the excavation, the computed results are presented in the form of nondimensional stability numbers. Failure mechanisms have also been investigated for a few typical cases. The results from the analysis have been compared with that evaluated separately with the application of the software OptumG2.

Range-Doppler Sidelobe Suppression for Pulsed Radar Based on Golay Complementary Codes

To relieve the interference caused by range-Doppler sidelobes in pulsed radars, we propose a new method to construct Doppler resilient complementary waveforms based on Golay codes. We design both the transmit pulse train and the receive pulse weights, so that the similarity between the pulse weights and a given window function is maximized and the constraints on Doppler null points and energy are met. That is summarized as a two-way partitioning problem, and then solved by semidefinite programming and randomization techniques. The novel waveform thus obtained has its range sidelobe outright suppressed in multiple and flexibly-adjustable Doppler zones, and performs well in Doppler sidelobe suppression, Doppler resolution and SNR. It shows great promise in detecting slightly-moving weak targets with the existence of dense interference.

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.

Nonlinear Tracking Control with Reduced Complexity of Serial Robots: A Robust Fuzzy Descriptor Approach

This paper presents a nonlinear tracking control approach for a two-degree-of-freedom serial manipulator. The design goal is to achieve a guaranteed $$\mathcal {H}_{\infty }$$ tracking performance while keeping the designed controller as simple as possible for real-time implementation. To this end, the descriptor Takagi–Sugeno fuzzy modeling is used to describe the nonlinear dynamics of the robot. Then, based on Lyapunov stability theory, we propose conditions to design fuzzy controllers for trajectory tracking purposes. The control design procedure is reformulated as an optimization problem under linear matrix inequality constraints which can be effectively solved with semidefinite programming technique. Numerical experiments carried out with the Simscape Multibody™ library of MATLAB® clearly demonstrate the effectiveness of the proposed approach in terms of tracking control and numerical simplicity.

Parity oblivious d-level random access codes and class of noncontextuality inequalities

The quantum ontological feature of contextuality apart from being central to foundations of quantum theory forms the basis of quantum advantage in a multitude of information processing tasks. In particular, the contextuality of preparation procedures was shown to power a particular two-party information processing task “parity oblivious multiplexing” (Spekkens et al. Phys Rev Lett 102:010401 (2009). Specifically, it was shown that there exists a limit to how well any preparation noncontextual theory can perform in this task. This limit constitutes a noncontextuality inequality. Moreover, the authors demonstrated quantum violation of this inequality along with preparation contextuality associated with the ontic description underlying two-level completely mixed quantum state. In this work, we extend these arguments to apply to arbitrary dimensions by introducing a class of two-party information processing tasks, namely d-level parity oblivious random access codes. We analytically obtain classical (or equivalently preparation noncontextual) bounds on the success probability for these tasks for arbitrary d. For each value of d, this bound constitutes a unique noncontextuality inequality. Remarkably, these bounds are independent of the amount of communication. Furthermore, we find a classical protocol utilizing a d_mathrm{c}=d-dimensional classical message which saturates this bound. In order to establish nontriviality of these inequalities, we provide evidence of significant quantum violations. Specifically, by numerical techniques, we show that for d=3,ldots ,10, the noncontextuality bound is violated by quantum theory. (1) We provide explicit quantum protocols which violate the associated noncontextuality inequality for d=3,4,5 employing d_mathrm{q}=d-leveled quantum systems. (2) Using see-saw semi-definite programming (SDP) technique, we find evidence (lower bounds) of significant quantum violation of these inequalities for d=3,ldots ,10. (3) With the help of state-of-the-art (NPA-hierarchy like) SDP technique, we provide upper bounds (independent of the dimension of the involved quantum systems) on quantum violation for d=3,ldots ,10. The introduced class of information tasks, thus, provides for operational depiction of preparation contextuality of the ontic description underlying mixed higher dimensional quantum systems.

Modeling optimal long-term investment strategies of hybrid wind-thermal companies in restructured power market

In this paper, a novel framework for the estimation of optimal investment strategies for combined wind-thermal companies is proposed. The medium-term restructured power market was simulated by considering the stochastic and rational uncertainties, the wind uncertainty was evaluated based on a data mining technique, and the electricity demand and fuel price were simulated using the Monte Carlo method. The Cournot game concept was used to determine the Nash equilibrium for each state and stage of the stochastic dynamic programming (DP). Furthermore, the long-term stochastic uncertainties were modeled based on the Markov chain process. The long-term optimal investment strategies were then solved for combined wind-thermal investors based on the semi-definite programming (SDP) technique. Finally, the proposed framework was implemented in the hypothetical restructured power market using the IEEE reliability test system (RTS). The conducted case study confirmed that this framework provides robust decisions and precise information about the restructured power market for combined wind-thermal investors.

A convergent hierarchy of non-linear eigenproblems to compute the joint spectral radius of nonnegative matrices

We show that the joint spectral radius of a finite collection of nonnegative matrices can be bounded by the eigenvalue of a non-linear operator. This eigenvalue coincides with the ergodic constant of a risk-sensitive control problem, or of an entropy game, in which the state space consists of all switching sequences of a given length. We show that, by increasing this length, we arrive at a convergent approximation scheme to compute the joint spectral radius. The complexity of this method is exponential in the length of the switching sequences, but it is quite insensitive to the size of the matrices, allowing us to solve very large scale instances (several matrices in dimensions of order 1000 within a minute). An idea of this method is to replace a hierarchy of optimization problems, introduced by Ahmadi, Jungers, Parrilo and Roozbehani, by a hierarchy of nonlinear eigenproblems. To solve the latter eigenproblems, we introduce a projective version of Krasnoselskii-Mann iteration. This method is of independent interest as it applies more generally to the nonlinear eigenproblem for a monotone positively homogeneous map. Here, this method allows for scalability by avoiding the recourse to linear or semidefinite programming techniques.

A Low-Complexity Parallelizable Numerical Algorithm for Sparse Semidefinite Programming

In the past two decades, the semidefinite programming (SDP) technique has been proven to be extremely successful in the convexification of hard optimization problems appearing in graph theory, control theory, polynomial optimization theory, and many areas in engineering. In particular, major power optimization problems, such as optimal power flow, state estimation, and unit commitment, can be formulated or well approximated as SDPs. However, the inability to efficiently solve large-scale SDPs is an impediment to the deployment of such formulations in practice. Motivated by the significant role of SDPs in revolutionizing the decision-making process for real-world systems, this paper designs a low-complexity numerical algorithm for solving sparse SDPs, using the alternating direction method of multipliers and the notion of tree decomposition in graph theory. The iterations of the designed algorithm are highly parallelizable and enjoy closed-form solutions, whose most expensive computation amounts to eigenvalue decompositions over certain submatrices of the SDP matrix. The proposed algorithm is a general-purpose parallelizable SDP solver for sparse SDPs, and its performance is demonstrated on the SDP relaxation of the optimal power flow problem for real-world benchmark systems with more than 13 600 nodes.

Non-Exhaustive, Overlapping Clustering.

Traditional clustering algorithms, such as K-Means, output a clustering that is disjoint and exhaustive, i.e., every single data point is assigned to exactly one cluster. However, in many real-world datasets, clusters can overlap and there are often outliers that do not belong to any cluster. While this is a well-recognized problem, most existing algorithms address either overlap or outlier detection and do not tackle the problem in a unified way. In this paper, we propose an intuitive objective function, which we call the NEO-K-Means (Non-Exhaustive, Overlapping K-Means) objective, that captures the issues of overlap and non-exhaustiveness in a unified manner. Our objective function can be viewed as a reformulation of the traditional K-Means objective, with easy-to-understand parameters that capture the degrees of overlap and non-exhaustiveness. By considering an extension to weighted kernel K-Means, we show that we can also apply our NEO-K-Means idea to overlapping community detection, which is an important task in network analysis. To optimize the NEO-K-Means objective, we develop not only fast iterative algorithms but also more sophisticated algorithms using low-rank semidefinite programming techniques. Our experimental results show that the new objective and algorithms are effective in finding ground-truth clusterings that have varied overlap and non-exhaustiveness; for the case of graphs, we show that our method outperforms state-of-the-art overlapping community detection algorithms.

Graphical Model Selection for Gaussian Conditional Random Fields in the Presence of Latent Variables

ABSTRACT We consider the problem of learning a conditional Gaussian graphical model in the presence of latent variables. Building on recent advances in this field, we suggest a method that decomposes the parameters of a conditional Markov random field into the sum of a sparse and a low-rank matrix. We derive convergence bounds for this estimator and show that it is well-behaved in the high-dimensional regime as well as “sparsistent” (i.e., capable of recovering the graph structure). We then show how proximal gradient algorithms and semi-definite programming techniques can be employed to fit the model to thousands of variables. Through extensive simulations, we illustrate the conditions required for identifiability and show that there is a wide range of situations in which this model performs significantly better than its counterparts, for example, by accommodating more latent variables. Finally, the suggested method is applied to two datasets comprising individual level data on genetic variants and metabolites levels. We show our results replicate better than alternative approaches and show enriched biological signal. Supplementary materials for this article are available online.

Moment-SOS relaxation of the medium term hydrothermal dispatch problem

This paper analyzes the application of moment-SOS (sum of squares) relaxation to solve the medium term hydrothermal dispatch (MTHD) problem for predominantly hydro systems. The optimal dispatch is formulated as a two stage stochastic programming problem, which is nonconvex due to constraints that represent hydro power production, and may have multiple solutions. The size and conditioning of the problem pose challenges to the use of the semidefinite programming (SDP) techniques that are discussed in the text. The size of the problem restricts this study to first order moment relaxations. Therefore, to improve the quality of relaxed solutions, additional meaningful constraints are introduced in the SDP problems. Subsequently, the problem re-scalded to achieve numerical stability in SDP relaxation. Results obtained for different test systems, most of them being partitions of the Brazilian hydroelectric system, indicate that the procedures used to improve the quality of SDP solutions are effective.

Received signal strength–based target localization under spatially correlated shadowing via convex optimization relaxation

Received signal strength–based target localization methods normally employ radio propagation path loss model, in which the log-normal shadowing noise is generally assumed to follow a zero-mean Gaussian distribution and is uncorrelated. In this article, however, we represent the simplified additive noise by the spatially correlated log-normal shadowing noise. We propose a new convex localization estimator in wireless sensor networks by using received signal strength measurements under spatially correlated shadowing environment. First, we derive a new non-convex estimator based on weighted least squares criterion. Second, by using the equivalence of norm, the derived estimator can be reformulated as its equivalent form which has no logarithm in the objective function. Then, the new estimator is relaxed by applying efficient convex relaxation that is based on second-order cone programming and semi-definite programming technique. Finally, the convex optimization problem can be efficiently solved by a standard interior-point method, thus to obtain the globally optimal solution. Simulation results show that the proposed estimator solves the localization problem efficiently and is close to Cramer–Rao lower bound compared with the state-of-the-art approach under correlated shadowing environment.

Secure Full-Duplex Two-Way Relaying for SWIPT

This letter studies bi-directional secure information exchange in a simultaneous wireless information and power transfer system enabled by a full-duplex multiple-input multiple-output amplify-and-forward (AF) relay. The AF relay injects artificial noise (AN) in order to confuse the eavesdropper. Specifically, we assume a zeroforcing (ZF) solution constraint to eliminate the residual self-interference. As a consequence, we address the optimal joint design of the ZF matrix and the AN covariance matrix at the relay node as well as the transmit power at the sources. We propose an alternating algorithm utilizing semi-definite programming technique and 1-D searching to achieve the optimal solution. Simulation results are provided to demonstrate the effectiveness of the proposed algorithm.

Upper-Bound Finite-Element Limit Analysis of Axisymmetric Problems for Mohr-Coulomb Materials Using Semidefinite Programming

An upper-bound formulation for performing finite-element limit analysis by using semidefinite programming (SDP) for an axisymmetric stability problem involving the Mohr-Coulomb yield criterion has been presented. The SDP technique has an advantage in that it can deal with the yield criterion directly in its native form in terms of principal stresses and strains without any smoothing of the parent yield surface. The associated flow rule and plastic power dissipation are expressed entirely in terms of principal plastic strain rates. Nodal velocities and element plastic strain rates were framed as basic governing variables without involving stresses. The solution was obtained by using the SDP solver MOSEK in MATLAB. The problem of finding the bearing capacity of a circular footing was dealt with by choosing a planar domain that was discretized with (1)three-noded constant strain finite elements and (2)six-noded linear strain finite elements. The results were obtained with and without the provision of the velocity discontinuities, and were compared with those reported in literature. It was learned that quite accurate solutions can be obtained with the application of six-noded linear strain triangular elements and using velocity discontinuities along all the elements' interfaces.

Expensive Control of Long-Time Averages Using Sum of Squares and Its Application to A Laminar Wake Flow

This paper presents a nonlinear state-feedback control design approach for long-time average cost control, where the control effort is assumed to be expensive. The approach is based on sum-of-squares and semidefinite programming techniques. It is applicable to dynamical systems whose right-hand side is a polynomial function in the state variables and the controls. The key idea, first described but not implemented by Chernyshenko et al. , is that the difficult problem of optimizing a cost function involving long-time averages is replaced by an optimization of the upper bound of the same average. As such, a controller design requires the simultaneous optimization of both the control law and a tunable function, similar to a Lyapunov function. This paper introduces a method resolving the well-known inherent nonconvexity of this kind of optimization. The method is based on the formal assumption that the control is expensive, from which it follows that the optimal control is small. The resulting asymptotic optimization problems are convex. The derivation of all the polynomial coefficients in the controller is given in terms of the solvability conditions of state-dependent linear and bilinear inequalities. The proposed approach is applied to the problem of designing a full-information feedback controller that mitigates vortex shedding in the wake of a circular cylinder in the laminar regime via rotary oscillations. Control results on a reduced-order model of the actuated wake and in direct numerical simulation are reported.

Compressive Acquisition and Least-Squares Reconstruction of Correlated Signals

This letter presents a framework for the compressive acquisition of correlated signals. We propose an implementable sampling architecture for the acquisition of ensembles of correlated (lying in an a priori unknown subspace) signals at a sub-Nyquist rate. The sampling architecture acquires structured compressive samples of the signals after preprocessing them with easy-to-implement components. Quantitatively, we show that an ensemble of $M$ correlated signals each of which is bandlimited to $W/2$ and can be decomposed as the linear combination of $R$ underlying signals can be acquired at roughly $RW$ (assuming without loss of generality $M ) samples per second. In the case, when $M \gg R$ , this results in significant reduction in the sampling rate compared to $MW$ samples per second dictated by the Shannon–Nyquist sampling theorem. The signal reconstruction is achieved by solving only a least-squares program, which is in stark contrast to the prohibitively computationally expensive semidefinite programming techniques suggested in the earlier literature. We also provide rigorous analytical results to support our claims.