Variable fixing method by weighted average for the continuous quadratic knapsack problem

<p style='text-indent:20px;'>We analyze the method of solving the separable convex continuous quadratic knapsack problem by weighted average from the viewpoint of variable fixing. It is shown that this method, considered as a variant of the variable fixing algorithms, and Kiwiel's variable fixing method generate the same iterates. We further improve the algorithm based on the analysis regarding the semismooth Newton method. Computational results are given and comparisons are made among the state-of-the-art algorithms. Experiments show that our algorithm has significantly good performance; it behaves very much like an <inline-formula><tex-math id="M1">\begin{document}$ O(n) $\end{document}</tex-math></inline-formula> algorithm with a very small constant.</p>

Dual applications of Chebyshev polynomials method: Efficiently finding thousands of central eigenvalues for many-spin systems

Computation of a large group of interior eigenvalues at the middle spectrum is an important problem for quantum many-body systems, where the level statistics provides characteristic signatures of quantum chaos. We propose an exact numerical method, dual applications of Chebyshev polynomials (DACP), to simultaneously find thousands of central eigenvalues, where the level space decreases exponentially with the system size. To disentangle the near-degenerate problem, we employ twice the Chebyshev polynomials, to construct an exponential semicircle filter as a preconditioning step and to generate a large set of proper basis states in the desired subspace. Numerical calculations on Ising spin chain and spin glass shards confirm the correctness and efficiency of DACP. As numerical results demonstrate, DACP is 30 times faster than the state-of-the-art shift-invert method for the Ising spin chain while 8 times faster for the spin glass shards. In contrast to the shift-invert method, the computation time of DACP is only weakly influenced by the required number of eigenvalues, which renders it a powerful tool for large scale eigenvalues computations. Moreover, the consumed memory also remains a small constant (5.6 GB) for spin-1/2 systems consisting of up to 20 spins, making it desirable for parallel computing.

The Dantzig selector: recovery of signal via ℓ 1 − αℓ 2 minimization

In the paper, we proposed the Dantzig selector based on the ℓ 1 − αℓ 2 (0 < α ⩽ 1) minimization for the signal recovery. In the Dantzig selector, the constraint ‖ A ⊤ ( b − Ax )‖∞ ⩽ η for some small constant η > 0 means the columns of A has very weakly correlated with the error vector e = Ax − b . First, recovery guarantees based on the restricted isometry property are established for signals. Next, we propose the effective algorithm to solve the proposed Dantzig selector. Last, we illustrate the proposed model and algorithm by extensive numerical experiments for the recovery of signals in the cases of Gaussian, impulsive and uniform noises. And the performance of the proposed Dantzig selector is better than that of the existing methods.

Optimizing over the Closure of Rank Inequalities with a Small Right-Hand Side for the Maximum Stable Set Problem via Bilevel Programming

In the context of the maximum stable set problem, rank inequalities impose that the cardinality of any set of vertices contained in a stable set be, at most, as large as the stability number of the subgraph induced by such a set. Rank inequalities are very general, as they subsume many classical inequalities such as clique, hole, antihole, web, and antiweb inequalities. In spite of their generality, the exact separation of rank inequalities has never been addressed without the introduction of topological restrictions on the induced subgraph and the tightness of their closure has never been investigated systematically. In this work, we propose a methodology for optimizing over the closure of all rank inequalities with a right-hand side no larger than a small constant without imposing any restrictions on the topology of the induced subgraph. Our method relies on the exact separation of a relaxation of rank inequalities, which we call relaxed k-rank inequalities, whose closure is as tight. We investigate the corresponding separation problem, a bilevel programming problem asking for a subgraph of maximum weight with a bound on its stability number, whose study could be of independent interest. We first prove that the problem is [Formula: see text]-hard and provide some insights on its polyhedral structure. We then propose two exact methods for its solution: a branch-and-cut algorithm (which relies on a family of faced-defining inequalities which we introduce in this paper) and a purely combinatorial branch-and-bound algorithm. Our computational results show that the closure of rank inequalities with a right-hand side no larger than a small constant can yield a bound that is stronger, in some cases, than Lovász’s Theta function, and substantially stronger than bounds obtained with standard inequalities that are valid for the stable set problem, including odd-cycle inequalities and wheel inequalities. Summary of Contribution: This paper proposes two original methods for solving a challenging cut-separation problem (of bilevel type) for a large class of inequalities valid for one of the key operations research problems, namely, the max stable set problem. An extensive set of experimental results validates the proposed methods. All the source code and data sets are available online on GitHub.

Exact and Approximate Algorithms for Computing Betweenness Centrality in Directed Graphs

Graphs (networks) are an important tool to model data in different domains. Realworld graphs are usually directed, where the edges have a direction and they are not symmetric. Betweenness centrality is an important index widely used to analyze networks. In this paper, first given a directed network G and a vertex r ∈ V (G), we propose an exact algorithm to compute betweenness score of r. Our algorithm pre-computes a set ℛ𝒱(r), which is used to prune a huge amount of computations that do not contribute to the betweenness score of r. Time complexity of our algorithm depends on |ℛ𝒱(r)| and it is respectively Θ(|ℛ𝒱(r)| · |E(G)|) and Θ(|ℛ𝒱(r)| · |E(G)| + |ℛ𝒱(r)| · |V(G)| log |V(G)|) for unweighted graphs and weighted graphs with positive weights. |ℛ𝒱(r)| is bounded from above by |V(G)| – 1 and in most cases, it is a small constant. Then, for the cases where ℛ𝒱(r) is large, we present a simple randomized algorithm that samples from ℛ𝒱(r) and performs computations for only the sampled elements. We show that this algorithm provides an (ɛ, δ)-approximation to the betweenness score of r. Finally, we perform extensive experiments over several real-world datasets from different domains for several randomly chosen vertices as well as for the vertices with the highest betweenness scores. Our experiments reveal that for estimating betweenness score of a single vertex, our algorithm significantly outperforms the most efficient existing randomized algorithms, in terms of both running time and accuracy. Our experiments also reveal that our algorithm improves the existing algorithms when someone is interested in computing betweenness values of the vertices in a set whose cardinality is very small.

An Improved PMU Data Manipulation Attack Model

The importance of Phasor Manipulation Unit (PMU) in the smart grid makes it a target for attackers who can create PMU Data Manipulation Attacks (PDMA) by adding a small constant to change the magnitude and angle of the voltage and current captured by the PMU. To prevent the attack result from being detected by PDMA detection based on the properties of equivalent impedance, this paper proposes a collaborative step attack. In this attack, the equivalent impedance’s value on the end of the transmission line is equal whether before or after been attack, which is taken as the constraint condition. The objective function of it is to minimize the number of the elements which is not 0 in attack vector but this number is not 0. Turn a vector construction problem into an optimization problem by building objective functions and constraints and then we use the Alternating Direction Method of Multipliers (ADMM) and Convex Relaxation (CR) to solve. The experiment verifies the feasibility of using the CR-ADMM algorithm to construct attack vectors from two aspects of attack vector construction time and vector sparsity. Further, it uses the constructed attack vectors to carry out attacks on PMU. The experimental results show that the measurement value of PMU will change after the attack, but the equivalent impedance value at both ends of the transmission line remains the same. The attack vector successfully bypasses the PDMA detection method based on the property of equivalent impedance and the attack model constructed based on this method was more covert than the original model.

On the smallest singular value of symmetric random matrices

We show that for an $n\times n$ random symmetric matrix $A_n$ , whose entries on and above the diagonal are independent copies of a sub-Gaussian random variable $\xi$ with mean 0 and variance 1, \begin{equation*}\mathbb{P}[s_n(A_n) \le \epsilon/\sqrt{n}] \le O_{\xi}(\epsilon^{1/8} + \exp(\!-\Omega_{\xi}(n^{1/2}))) \quad \text{for all } \epsilon \ge 0.\end{equation*} This improves a result of Vershynin, who obtained such a bound with $n^{1/2}$ replaced by $n^{c}$ for a small constant c, and $1/8$ replaced by $(1/8) - \eta$ (with implicit constants also depending on $\eta > 0$ ). Furthermore, when $\xi$ is a Rademacher random variable, we prove that \begin{equation*}\mathbb{P}[s_n(A_n) \le \epsilon/\sqrt{n}] \le O(\epsilon^{1/8} + \exp(\!-\Omega((\!\log{n})^{1/4}n^{1/2}))) \quad \text{for all } \epsilon \ge 0.\end{equation*} The special case $\epsilon = 0$ improves a recent result of Campos, Mattos, Morris, and Morrison, which showed that $\mathbb{P}[s_n(A_n) = 0] \le O(\exp(\!-\Omega(n^{1/2}))).$ Notably, in a departure from the previous two best bounds on the probability of singularity of symmetric matrices, which had relied on somewhat specialized and involved combinatorial techniques, our methods fall squarely within the broad geometric framework pioneered by Rudelson and Vershynin, and suggest the possibility of a principled geometric approach to the study of the singular spectrum of symmetric random matrices. The main innovations in our work are new notions of arithmetic structure – the Median Regularized Least Common Denominator (MRLCD) and the Median Threshold, which are natural refinements of the Regularized Least Common Denominator (RLCD)introduced by Vershynin, and should be more generally useful in contexts where one needs to combine anticoncentration information of different parts of a vector.

Allergic reaction and metal hypersensitivity after shoulder joint replacement

Purpose Metal ion release may cause local and systemic effects and induce hypersensitivity reactions. The aim of our study is first to determine if implant-related hypersensitivity correlates to patient symptoms or not; second, to assess the rate of hypersensitivity and allergies in shoulder arthroplasty. Methods Forty patients with shoulder replacements performed between 2015 and 2017 were studied with minimum 2-year follow-up; no patient had prior metal implants. Each patient underwent radiographic and clinical evaluation using the Constant-Murley Score (CMS), 22 metal and cement haptens patch testing, serum and urine tests to evaluate 12 metals concentration, and a personal occupational medicine interview. Results At follow-up (average 45 ± 10.7 months), the mean CMS was 76 ± 15.9; no clinical complications or radiographic signs of loosening were detected; two nickel sulfate (5%), 1 benzoyl peroxide (2.5%) and 1 potassium dichromate (2.5%) positive findings were found, but all these patients were asymptomatic. There was an increase in serum aluminum, urinary aluminum and urinary chromium levels of 1.74, 3.40 and 1.83 times the baseline, respectively. No significant difference in metal ion concentrations were found when patients were stratified according to gender, date of surgery, type of surgery, and type of implant. Conclusions Shoulder arthroplasty is a source of metal ion release and might act as a sensitizing exposure. However, patch test positivity does not seem to correlate to hypersensitivity cutaneous manifestations or poor clinical results. Laboratory data showed small constant ion release over time, regardless of gender, type of shoulder replacement and implant used. Levels of evidence Level II.