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The block coordinate descent (BCD) method is widely used for minimizing a continuous function $f$ of several block variables. At each iteration of this method, a single block of variables is optimized, while the remaining variables are held fixed. To ensure the convergence of the BCD method, the subproblem of each block variable needs to be solved to its unique global optimal. Unfortunately, this requirement is often too restrictive for many practical scenarios. In this paper, we study an alternative inexact BCD approach which updates the variable blocks by successively minimizing a sequence of approximations of $f$ which are either locally tight upper bounds of $f$ or strictly convex local approximations of $f$. The main contributions of this work include the characterizations of the convergence conditions for a fairly wide class of such methods, especially for the cases where the objective functions are either nondifferentiable or nonconvex. Our results unify and extend the existing convergence results for many classical algorithms such as the BCD method, the difference of convex functions (DC) method, the expectation maximization (EM) algorithm, as well as the block forward-backward splitting algorithm, all of which are popular for large scale optimization problems involving big data. [PUBLICATION ABSTRACT]

In the paper, the peak solutions of a class equation is studied, the peak solutions are the maximal and minimal solutions. There is an iterative algorithm given for the solutions of the class equation. First, the existence of the peak solutions of the class equations is obtained. Second, when the peak solutions exist, an iterative algorithm is established to converge to the peak solutions of the class equation. By an upper bound and a lower bound of the solutions of the equation solution as the initial matrix, the iterative algorithm of the paper converges to the peak solutions of the class equation. The convergence problem of the algorithm is proved by the mathematical induction in the paper. The above results are verified by the examples.

In this article, we prove that the Riemann hypothesis implies a conjecture of Chandee on shifted moments of the Riemann zeta function. The proof is based on ideas of Harper concerning sharp upper bounds for the $2k$ th moments of the Riemann zeta function on the critical line.

Identifying all frequent high utility occupancy itemsets (FHUOIs) in a quantitative transaction dataset is a new trend in data mining. By combining both factors of frequency and utility occupancy, these patterns are more suitable for several applications in the real world. These patterns not only reflect the interests of most users but also contribute a high proportion of the utility in supporting transactions. Nonetheless, the set of all discovered FHUOIs may be very large, especially for large and dense datasets or for low values of predefined minimum thresholds. For this reason, it is often quite challenging for users to analyze and use the obtained patterns. To address this issue, this paper proposes two novel algorithms named MaxCloFHUOIM and CloFHUOIM to extract compact representations of FHUOIs. The former is designed to simultaneously mine two concise representations of FHUOIs that consist of all closed FHUOIs and all maximal FHUOIs, whereas the latter only discovers the closed FHUOIs, which provide a lossless summary of all FHUOIs. The proposed algorithms rely on a novel weak upper bound on utility occupancy, to reduce the search space by quickly pruning itemsets with low utility occupancy. Especially, the algorithms integrate two new efficient strategies to prune non-closed FHUOI candidate branches early in the prefix search tree. Results from an in-depth experimental evaluation conducted on several benchmark real-life and synthetic quantitative datasets demonstrate that MaxCloFHUOIM and CloFHUOIM have excellent performance in terms of runtime, memory usage, and scalability. In particular, the proposed algorithms are up to two orders of magnitude faster than a baseline algorithm.

Let G be a graph with no isolated vertex. A semitotal forcing set of G is a (zero) forcing set S such that every vertex in S is within distance 2 of another vertex of S. The semitotal forcing number $F_{t2}(G)$ is the minimum cardinality of a semitotal forcing set in G. In this paper, we prove that it is NP-complete to determine the semitotal forcing number of a graph. We also prove that if $G\\neq K_n$ is a connected graph of order $n\\geq 4$ with maximum degree $\\Delta \\geq 2$ , then $F_{t2}(G)\\leq (\\Delta-1)n/\\Delta$ , with equality if and only if either $G=C_{4}$ or $G=P_{4}$ or $G=K_{\\Delta ,\\Delta }$ .

This paper proposes a novel adaptive observer technique for estimating the state and disturbance of uncertain nonlinear systems. To remove the knowledge of the upper bounds of the disturbance and its derivative, a leakage-type (LT) algorithm is introduced to approximate the variations of their bounds. A state observer is first provided based on a conventional Walcott–Zak observer structure, and then, a disturbance observer is proposed by introducing an auxiliary dynamics. Due to the features of the LT adaptive law, the estimation error of the system state or the disturbance is bounded in a small neighborhood around zero in finite time. In addition, since the switching gain is automatically adapted to the disturbance change, the chattering in the estimation signal is effectively suppressed that is useful for the estimation precision in a practical system. Another important advantage of the proposed method lies in its simple structure compared to the existing finite-time observers. Lyapunov analysis demonstrates that for both types of observers, the estimation error is achieved to be globally uniformly ultimately bounded. To demonstrate the proposed method, simulation examples are separately carried out on a vehicle system and a linear motor system.

The assessment of uncertainty prediction has become a necessity for most modeling studies within the hydrology community. This paper addresses uncertainty analysis on a novel hybrid double feedforward neural network (HDFNN) model for generating the sediment load prediction interval (PI). By using the Lower Upper Bound Estimation (LUBE) method, the lower and upper bounds are directly generated as outputs of neural network based models. Coverage Width-based Criterion (CWC) is employed as an objective function for searching high quality PIs. The LUBE-based model is then applied to estimate sediment loads of Muddy Creek in Montana of USA. Results demonstrate the suitability of HDFNN-LUBE model in producing PI in both 90% and 95% confidence levels (CL). It is capable of generating appropriate lower bounds of PIs with narrow intervals. Partitioning analysis reveals consistently excellent performances of HDFNN model in constructing PI in terms of low, medium and high loads. These results therefore verify the reliability and potentiality of the HDFNN model for sediment load estimation with uncertainty. LUBE shows its efficiency in uncertainty prediction as well, which could be used to quantify total uncertainty of data-driven models.

Sylvester equation is widely used to study the stability of a nonlinear system in the control field. In this paper, a finite-time Zhang neural network (FTZNN) is proposed and applied to online solution of time-varying Sylvester equation. Differing from the conventional accelerating method, the design of the proposed FTZNN model is based on a new evolution formula, which is presented and studied to accelerate the convergence speed of a recurrent neural network. Compared with the original Zhang neural network (ZNN) for time-varying Sylvester equation, the FTZNN model can converge to the theoretical time-varying solution within finite time, instead of converging exponentially with time. Besides, we can obtain the upper bound of the finite convergence time for the FTZNN model in theory. Simulation results show that the proposed FTZNN model achieves the better performance as compared with the original ZNN model for solving online time-varying Sylvester equation.

High utility sequence mining is a popular data mining task, which aims at finding sequences having a high utility (importance) in a quantitative sequence database. Though it has several applications, state-of-the-art algorithms have one or more of the following limitations: (1) they rely on a utility function that tends to be biased toward finding long patterns, (2) some algorithms do take pattern length into account using an average-utility function but they adopt an optimistic perspective that can be risky or misleading for some applications, (3) they do not let the user specify additional constraints on patterns to be found. To address these three limitations, this paper defines a novel task of mining frequent high minimum average-utility sequences (FHAUS) with constraints in a quantitative sequence database. This task has the following benefits. First, it uses the average-utility au function based on the minimum utility, which takes the length of a pattern into account to calculate its utility. This helps finding short patterns missed by traditional algorithms and it is based on more safe pessimistic utility calculations. Second, the user can specify a set of monotonic and anti-monotonic constraints C on patterns to filter irrelevant patterns and improve the performance of the mining process. To efficiently find all FHAUSs with constraints, this paper first proposes some novel upper bounds (UBs) and weak upper bounds (WUBs) on the average-utility, which satisfy downward-closure (DC) properties or DC-like properties. Then, to effectively reduce the search space, the paper designs novel width pruning, depth pruning, reducing, and tightening strategies based on the proposed bounds. These proposed novel theoretical results are integrated into an algorithm named C-FHAUSPM (Constrained Frequent High minimum Average-Utility Sequential Pattern Mining) for efficiently discovering all FHAUSs with constraints. Results from extensive experiments on both real-life and synthetic quantitative sequence databases show that C-FHAUSPM is highly efficient in terms of runtime and memory usage.

The plate curvature model plays an important role in the prediction and control of the double layered clad plate. However, there is currently no such a fast, precise and stable plate curvature model, and the actual shear stress on the cross section is uneven distributed actually which is conflicted with the uniform assumption of the traditional model. Based on the non-uniform distribution, this paper innovatively presents a theoretical plate curvature model using the flow function method and the upper bound method together without the complex stress analysis. The inlet boundary conditions of the model are more accurately modified and optimized as nonlinear functions. The velocity field and strain rate field were built depending on the flow function field. The plastic deformation, shear and friction power were developed by the five-node Gauss–Legendre quadrature method. The post-rolling strain model was obtained by integrating the positive strain rate and shear strain rate with time. The curvature model was constructed by calculating the curvature due to linear and shear strain differences. To demonstrate the validity of the theoretical model, the experiments and the simulation were conducted. The results showed that the relative deviation of the numerical values v a and v b is less than 10.0%, accounting for 95.1% and 95.4%, compared with theoretical values. The shear power difference between layer a and b is the main reason for bending. The bending phenomenon appeared at the outlet section of the plate, and the velocity difference provided the shear stress required for bending. The increase of the diameter ratios leads to the increase of the shear strain difference and shear power, and it leads to an increase in the total curvature. To certify the precision of the curvature model, the deviation of the theoretical peak curvature values is 12.05%, and the average deviation is 5.92%, compared with the simulation and experiments. The model can quickly grasp the affecting law of key rolling process parameters on curvature, which will provide theoretical and technical basis for rolling process.