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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]

This paper establishes some Gronwall–Bellman–Pachpatte type integral inequalities that generalize some well-known earlier inequalities. It also shows that we can deal with such inequalities in various ways, resulting in different upper bounds. These techniques are also applicable to similar inequalities.

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.

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.

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 $G=K_{\\Delta ,\\Delta }$ .

Casualty rescue is always a critical issue in emergency departments when the buildings collapsing, bridges or roads damaged happen in the earthquakes. However, previous studies essentially considered such a problem by assuming fixed travel time for the ambulance. This paper aims to involve road failure during the earthquake by considering the stochastic durations for each road. The probability that the casualty can be successfully rescued to the casualty collection point (CCP) provides further medical treatment for the casualties. Such a probability is named network reliability which is defined as the probability that the number of casualties can be successfully transported to the CCP in a given rescue time. To evaluate the network reliability, the data transformation procedure is firstly developed to convert the road data into duration probability table, which addresses the travel time and corresponding probability of each road. Second, the multi-state rescue network is established, and an algorithm is constructed to obtain all upper bound vectors meeting the demand and time constraint. The network reliability can be computed by obtained upper bound vectors using the recursive sum disjoint product method. An example of a real earthquake disaster in Tainan City, Taiwan is adopted to demonstrate the practicality of the proposed algorithm. Finally, the experimental results with different number of ambulances and times can provide the commander decision recommendations for immediate emergency responses.

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.

We introduce a new unified extension of the integral form of Euler’s beta function with a MacDonald function in the integrand and establish functional upper bounds for it. We use this definition to extend as well the Gaussian and Kummer’s confluent hypergeometric functions, for which we provide bounding inequalities. Moreover, we use our extension of the beta function to define a new probability distribution, for which we establish raw moments and moment inequalities and, as by-products, Turán inequalities for the initially defined extended beta function.