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Many papers in the field of integer linear programming (ILP, for short) are devoted to problems of the type $\max\{c^\top x \colon A x = b,\, x \in \mathbb{Z}^n_{\geq 0}\}$, where all the entries of $A,b,c$ are integer, parameterized by the number of rows of $A$ and $\|A\|_{\max}$. This class of problems is known under the name of ILP problems in the standard form, adding the word "bounded" if $x \leq u$, for some integer vector $u$. Recently, many new sparsity, proximity, and complexity results were obtained for bounded and unbounded ILP problems in the standard form. In this paper, we consider ILP problems in the canonical form $$\max\{c^\top x \colon b_l \leq A x \leq b_r,\, x \in \mathbb{Z}^n\},$$ where $b_l$ and $b_r$ are integer vectors. We assume that the integer matrix $A$ has the rank $n$, $(n + m)$ rows, $n$ columns, and parameterize the problem by $m$ and $\Delta(A)$, where $\Delta(A)$ is the maximum of $n \times n$ sub-determinants of $A$, taken in the absolute value. We show that any ILP problem in the standard form can be polynomially reduced to some ILP problem in the canonical form, preserving $m$ and $\Delta(A)$, but the reverse reduction is not always possible. More precisely, we define the class of generalized ILP problems in the standard form, which includes an additional group constraint, and prove the equivalence to ILP problems in the canonical form. We generalize known sparsity, proximity, and complexity bounds for ILP problems in the canonical form. Additionally, sometimes, we strengthen previously known results for ILP problems in the canonical form, and, sometimes, we give shorter proofs. Finally, we consider the special cases of $m \in \{0,1\}$. By this way, we give specialised sparsity, proximity, and complexity bounds for the problems on simplices, Knapsack problems and Subset-Sum problems.

This paper develops an asymptotic theory for estimating the time-varying characteristics of locally stationary functional time series. We introduce a kernel-based method to estimate the time-varying covariance operator and the time-varying mean function of a locally stationary functional time series. Subsequently, we derive the convergence rate of the kernel estimator of the covariance operator and associated eigenvalue and eigenfunctions. We also establish a central limit theorem for the kernel-based locally weighted sample mean. As applications of our results, we discuss the prediction of locally stationary functional time series and methods for testing the equality of time-varying mean functions in two functional samples.

We propose in this paper New Q-Newton's method. The update rule is very simple conceptually, for example $x_{n+1}=x_n-w_n$ where $w_n=pr_{A_n,+}(v_n)-pr_{A_n,-}(v_n)$, with $A_n=\nabla ^2f(x_n)+\delta _n||\nabla f(x_n)||^2.Id$ and $v_n=A_n^{-1}.\nabla f(x_n)$. Here $\delta _n$ is an appropriate real number so that $A_n$ is invertible, and $pr_{A_n,\pm}$ are projections to the vector subspaces generated by eigenvectors of positive (correspondingly negative) eigenvalues of $A_n$. The main result of this paper roughly says that if $f$ is $C^3$ (can be unbounded from below) and a sequence $\{x_n\}$, constructed by the New Q-Newton's method from a random initial point $x_0$, {\bf converges}, then the limit point is a critical point and is not a saddle point, and the convergence rate is the same as that of Newton's method. The first author has recently been successful incorporating Backtracking line search to New Q-Newton's method, thus resolving the convergence guarantee issue observed for some (non-smooth) cost functions. An application to quickly finding zeros of a univariate meromorphic function will be discussed. Various experiments are performed, against well known algorithms such as BFGS and Adaptive Cubic Regularization are presented.

The O'Sullivan penalized splines approach is a popular frequentist approach for nonparametric regression. Thereby, the unknown regression function is expanded in a rich spline basis and a roughness penalty based on the integrated squared $q$th derivative is used for regularization. While the asymptotic properties of O'Sullivan penalized splines in a frequentist setting have been investigated extensively, the theoretical understanding of the Bayesian counterpart has been missing so far. In this paper, we close this gap and study the asymptotics of the Bayesian counterpart of the frequentist O-splines approach. We derive sufficient conditions for the entire posterior distribution to concentrate around the true regression function at near optimal rate. Our results show that posterior concentration at near optimal rate can be achieved with a faster rate for the number of spline knots than the slow regression spline rate that is commonly being used. Furthermore, posterior concentration at near optimal rate can be achieved with several different hyperpriors on the smoothing variance such as a Gamma and a Weibull hyperprior.

Regula Falsi, or the method of false position, is a numerical method for finding an approximate solution to f(x) = 0 on a finite interval [a, b], where f is a real-valued continuous function on [a, b] and satisfies f(a)f(b) < 0. Previous studies proved the convergence of this method under certain assumptions about the function f, such as both the first and second derivatives of f do not change the sign on the interval [a, b]. In this paper, we remove those assumptions and prove the convergence of the method for all continuous functions.

The tube method or the volume-of-tube method approximates the tail probability of the maximum of a smooth Gaussian random field with zero mean and unit variance. This method evaluates the volume of a spherical tube about the index set, and then transforms it to the tail probability. In this study, we generalize the tube method to a case in which the variance is not constant. We provide the volume formula for a spherical tube with a non-constant radius in terms of curvature tensors, and the tail probability formula of the maximum of a Gaussian random field with inhomogeneous variance, as well as its Laplace approximation. In particular, the critical radius of the tube is generalized for evaluation of the asymptotic approximation error. As an example, we discuss the approximation of the largest eigenvalue distribution of the Wishart matrix with a non-identity matrix parameter. The Bonferroni method is the tube method when the index set is a finite set. We provide the formula for the asymptotic approximation error for the Bonferroni method when the variance is not constant.

In this work, we study the $k$-means cost function. Given a dataset $X \subseteq \mathbb{R}^d$ and an integer $k$, the goal of the Euclidean $k$-means problem is to find a set of $k$ centers $C \subseteq \mathbb{R}^d$ such that $\Phi(C, X) \equiv \sum_{x \in X} \min_{c \in C} ||x - c||^2$ is minimized. Let $\Delta(X,k) \equiv \min_{C \subseteq \mathbb{R}^d} \Phi(C, X)$ denote the cost of the optimal $k$-means solution. For any dataset $X$, $\Delta(X,k)$ decreases as $k$ increases. In this work, we try to understand this behaviour more precisely. For any dataset $X \subseteq \mathbb{R}^d$, integer $k \geq 1$, and a precision parameter $\varepsilon > 0$, let $L(X, k, \varepsilon)$ denote the smallest integer such that $\Delta(X, L(X, k, \varepsilon)) \leq \varepsilon \cdot \Delta(X,k)$. We show upper and lower bounds on this quantity. Our techniques generalize for the metric $k$-median problem in arbitrary metric spaces and we give bounds in terms of the doubling dimension of the metric. Finally, we observe that for any dataset $X$, we can compute a set $S$ of size $O \left(L(X, k, \varepsilon/c) \right)$ using $D^2$-sampling such that $\Phi(S,X) \leq \varepsilon \cdot \Delta(X,k)$ for some fixed constant $c$. We also discuss some applications of our bounds.

In this work, we develop a high-order pressure-robust method for the rotation form of the stationary incompressible Navier-Stokes equations. The original idea is to change the velocity test functions in the discretization of trilinear and right hand side terms by using an H(div)-conforming velocity reconstruction operator. In order to match the rotation form and error analysis, a novel skew-symmetric discrete trilinear form containing the reconstruction operator is proposed, in which not only the velocity test function is changed. The corresponding well-posed discrete weak formulation stems straight from the classical inf-sup stable mixed conforming high-order finite elements, and it is proven to achieve the pressure-independent velocity errors. Optimal convergence rates of H1, L2-error for the velocity and L2-error for the Bernoulli pressure are completely established. Adequate numerical experiments are presented to demonstrate the theoretical results and the remarkable performance of the proposed method.

The problem of constructing a simultaneous confidence band for the mean function of a locally stationary functional time series $ \{ X_{i,n} (t) \}_{i = 1, \ldots, n}$ is challenging as these bands can not be built on classical limit theory. On the one hand, for a fixed argument $t$ of the functions $ X_{i,n}$, the maximum absolute deviation between an estimate and the time dependent regression function exhibits (after appropriate standardization) an extreme value behaviour with a Gumbel distribution in the limit. On the other hand, for stationary functional data, simultaneous confidence bands can be built on classical central theorems for Banach space valued random variables and the limit distribution of the maximum absolute deviation is given by the sup-norm of a Gaussian process. As both limit theorems have different rates of convergence, they are not compatible, and a weak convergence result, which could be used for the construction of a confidence surface in the locally stationary case, does not exist. In this paper we propose new bootstrap methodology to construct a simultaneous confidence band for the mean function of a locally stationary functional time series, which is motivated by a Gaussian approximation for the maximum absolute deviation. We prove the validity of our approach by asymptotic theory, demonstrate good finite sample properties by means of a simulation study and illustrate its applicability analyzing a data example.

In this note we prove that the version of Newton algorithm with line search we used in [2] converges quadratically.