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We discuss interior point methods for large-scale linear programming, with an emphasis on methods that are useful for problems arising in telecommunications. We give the basic framework of a primal-dual interior point method, and consider the numerical issues involved in calculating the search direction in each iteration, including the use of factorization methods and/or preconditioned conjugate gradient methods. We also look at interior point column generation methods which can be used for very large scale linear programs or for problems where the data is generated only as needed.

This paper presents a load flow solution for radial and weakly meshed distribution system. The problem is formulated as an optimization problem based on minimization of L1 norm of a vector consisting of power mismatch at every node in a distribution system. The optimization problem is solved using interior point (IP) method. The advantage of interior point method is that it converges in polynomial time, and can handle the numerical ill conditioning. A primal-dual interior point (PDIP) method is demonstrated in this paper and performance of the method is evaluated on standard radial and weakly meshed test systems.

The conventional wisdom states that the interior problem (reconstruction of an interior region from projection data along lines only through that region) is NOT uniquely solvable. While it remains correct, our recent theoretical and numerical results demonstrated that this interior problem CAN be solved in a theoretically exact and numerically stable fashion if a sub-region within the interior region is known. In contrast to the well-established lambda tomography, the studies on this type of exact interior reconstruction are referred to as "interior tomography". In this paper, we will overview the development of interior tomography, involving theory, algorithms and applications. The essence of interior tomography is to find the unique solution from highly truncated projection data via analytic continuation. Such an extension can be done either in the filtered backprojection or backprojection filtration formats. The key issue for the exact interior reconstruction is how to invert the truncated Hilbert transform. We have developed a projection onto convex set (POCS) algorithm and a singular value decomposition (SVD) method and produced excellent results in numerical simulations and practical applications.

Design of an interior point method for linear programming is discussed, and results of a simulation study reported. Emphasis is put on guessing the optimal vertex at as early a stage as possible.

A procedure is given which relates various linear boundary value problems. It is shown that a relations exists between certain ‘exterior’ and ‘interior’ problems. For computational purposes an ‘interior’ problem is more desirable. The method is restricted by the requirement that the Neumann function, or its equivalent, be known for the problem being considered.

According to the limitation of principal components analysis in dealing with the nonlinear data, connecting with the linear programming techniques for multidimensional analysis of preference, this paper presents the kernel principal components analysis-linear programming techniques for multidimensional analysis of preference evaluation model. Kernel function maps linear inseparable input data into a high dimensional linear separable feature space via a nonlinear mapping technique. Then it carries on the linear principal components analysis in the high dimensional feature space. In addition, the weight of each index can be obtained in this model, thus it makes up another shortage of principal components analysis. In the wastewater evaluation, the indices are numerous and the degree of correlation is not high, therefore, this model is more appropriate. Finally, this paper applies the model to the wastewater evaluation in Shanghai, and we obtain better evaluation results.

We consider shape optimization of Stokes flow in channels where the objective is to design the lateral walls of the channel in such a way that a desired velocity profile is achieved. This amounts to the solution of a PDE constrained optimization problem with the state equation given by the Stokes system and the design variables being the control points of a Bezier curve representation of the lateral walls subject to bilateral constraints. Using a finite element discretization of the problem by Taylor-Hood elements, the shape optimization problem is solved numerically by a path-following primal-dual interior-point method applied to the parameter dependent nonlinear system representing the optimality conditions. The method is an all-at-once approach featuring an adaptive choice of the continuation parameter, inexact Newton solves by means of right-transforming iterations, and a monotonicity test for convergence monitoring. The performance of the adaptive continuation process is illustrated by several numerical examples.

BACKGROUND: Fatigue is a debilitating condition with a significant impact on patients' quality of life. Fatigue is frequently reported by patients suffering from primary Sjögren's Syndrome (pSS), a chronic autoimmune condition characterised by dryness of the eyes and the mouth. However, although fatigue is common in pSS, it does not manifest in all sufferers, providing an excellent model with which to explore the potential underpinning biological mechanisms.METHODS: Whole blood samples from 133 fully-phenotyped pSS patients stratified for the presence of fatigue, collected by the UK primary Sjögren's Syndrome Registry, were used for whole genome microarray. The resulting data were analysed both on a gene by gene basis and using pre-defined groups of genes. Finally, gene set enrichment analysis (GSEA) was used as a feature selection technique for input into a support vector machine (SVM) classifier. Classification was assessed using area under curve (AUC) of receiver operator characteristic and standard error of Wilcoxon statistic, SE(W).RESULTS: Although no genes were individually found to be associated with fatigue, 19 metabolic pathways were enriched in the high fatigue patient group using GSEA. Analysis revealed that these enrichments arose from the presence of a subset of 55 genes. A radial kernel SVM classifier with this subset of genes as input displayed significantly improved performance over classifiers using all pathway genes as input. The classifiers had AUCs of 0.866 (SE(W) 0.002) and 0.525 (SE(W) 0.006), respectively.CONCLUSIONS: Systematic analysis of gene expression data from pSS patients discordant for fatigue identified 55 genes which are predictive of fatigue level using SVM classification. This list represents the first step in understanding the underlying pathophysiological mechanisms of fatigue in patients with pSS.

In classifier fusion models, classifiers outputs are combined to achieve a group decision. The most often used classifiers fusion models are majority vote, probability schemes, weighted averaging and Bayes approach to name few. We propose a model of classifiers fusion by combining the mathematical belief of classifiers. We used Dempster-shafer theory of evidence to determine the mathematical belief of classifiers. Support Vector Machine (SVM) with linear, polynomial and radial kernel has been employed as classifiers. The output of classifiers used as basis for computing beliefs. We combined these beliefs to arrive at one final decision. Our experimental results have shown that the new proposed classifiers fusion methodology have outperforms single classification models.

In this work the authors show how the zero dynamics design methodology, developed in a series of papers for boundary control of distributed parameter systems, can be extended to include interior point control for tracking problems and disturbance rejection for a one dimensional heat equation. In particular we demonstrate how simple control laws can be obtained for solving MIMO set-point control problems using co-located interior point control and actuation. The results apply also to a much wider class on one dimensional problems and also to some interesting nonlinear problems. In this conference work we restrict to the case of the heat equation and present two examples. In the first example we consider a problem with two interior controls. The tracking problem consists of a set-point control at one interior point while tracking a sinusoid at another interior point. In our second example we consider a multivariable set-point control problem.

The discrimination of ventricular tachycardia (VT) with 1:1 retrograde conduction from sinus tachycardia often remains a challenge for rate-based arrhythmia recognition algorithms commonly used in dual-chamber implantable cardioverter defibrillators (ICDs). In this study, we propose hybrid wavelet packet-support vector classifiers for the recognition of atrial activation patterns as rate independent approach. Consecutive beats representing antegrade (AA) and retrograde atrial (RA) activation patterns within data segments of 10s duration were supplied to a binary hybrid wavelet packet-support vector classifier with radial kernel and morphologically adapted wavelet packets followed by a regularization scheme to exclude ectopic beats and artifacts. This system utilizes an optimal representation of the EE waveforms and inherently minimizes an upper bound on the generalization error All data segments of an independent test set with AA and RA sequences Were classified correctly by the proposed scheme. The possible use of this scheme in dual-chamber ICDs may increase the specificity of the VT detection by these devices.

It is well-known that electromagnetic solutions of boundary integral equations for perfectly electrically conducting scatterers are nonunique for those frequencies which correspond to interior resonances of the scatterer. In this paper a simple and efficient computational method is developed, in which the interior integral representations, required to hold on an interior closed surface, are used as a sufficient constraint to restore uniqueness. We use the interior equations together with the second kind magnetic field integral equation, so that the ill-posedness of the interior equations does not give a problem. We develop a constrained conjugate gradient method that minimizes a cost functional consisting of two terms. The first term is the error norm with respect to the magnetic field boundary integral equation, while the second term is the error norm with respect to the interior equations over a closed interior surface, which is chosen as small as possible. Some numerical examples show the robustness and efficiency of the pertaining computational procedure.

A novel interior-point algorithm related to entropy barrier functions is considered. An exponential rate of convergence to the optimal solution of corresponding vector fields is proved. >

Consider the linear program in standard form $$\min {{\mathbf{c}}^T}{\mathbf{x}}\;s.t.\;{\mathbf{Ax}} = {\mathbf{b}}\;and\;{\mathbf{x}} \geqslant {\mathbf{0}}$$ (6.1) and its dual $$\max {{\mathbf{b}}^T}{\mathbf{y}}\;s.t.\;{\mathbf{s}} = {\mathbf{c}} - {{\mathbf{A}}^T}{\mathbf{y}} \geqslant {\mathbf{0}}$$ (6.2) with \({\mathbf{A}} \in {\mathbb{R}^{m \times n}},{\mathbf{c}} \in {\mathbb{R}^n}\;and\;{\mathbf{b}} \in {\mathbb{R}^m}\).

In this chapter we develop a new approach to truncated potentials. We introduce the extension of truncated potentials and prove necessary and sufficient conditions for boundedness from L p (R n ) into L v q (R n ), when 1 n/p. A generalization of Sawyer’s result [258] is presented. Then a compactness criterion for this operator is proved, and upper and lower estimates of its distance from the class of compact operators are derived.

We derive optimal radial kernel in the radial basis function network applied in nonlinear function learning and classification.

An interior point of a finite planar point set is a point of the set that is not on the boundary of the convex hull of the set. For any integer k ≥ 1, let h(k) be the smallest integer such that every set of points in the plane, no three collinear, containing at least h(k) interior points has a subset of points containing k or k + 1 interior points. We proved that h(3) =3 in an earlier paper. In this paper we prove that h(4) = 7.

Let us assume that a 2-torus homeomorphism $f$ isotopic to the identity has a segment of irrational slope as its rotation set $\rho(F)$. We prove that if the chain recurrent set $R(f)$ of $f$ is not chain transitive, then $\rho(F)$ has a rational point realized by a periodic point.

In this paper we propose an interior point method for solving the dual form of min-max type problems. The dual variables are updated by means of a scaling supergradient method. The boundary of the dual feasible region is avoided by the use of a logarithmic barrier function. A major difference with other interior point methods is the nonsmoothness of the objective function.

We discuss some aspects of the interior geometry of a metric space X. The metric on X is denoted d, the open respectively closed metric ball about a point x ∈ X is denoted B(x,r) or B r (x) respectively \( \overline B {\rm{ }}(x,r){\rm{ or }}{\overline B _r}{\rm{(}}x{\rm{)}} \).

We describe a new general approach for interior points algorithms in nonlinear constrained optimization. It consists on the iterative solution, in the primal and dual variables, of Karush — Kuhn — Tucker first order optimality conditions. Based on this approach, different algorithms can be stated by taking advantage of the particular characteristics of the problem in consideration and of the order of the available information. This method is very strong and efficient, since at each iteration it only requires the solution of two linear systems with the same matrix. It is also particularly appropriated for Engineering Design Optimization, since feasible designs are obtained. We present a basic algorithm for inequality constrained problems and two of the possible particular versions. The first one is a first order algorithm and the second one uses a quasi — Newton approximation of the second derivative of the Lagrangian, in order to have superlinear asymptotic convergence. Equality constraints are introduced later.

A method based on adaptive kernel probablity density estimators is proposed in order to design optimal receivers from finite sets of examples. Performances are shown for the classification of underwater transient signals.

When the boundary integral -equation method is applied to exterior acoustics problems, singularities occur in the resulting algebraic equations at various frequencies associated with the eigenvalues of an interior problem. These frequencies are referred to as “forbidden”, and various methods have been devised to overcome the computational difficulties presented at these frequencies. The work presented here is an extension to the CHIEF method in that higher derivatives, in addition to the function itself, are constrained to be zero at selected points in the interior domain. Whereas the relative success of either method depends on the quantity and selection of interior points, the SuperCHIEF method requires fewer interior points and is less sensitive to point selection, resulting in improved robustness without significant increase in computational complexity.

PURPOSE. Machine learning techniques were used to identify which of 14 algorithms best predicts the genetic risk for development of proliferative vitreoretinopathy (PVR) in patients who are experiencing primary rhegmatogenous retinal detachment (RD). METHOD Data from a total of 196 single nucleotide polymorphisms in 30 candidate genes were used. The genotypic profile of 138 patients with PVR following primary rhegmatogenous RD and 312 patients without PVR RD were analyzed. Machine learning techniques were used to develop statistical predictive models. Fourteen models were assessed. Their reproducibility was evaluated by an internal cross-validation method. RESULTS The three best predictive models were the lineal kernel based on the Support Vector Machine (SMV), the radial kernel based on the SVM, and the Random Forest. Accuracy values were 78.4%, 70.3%, and 69.3%, respectively. The more accurate, although complex, algorithm uses 42 SNPs, whereas the simpler one uses only two SNPs, which makes them more suitable for routine diagnostic work. The radial kernel based on SVM uses 10 SNPs. The best individually predictor marker was rs2229094 in the tumor necrosis factor locus. CONCLUSION Genetic variables may be useful to predict the likelihood of the development of PVR. The predictive capabilities of these models are as good as those observed with clinical approaches. These results need external validation to estimate the true predictive capability and select the most appropriate ones for clinical use. © Association for Research in Vision and Ophthalmology.

This paper gives a necessary and sufficient condition for a D -stable matrix to be in the topological interior of the set of D -stable matrices.

We approximate the effective potential appearing in the radial part of the massless Klein-Gordon equation for the interior of a charged black hole by a second-order polynomial. This approximation allows an analytic treatment of the equation. Further we recover all the relevant properties in the interior region. For the interior normal modes the comparison between the analytical and numerical results yields an excellent agreement when the ratio of charge to mass for the black hole is close to unity. We use two theorems to bound the difference between the exact and approximate eigenvalues. A complementary scheme using both the analytical and numerical methods is proposed to solve related problems in the Kerr metric.

in the sense of "all, except those in a set of first Baire category". The space of all convex surfaces in IR", endowed with Hausdorfl's metric, is a Baire space. We shall see how abnormal convex surfaces may be, by proving that most of them are so.

The problem of optimal analytical extrapolation of holomorphic functions from a finite set of interior data points to another interior point is completely solved in the general case of data known with unequal errors. Simple and easy to handle algorithms are obtained.

Interior solutions for a static, axially symmetric family of solutions of Einstein's equations are described. The interior solutions correspond to spatially bound matter and are properly matched to an exterior vacuum solution. The family of solutions discussed include the Schwarzschild solution as a special case. A general method is exhibited for transforming any spherically symmetric interior solution to an interior for the other members of the family of solutions. The energy density remains positive for at least a finite range of the parameter that describes the family of solutions. Two solutions are explicitly exhibited. One is transformed from the constant density Schwarzschild interior solution and one from the Adler interior solution. The first solution would be expected to be unstable under adiabatic perturbations of the matter, the second would be expected to be stable.

For 1 > p ⩽ q > ∞ , 0 > α > n 1 > p \leqslant q > \infty ,0 > \alpha > n and w ( x ) , υ ( x ) w(x),\upsilon (x) nonnegative weight functions on R n {R^n} we show that the weak type inequality \[ ∫ { T α f > λ } w ( x ) d x ⩽ A λ − q ( ∫ | f ( x ) | p υ ( x ) d x ) q / p \int _{\{ {T_\alpha }f > \lambda \} }\,w(x)\;dx \leqslant A{\lambda ^{ - q}}{\left ( \int |f(x){|^p}\;\upsilon (x)\;dx \right )^{q/p}} \] holds for all f ⩾ 0 f \geqslant 0 if and only if \[ ∫ Q [ T α ( χ Q w ) ( x ) ] p ′ υ ( x ) 1 − p ′ d x ⩽ B ( ∫ Q w ) p ′ / q ′ > ∞ \int _Q\,[{T_\alpha }({\chi _Q}w)\,(x)]^{p’}\upsilon (x)^{1 - p’}\,dx \leqslant B\left ( \int _Qw \right )^{p’/q’} > \infty \] for all cubes Q Q in R n {R^n} . Here T α {T_\alpha } denotes the fractional integral of order α , T α f ( x ) = ∫ | x − y | α − n f ( y ) d y \alpha ,{T_\alpha }f(x) = \int |x - y{|^{\alpha - n}}f(y)\,dy . More generally we can replace T α {T_\alpha } by any suitable convolution operator with radial kernel decreasing in | x | |x| .

Algebraic method to construct irreducible representations of space groups is presented by making an example of white tin having screw axes and glide planes in the symmetry operations. In present method it is enough to treat explicitly at most only symmetry elements equal in number to the operations of the point group to which concerned lattice belongs, regardless of the cases with or without spin and also of interior points or the points of the Brillouin zone boundary. This becomes very simplification compared to usual group-theoretical method.

In the literature there are multiple machine learning techniques that have been used successfully in clinical data analysis. However, there is little information about the parameter configurations, the required data transformations to prepare the data used to train and evaluate the models and the impact of these decisions in the accuracy of the predictive model. This research tackles these issues, using the clinical data of MIMICII to build features from physiological measure patterns to predict the decease of patients inside the hospital in the next 24 hours, building predictive models based on Logistic Regression, Neural Networks, Decision Trees and Nearest Neighbors. In particular, we use data associated to physiological measures of 3220 patients, where 2385 left the hospital alive and 835 passed in the hospital. The results show that the chosen strategy for building features from physiological data gives good results with Neural Networks and Logistic Regression with radial kernel models and the parameter configuration plays a fundamental role in the models performance.

In this study, we present some results on the weighted pointwise convergence of a family of singular integral operators with radial kernels given in the following form: $$\begin{aligned} L_{\lambda }\left( f;x,y\right) =\underset{ \mathbb {R} ^{2}}{\iint }f\left( t,s\right) H_{\lambda }\left( t-x,s-y\right) \mathrm{d}s\,\mathrm{d}t,\quad \left( x,y\right) \in \mathbb {R} ^{2},\quad \lambda \in \Lambda , \end{aligned}$$ where $$\Lambda$$ is a set of non-negative numbers with accumulation point $$\lambda _{0}$$ , and the function f is measurable on $$\mathbb {R} ^{2}$$ in the sense of Lebesgue.

In this paper, a new algorithm for tracing the combined homotopy path of the non-convex nonlinear programming problem is proposed. The algorithm is based on the techniques of β-cone neighborhood and a combined homotopy interior point method. The residual control criteria, which ensures that the obtained iterative points are interior points, is given by the condition that ensures the β-cone neighborhood to be included in the interior part of the feasible region. The global convergence and polynomial complexity are established under some hypotheses.

In this paper, we present a new barrier function for primal–dual interior-point methods in linear optimization. The proposed kernel function has a trigonometric barrier term. It is shown that in the interior-point methods based on this function for large-update methods, the iteration bound is improved significantly. For small-update interior-point methods, the iteration bound is the best currently known bound for primal–dual interior-point methods.

This paper concerns the generation of support vector machine classifiers for solving the pattern recognition problem in machine learning. A method is proposed based on interior-point methods for convex quadratic programming. This interior-point method uses a linear preconditioned conjugate gradient method with a novel preconditioner to compute each iteration from the previous. An implementation is developed by adapting the object-oriented package OOQP to the problem structure. Numerical results are provided, and computational experience is discussed.

Interior point methods are not only the most effective methods for solving optimisation problems in practice but they also have polynomial time complexity. However, there is still a gap between the practical behavior of the interior point method algorithms and their theoretical complexity results. In this paper, by focusing on linear programming problems, we introduce a new family of kernel functions that have some simple and easy to check properties. We present a simplified analysis to obtain the complexity of generic interior point methods based on the proximity functions induced by these kernel functions. Finally, we prove that this family of kernel functions leads to improved iteration bounds of the large-update interior point methods.

We present a framework for designing and analyzing primal-dual interior-point methods for convex optimization. We assume that a self-concordant barrier for the convex domain of interest and the Legendre transformation of the barrier are both available to us. We directly apply the theory and techniques of interior-point methods to the given good formulation of the problem (as is, without a conic reformulation) using the very usual primal central path concept and a less usual version of a dual path concept. We show that many of the advantages of the primal-dual interior-point techniques are available to us in this framework and therefore, they are not intrinsically tied to the conic reformulation and the logarithmic homogeneity of the underlying barrier function.

In the present work we prove the pointwise convergence and the rate of pointwise convergence for a family of singular integral operators with radial kernel in two-dimensional setting in the following form: $L_{\lambda} ( f;x,y ) =\iint_{D}f ( t,s ) H_{\lambda} ( t-x,s-y ) \,dt\,ds$ , $( x,y ) \in D$ , where $D= \langle a,b \rangle\times \langle c,d \rangle$ is an arbitrary closed, semi-closed or open region in $\mathbb{R}^{2}$ and $\lambda\in\Lambda$ , Λ is a set of non-negative numbers with accumulation point $\lambda_{0}$ . Also we provide an example to justify the theoretical results.