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Chamfer (or weighted) distances are popular digital distances used in various grids. They are based on the weights assigned to steps to various neighborhoods. In the triangular grid there are three usually used neighbor relations, consequently, chamfer distances based on three weights are used. A chamfer (or digital) disk of a grid is the set of the pixels which have distance from the origin that is not more than a given finite bound called radius. These disks are well known and well characterized on the square grid. Using the two basic (i.e., the cityblock and the chessboard) neighbors, the convex hull of a disk is always an octagon (maybe degenerated). Recently, these disks have been defined on the triangular grid; their shapes have a great variability even with the traditional three type of neighbors, but their complete characterization is still missing. Chamfer balls are convex hulls of integer points that lie in polytopes defined by linear inequalities, and thus can be computed through a linear integer programming approach. Generally, the integer hull of a polyhedral set is the convex hull of the integer points of the set. In most of the cases, for example when the set is bounded, the integer hull is a polyhedral set, as well. The integer hull can be determined in an iterative way by Chvátal cuts. In this paper, sides of the chamfer disks are determined by the inequalities with their Chvátal rank 1. The most popular coordinate system of the triangular grid uses three coordinates. By giving conditions depending only a coordinate, the embedding hexagons of the shapes are obtained. These individual bounds are described completely by Chvátal cuts. They also give the complete description of some disks. Further inequalities having Chvátal rank 1 are also discussed. [ABSTRACT FROM AUTHOR]

Emerging distribution systems with a proliferation of distributed energy resources are facing with new challenges, such as voltage collapse and power flow congestion in unsymmetrical network configurations. As a fundamental tool that could help quantify these new challenges and further mitigate their impacts on the secure and economic operation of distribution systems, effective AC optimal power flow (ACOPF) models and solution approaches are in urgent need. This study focuses on ACOPF of three‐phase four‐conductor configured distribution systems, in which neutral conductors and ground resistances are modelled explicitly to reflect practical situation. In addition, by leveraging the Kirchhoff's current law (KCL) theorem and the effect of zero injections, voltage variables of neutrals and zero‐injection phases can be effectively eliminated. The ACOPF problem is formulated as a convex semidefinite programming (SDP) relaxation model in complex domain. In recognising possible solution inexactness of SDP relaxation model, a Karush–Kuhn–Tucker condition based process is further proposed to effectively recover feasible solutions to the original ACOPF problem by calculating a set of computational‐inexpensive non‐linear equations. Numerical studies on a modified IEEE 123‐bus system show the effectiveness of the proposed SDP relaxation model with variable reductions and the feasible solution recovery process for three‐phase four‐conductor configured distribution systems. [ABSTRACT FROM AUTHOR]

We discuss a control constrained boundary optimal control problem for the Boussinesq-type system arising in the study of the dynamics of an arterial network. We suppose that a control object is described by an initial-boundary value problem for 1D system of pseudo-parabolic nonlinear equations with an unbounded coefficient in the principal part and Robin boundary conditions. The main question we discuss in this part of paper is about topological and algebraical properties of the set of feasible solutions. Following Faedo-Galerkin method, we establish the existence of weak solutions to the corresponding initial-boundary value problem and show that these solutions possess some special extra regularity properties which play a crucial role in the proof of solvability of the original optimal control problem. [ABSTRACT FROM AUTHOR]

Evaluation of uncertainties due to rotational ambiguities and noise propagation is essential to ascertain the reliability of Multivariate Curve Resolution estimations. When ambiguity is present, every resolved profile can be represented by a band of feasible solutions instead of by a unique profile. In the presence of experimental noise the estimation of this band of feasible solutions is more difficult and uncertain. The aim of this work is to show how experimental noise and profiles overlapping affect the reliability of feasible solutions. For this purpose, feasible solutions of several two and three components data systems with different profiles overlapping and noise levels have been systematically investigated. Results obtained using simulated and experimental data showed that increasing noise levels and profiles overlapping in one mode (e.g. concentration profiles) make more uncertain the estimation of feasible solutions in the other mode (e.g. spectral profiles), and this uncertainty is propagated in a non-uniform and complex way. [ABSTRACT FROM AUTHOR]

As a novel evolutionary searching technique, ant colony optimization (ACO) has gained wide research attention and can be used as a tool for optimizing an array of mathematical functions. In transportation systems, when ACO is applied to solve the vehicle routing problem (VRP), the path of each ant is only “part” of a feasible solution. In other words, multiple ants' paths may constitute one feasible solution. Previous works mainly focus on the algorithm itself, such as revising the pheromone updating scheme and combining ACO with other optimization methods. However, this body of literature ignores the important procedure of constructing feasible solutions with those “parts”. To overcome this problem, this paper presents a novel ACO algorithm (called AMR) to solve the VRP. The proposed algorithm allows ants to go in and out the depots more than once until they have visited all customers, which simplifies the procedure of constructing feasible solutions. To further enhance AMR, we propose two extensions (AMR-SA and AMR-SA-II) by integrating AMR with other saving algorithms. The computational results for standard benchmark problems are reported and compared with those from other ACO methods. Experimental results indicate that the proposed algorithms outperform the existing ACO algorithms. [ABSTRACT FROM PUBLISHER]

The benchmark functions and some of the algorithms proposed for the special session on real parameter optimization of the 2005 IEEE Congress on Evolutionary Computation (CEC'05) have played and still play an important role in the assessment of the state of the art in continuous optimization. In this article, we show that if bound constraints are not enforced for the final reported solutions, state-of-the-art algorithms produce infeasible best candidate solutions for the majority of functions of the IEEE CEC'05 benchmark function suite. This occurs even though the optima of the CEC'05 functions are within the specified bounds. This phenomenon has important implications on algorithm comparisons, and therefore on algorithm designs. This article's goal is to draw the attention of the community to the fact that some authors might have drawn wrong conclusions from experiments using the CEC'05 problems. [ABSTRACT FROM AUTHOR]

Highlights: [•] The replicate spectra are used to build a virtual two-way data. [•] This data matrix is rank deficient. [•] Augmentation of this data with standard addition data will break the rank deficiency. [•] The algorithm MCR-BANDS has been employed to evaluate the rotational ambiguity. [ABSTRACT FROM AUTHOR]

Abstract: In this study the effect of the rotational ambiguity on the accuracy of quantitative results obtained with the help of MCR methods is investigated in detail by simulating different examples chosen from real systems analyzed in previous reported researches. In order to analyze the rotational ambiguity, the systematic grid search minimization algorithm in two and three component systems is employed. The main objective here is to demonstrate that applying not enough constraint to restrict the feasible region makes the estimated concentration for the analyte of interest deviate considerably from the true solution. [Copyright &y& Elsevier]

Abstract: The presence of rotation ambiguities and unique solutions in Multivariate Curve Resolution (MCR) chemometric methods is discussed in detail. Using recently proposed graphical approaches to display the bands and areas of feasible solutions in a subspace of reduced dimensions, the results obtained by different MCR methods are compared. These results show that in the presence of rotation ambiguities and under a particular set of constraints, the solutions obtained by the different MCR methods can differ among them and also from the true solution depending on initial estimates and on the applied algorithm. In absence of rotational ambiguities, all MCR methods should give the same unique solution which should be equal to the true one. Many of the MCR methods proposed in the literature like MCR-ALS, RFA, MCR-FMIN, or MCR-BANDS are confirmed to give a valid solution within the band or area of feasible solutions. On the contrary, and according to the results of this study, in its present implementation, the minimum volume simplex analysis, MVSA method can give unfeasible solutions when resolving bilinear data systems with more than two components, because it only applies non-negativity constraints to concentration profiles and not to spectral profiles. [Copyright &y& Elsevier]

Since the combinatorial version of the process network synthesis (PNS) problem is NP-complete, it is important to have fast enumeration procedures to generate its feasible solutions or its optimal solutions. In this work, a partial enumeration procedure is presented which is based on the merging reduction. This algorithm generates such a subset of the feasible solutions which tontains all of the optimal solutions of problem considered. The power of this enumeration procedure is illustrated by an empirical analysis. [Copyright &y& Elsevier]

Multivariate self-modeling curve resolution (SMCR) methods are the best choice for analyzing chemical data when there is not any prior knowledge about the chemical or physical model of the process under investigation [ 1Q3: The reference '1' is only cited in the abstract and not in the text. Please introduce a citation in the text. ]. However, the rotational ambiguity is the main problem of SMCR methods, yielding a range of feasible solutions. It is, therefore, important to determine the range of all feasible solutions of SMCR methods. Different methods have been presented in the literature to find feasible solutions of two, three, and four component systems. Here, a novel simple SMCR method is presented for calculating the boundaries of feasible solutions of two-component systems. At first, the simple strategy is presented for calculating the feasible solutions of two-component systems. Next, four different experimental two-component systems are analyzed in detail for calculating the boundaries of feasible solutions in both spaces, including complex formation equilibrium, keto-enol tautomerization kinetic, lipidomics data, and a case for quantification of an analyte in gray systems. In all cases, the boundaries of range of feasible solutions are properly determined by the proposed simple strategy. [Display omitted] • By applying different constraints, Self-modeling curve resolution (SMCR) techniques produce a range of feasible solutions for every component in a system. It is important to determine the range of all feasible solutions of SMCR methods. • A simple procedure for determining the boundaries of feasible solutions of two-component systems using the micro-structure of data is proposed. • The proposed method is based on the fact that the solutions of non-negative bilinear decomposition of two-component systems are always present as two rows and columns of the measured data. [ABSTRACT FROM AUTHOR]

We generalize the successive continuation paradigm introduced by Kernévez and Doedel [1] for locating locally optimal solutions of constrained optimization problems to the case of simultaneous equality and inequality constraints. The analysis shows that potential optima may be found at the end of a sequence of easily-initialized separate stages of continuation, without the need to seed the first stage of continuation with nonzero values for the corresponding Lagrange multipliers. A key enabler of the proposed generalization is the use of complementarity functions to define relaxed complementary conditions, followed by the use of continuation to arrive at the limit required by the Karush-Kuhn-Tucker theory. As a result, a successful search for optima is found to be possible also from an infeasible initial solution guess. The discussion shows that the proposed paradigm is compatible with the staged construction approach of the coco software package. This is evidenced by a modified form of the coco core used to produce the numerical results reported here. These illustrate the efficacy of the continuation approach in locating optimal solutions of an objective function along families of two-point boundary value problems and in optimal control problems. [ABSTRACT FROM AUTHOR]