Computational complexity of LCPs associated with positive definite symmetric matrices

Murty in a recent paper has shown that the computational effort required to solve a linear complementarity problem (LCP), by either of the two well known complementary pivot methods is not bounded above by a polynomial in the size of the problem. In that paper, by constructing a class of LCPs—one of order n for n ≥ 2—he has shown that to solve the problem of order n , either of the two methods goes through 2 n pivot steps before termination.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47905/1/10107_2005_Article_BF01588254.pd

Computational complexity of parametric linear programming

We establish that in the worst case, the computational effort required for solving a parametric linear program is not bounded above by a polynomial in the size of the problem.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47907/1/10107_2005_Article_BF01581642.pd

Clustering problems in optimization models

We discuss a variety of clustering problems arising in combinatorial applications and in classifying objects into homogenous groups. For each problem we discuss solution strategies that work well in practice. We also discuss the importance of careful modelling in clustering problems.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/44350/1/10614_2004_Article_BF00121636.pd

CP-rays in simplicial cones

An interior point of a triangle is called CP-point if its orthogonal projection on the line containing each side lies in the relative interior of that side. In classical mathematics, interest in the concept of regularity of a triangle is mainly centered on the property of every interior point of the triangle being a CP-point. We generalize the concept of regularity using this property, and extend this work to simplicial cones in ℝ n , and derive necessary and sufficient conditions for this property to hold in them. These conditions highlight the geometric properties of Z-matrices. We show that these concepts have important ramifications in algorithmic studies of the linear complementarity problem. We relate our results to other well known properties of square matrices.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47921/1/10107_2005_Article_BF01582265.pd

Local decomposition of gray-scale morphological templates

Template decomposition techniques can be useful for improving the efficiency of imageprocessing algorithms. The improved efficiency can be realized either by reorganizing a computation to fit a specialized structure, such as an image-processing pipeline, or by reducing the number of operations used. In this paper two techniques are described for decomposing templates into sequences of 3×3 templates with respect to gray-scale morphological operations. Both techniques use linear programming and are guaranteed to find a decomposition of one exists.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46623/1/10851_2004_Article_BF00123880.pd

Suppression of σ-phase in nanocrystalline CoCrFeMnNiV high entropy alloy by unsolicited contamination during mechanical alloying and spark plasma sintering

CoCrFeMnNiV high entropy alloy (HEA) exhibits a high content of σ-phase (70 vol%) when produced by casting route. In the present work, a combination of mechanical alloying (MA) and spark plasma sintering (SPS) has been used to synthesize nanocrystalline CoCrFeMnNiV HEA where the formation of σ-phase has been avoided. Electron microscopy and atom probe tomography analysis indicated the formation of FCC structured HEA matrix along with (Cr,V) carbide (15 vol%) precipitation, without the presence of σ-phase in SPS processed alloy. Gibbs energy vs composition (G-x) diagrams of binary subsystems and possible carbides and oxides substantiate the absence of σ-phase during SPS of CoCrFeMnNiV alloy. Thus, the unsolicited contamination during MA-SPS route proves beneficial in suppressing the complex phase formation. © 2020 Elsevier B.V

A finite characterization of K -matrices in dimensions less than four

The class of real n × n matrices M , known as K -matrices, for which the linear complementarity problem w − Mz = q, w ≥ 0, z ≥ 0, w T z =0 has a solution whenever w − Mz =q, w ≥ 0, z ≥ 0 has a solution is characterized for dimensions n <4. The characterization is finite and ‘practical’. Several necessary conditions, sufficient conditions, and counterexamples pertaining to K -matrices are also given. A finite characterization of completely K -matrices ( K -matrices all of whose principal submatrices are also K -matrices) is proved for dimensions <4.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47913/1/10107_2005_Article_BF01589438.pd

Pivoting in Linear Complementarity: Two Polynomial-Time Cases

We study the behavior of simple principal pivoting methods for the P-matrix linear complementarity problem (P-LCP). We solve an open problem of Morris by showing that Murty’s least-index pivot rule (under any fixed index order) leads to a quadratic number of iterations on Morris’s highly cyclic P-LCP examples. We then show that on K-matrix LCP instances, all pivot rules require only a linear number of iterations. As the main tool, we employ unique-sink orientations of cubes, a useful combinatorial abstraction of the P-LCP

Профилактика болезней периодонта на начальных этапах ортодонтического лечения

ПЕРИОДОНТА БОЛЕЗНИБРЕКЕТ-СИСТЕМЫСТОМАТОЛОГИЯ ЛЕЧЕБНО-ВОССТАНОВИТЕЛЬНАЯОРТОДОНТИЧЕСКАЯ СТОМАТОЛОГИЯГЕКСАЛИЗГИНГИВИ