With the technology development in cyber physical systems and big data, there are huge potential to apply them to achieve personalization and improve resource efficiency in Industry 4.0. As Industr...

Big data for cyber physical systems in industry 4.0: a survey

The purpose of this paper is to discuss the various pivot rules of the simplex method and its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with finiteness properties of simplex type pivot rules. Well known classical results concerning the simplex method are not considered in this survey, but the connection between the new pivot methods and the classical ones, if there is any, is discussed. In this paper we discuss three classes of recently developed pivot rules for linear programming. The first and largest class is the class of essentially combinatorial pivot rules including minimal index type rules and recursive rules. These rules only use labeling and signs of the variables. The second class contains those pivot rules which can actually be considered as variants or generalizations or specializations of Lemke's method, and so they are closely related to parametric programming. The last class has the common feature that the rules all have close connections to certain interior point methods. Finally, we mention some open problems for future research.

Pivot rules for linear programming: A survey on recent theoretical developments

This paper pursues a number of theoretical explorations and conjectures pertaining to the uncovered set in spatial voting games. It was stimulated by the article ‘‘The Uncovered Set and the Limits of Legislative Action’’ by W. T. Bianco, I. Jeliazkov, and I. Sened (2004, Political Analysis 12:256–78) that employed a grid-search computational algorithm for estimating the size, shape, and location of the uncovered set, and it has been greatly facilitated by access to the CyberSenate spatial voting software being developed by Joseph Godfrey. I bring to light theoretical considerations that account for important features of the Bianco, Jeliazkov, and Sened results (e.g., the straight-line boundaries of uncovered sets displayed in some of their figures, the ‘‘unexpectedly large’’ uncovered sets displayed in other figures, and the apparent sensitivity of the location of uncovered sets to small shifts in the relative sizes of party caucuses) and present theoretical insights of more general relevance to spatial voting theory.

/pdf/in-search-of-the-uncovered-set-ojcv4r4lct.pdf

In Search of the Uncovered Set

Publisher Summary This chapter discusses parts of convex geometry that are important for optimization in mathematical programming. Mathematical programming consists of methods for finding or approximating the global optimum of a real-valued function φ over a domain D in real n -space ℝ d or over D ⋂ℚ n or D ⋂ℤ n (the rational or integer points in D ). Linear programming is generally understood to imply that the objective function φ is linear, that the domain D is a convex polyhedron given as the intersection of a finite number of closed half spaces, and that the optimization is over D or D ⋂ℚ n . The chapter discusses some important and contrasting aspects of convex minimization and convex maximization—(1) relationship between local and global extrema; (2) extent to which optimization is an interior- or boundary-point phenomenon; (3) extent to which optimization reduces to one-dimensional considerations; and (4) complexity of quadratic optimization over polytopes. If φ is an affine function, then φ is convex, so the problem of minimizing φ has some aspects in common with convex minimization. Similarly, if φ is concave, then the problem of minimizing φ has aspects in common with concave minimization, or, equivalently, with convex maximization.

Mathematical Programming and Convex Geometry

The yolk, developed in Ferejohn, McKelvey and Packel (1984) and McKelvey (1986), is a key solution concept in the Euclidean spatial model as the region of policies where a dynamic voting game will tend to reside. However, determining the yolk is NP-hard for arbitrary dimension. This paper derives an algorithm to compute the yolk in polynomial time for any fixed dimension.

A polynomial-time algorithm for computing the yolk in fixed dimension

The yolk, an important concept in the spatial model of voting, is defined in two dimensions as the smallest circle intersecting all median lines. In the literature one finds the assumption that the limiting median lines, i.e., those that pass through two voter ideal points, suffice to determine the yolk. Counterex-amples are given here which invalidate this assumption.

Limiting median lines do not suffice to determine the yolk

The projective scaling algorithm in Karmarkar [Combinatorica, 4 (1984), pp. 373–395] and Dantzig’s simplex method (see [Linear Programming and Extensions, Princeton University Press, 1963]) are usually thought of as fundamentally different approaches to linear programming. When viewed in Dantzig’s column space geometry, however, both algorithms turn out to be iteratively reweighted least squares methods. The projective scaling algorithm (and the affine scaling method in Dikin [Soviet Math. Dokl., 8 (1967), pp. 674–675]) can then be derived as a natural generalization of the simplex method. This derivation shows how the dual variables arise in the interior methods. The insight is essentially geometric; suitable figures are provided.

https://www.jstor.org/stable/pdfplus/2031142.pdf

The Simplex and Projective Scaling Algorithms as Iteratively Reweighted Least Squares Methods

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Limiting median lines do not suffice to determine the yolk

The Simplex and Projective Scaling Algorithms as Iteratively Reweighted Least Squares Methods