scholarly journals Local and global extrema for functions of several variables

1980 ◽  
Vol 29 (3) ◽  
pp. 362-368 ◽  
Author(s):  
Bruce Calvert ◽  
M. K. Vamanamurthy
Keyword(s):  
Critical Point ◽  
Local Minimum ◽  
Global Minimum ◽  
Sufficient Conditions ◽  
Subject Classification ◽  
General Function ◽  
Several Variables ◽  
Open Question

AbstractLet p: R2 → R be a polynomial with a local minimum at its only critical point. This must give a global minimum if the degree of p is < 5, but not necessarily if the degree is ≥ 5. It is an open question what the result is for cubics and quartics in more variables, except cubics in three variables. Other sufficient conditions for a global minimum of a general function are given.1980 Mathematics subject classification (Amer. Math. Soc.): 26 B 99, 26 C 99.

scholarly journals New Results on Degree Sequences of Uniform Hypergraphs

10.37236/3414 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Sarah Behrens ◽  
Catherine Erbes ◽  
Michael Ferrara ◽  
Stephen G. Hartke ◽  
Benjamin Reiniger ◽  
...  
Keyword(s):  
Efficient Algorithm ◽  
Necessary Conditions ◽  
Degree Sequence ◽  
Sufficient Conditions ◽  
Degree Sequences ◽  
Uniform Hypergraph ◽  
Graphic Sequence ◽  
Uniform Hypergraphs ◽  
Open Question

A sequence of nonnegative integers is $k$-graphic if it is the degree sequence of a $k$-uniform hypergraph. The only known characterization of $k$-graphic sequences is due to Dewdney in 1975. As this characterization does not yield an efficient algorithm, it is a fundamental open question to determine a more practical characterization. While several necessary conditions appear in the literature, there are few conditions that imply a sequence is $k$-graphic. In light of this, we present sharp sufficient conditions for $k$-graphicality based on a sequence's length and degree sum.Kocay and Li gave a family of edge exchanges (an extension of 2-switches) that could be used to transform one realization of a 3-graphic sequence into any other realization. We extend their result to $k$-graphic sequences for all $k \geq 3$. Finally we give several applications of edge exchanges in hypergraphs, including generalizing a result of Busch et al. on packing graphic sequences.

scholarly journals MINIMIZATION OF REAL POWER LOSS BY ENHANCED GRAVITATIONAL SEARCH ALGORITHM

2017 ◽  
Vol 5 (7) ◽  
pp. 623-630
Author(s):  
K. Lenin
Keyword(s):  
Local Minimum ◽  
Global Minimum ◽  
Reactive Power ◽  
Search Algorithm ◽  
Gravitational Search Algorithm ◽  
Gravitational Forces ◽  
Bee Colony ◽  
Gravitational Search ◽  
The Masses ◽  
Power Problem

In this paper, Enhanced Gravitational Search (EGS) algorithm is proposed to solve the reactive power problem. Gravitational search algorithm (GSA) results are improved by using artificial bee colony algorithm (ABC). In GSA, solutions are fascinated towards each other by applying gravitational forces, which depending on the masses assigned to the solutions, to each other. The heaviest mass will move slower than other masses and pull others. Due to nature of gravitation, GSA may pass global minimum if some solutions stuck to local minimum. ABC updates the positions of the best solutions that have obtained from GSA, preventing the GSA from sticking to the local minimum by its strong penetrating capability. The proposed algorithm improves the performance of GSA in greater level. In order to evaluate the performance of the proposed EGS algorithm, it has been tested on IEEE 57,118 bus systems and compared to other standard algorithms.

More General Surplus

2016 ◽  
Author(s):  
Alfred Galichon
Keyword(s):  
General Result ◽  
Convex Analysis ◽  
Scalar Product ◽  
Sufficient Conditions ◽  
General Function ◽  
Existence Of A Solution ◽  
Generalize Convex ◽  
Monge Problem ◽  
Natural Way

This chapter considers a case with a more general surplus function. It shows that when the scalar-product surplus is replaced by a more general function, much of the machinery put in place in Chapter 6 goes through. In particular, it is possible to generalize convex analysis in a natural way, and to obtain generalized notions of convex conjugates, of convexity, and of a subdifferential that are perfectly suited to the problem. A general result on the existence of dual minimizers is given, as well as sufficient conditions for the existence of a solution to the Monge problem.

Designs Robust Against Aberrations

1993 ◽  
Vol 43 (1-2) ◽  
pp. 95-108 ◽  
Author(s):  
N. K. Mandal ◽  
K. R. Shah
Keyword(s):  
Sufficient Conditions ◽  
Subject Classification ◽  
Alternative Formulation ◽  
Block Designs ◽  
Ams Subject Classification ◽  
Secondary 62K05

In this paper, we obtain sufficient conditions for a design to be robust against aberrations in the sense of Box and Draper. Block designs, row-column designs and fractional designs are considered here. An alternative formulation of robustness is also presented. AMS Subject Classification: Primary 62K99; Secondary 62K05.

scholarly journals Functional advantages of Lévy walks emerging near a critical point

2020 ◽  
Vol 117 (39) ◽  
pp. 24336-24344 ◽  
Author(s):  
Masato S. Abe
Keyword(s):  
Critical Point ◽  
Dynamic Range ◽  
Movement Trajectories ◽  
Lévy Walks ◽  
Optimal Dynamic ◽  
Freely Moving ◽  
High Flexibility ◽  
Open Question ◽  
The Brain ◽  
Power Law Distributions

A special class of random walks, so-called Lévy walks, has been observed in a variety of organisms ranging from cells, insects, fishes, and birds to mammals, including humans. Although their prevalence is considered to be a consequence of natural selection for higher search efficiency, some findings suggest that Lévy walks might also be epiphenomena that arise from interactions with the environment. Therefore, why they are common in biological movements remains an open question. Based on some evidence that Lévy walks are spontaneously generated in the brain and the fact that power-law distributions in Lévy walks can emerge at a critical point, we hypothesized that the advantages of Lévy walks might be enhanced by criticality. However, the functional advantages of Lévy walks are poorly understood. Here, we modeled nonlinear systems for the generation of locomotion and showed that Lévy walks emerging near a critical point had optimal dynamic ranges for coding information. This discovery suggested that Lévy walks could change movement trajectories based on the magnitude of environmental stimuli. We then showed that the high flexibility of Lévy walks enabled switching exploitation/exploration based on the nature of external cues. Finally, we analyzed the movement trajectories of freely moving Drosophila larvae and showed empirically that the Lévy walks may emerge near a critical point and have large dynamic range and high flexibility. Our results suggest that the commonly observed Lévy walks emerge near a critical point and could be explained on the basis of these functional advantages.

Lagrangean conditions and quasiduality

1977 ◽  
Vol 16 (3) ◽  
pp. 325-339 ◽  
Author(s):  
B.D. Craven
Keyword(s):  
Critical Point ◽  
Critical Points ◽  
Local Minimum ◽  
Constraint Qualification ◽  
Necessary Conditions ◽  
Weak Duality ◽  
Partially Ordered Vector Space ◽  
Counter Example ◽  
Cone Constraints ◽  
Constrained Minimization Problem

For a constrained minimization problem with cone constraints, lagrangean necessary conditions for a minimum are well known, but are subject to certain hypotheses concerning cones. These hypotheses are now substantially weakened, but a counter example shows that they cannot be omitted altogether. The theorem extends to minimization in a partially ordered vector space, and to a weaker kind of critical point (a quasimin) than a local minimum. Such critical points are related to Kuhn-Tucker conditions, assuming a constraint qualification; in certain circumstances, relevant to optimal control, such a critical point must be a minimum. Using these generalized critical points, a theorem analogous to duality is proved, but neither assuming convexity, nor implying weak duality.

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