Multi-Objective Optimization Algorithm for Grouping Decision Variables Based on Extreme Point Pareto Frontier

The existing algorithms for solving multi-objective optimization problems fall into three main categories:Decomposition-based,dominance-based,and indicator-based.Traditional multi-objective optimization problemsmainly focus on objectives,treating decision variables as a total variable to solve the problem without consideringthe critical role of decision variables in objective optimization.As seen,a variety of decision variable groupingalgorithms have been proposed.However,these algorithms are relatively broad for the changes of most decisionvariables in the evolution process and are time-consuming in the process of finding the Pareto frontier.To solvethese problems,a multi-objective optimization algorithm for grouping decision variables based on extreme pointPareto frontier(MOEA-DV/EPF)is proposed.This algorithm adopts a preprocessing rule to solve the Paretooptimal solution set of extreme points generated by simultaneous evolution in various target directions,obtainsthe basic Pareto front surface to determine the convergence effect,and analyzes the convergence and distributioneffects of decision variables.In the later stages of algorithm optimization,different mutation strategies are adoptedaccording to the nature of the decision variables to speed up the rate of evolution to obtain excellent individuals,thusenhancing the performance of the algorithm.Evaluation validation of the test functions shows that this algorithmcan solve the multi-objective optimization problem more efficiently.

EXTREME POINTS AND SUPPORT POINTS OF A CLASS OF ANALYTIC FUNCTIONS

Suppose that {b(n)} and {c(n)} are two positive sequences. Let F({b(n)}, {c(n)}) = {f(z) : f(z) is analytic in \z\ < 1, f(z) = z - Sigma(n=2)(+infinity) a(n)z(n), a(n) greater than or equal to 0, Sigma(n=2)(+infinity) b(n)a(n) less than or equal to 1 and Sigma(n=2)(+infinity) c(n)a(n) less than or equal to 1}. This article obtains the extreme points and support points of F({b(n)}, {c(n)}).

THE EXTREME POINTS OF A CLASS OF ANALYTIC FUNCTIONS WITH POSITIVE REAL PART AND A PRESCRIBED SET OF VALUES

Letζ =（0,z1,z2,···,zn） with ｜zj｜〈1for1≤j≤n,ω=（1,w1,w2,···,wn）,and P（ζ,ω） denote the set of functions p（z） that are analytic in D={z：｜z｜〈1} and satisfy Rep（z）〉0（｜z｜〈1）,p（0）=1,p（zj）=wj,j=1,2,···,n.In this article we investigate the extreme points of P（ζ,ω）.

THE EXTREME POINTS OF SEVERAL CLASSES OFANALYTIC FUNCTIONS

Let P-n(c(1),c(2),...,c(n-1)) = {p(z) : p(z) is analytic in \z\ < 1 with Rep(z) > 0 and p(z) = 1 + c(1)z + c(2)z(2) +...+ c(n-1)z(n-1) + d(n)z(n) +..., where c(1),c(2),...,c(n-1) are fixed complex constants}. Let P-R,P-n(b(1),b(2),...,b(n-1)) = {p(z) : p(z) is analytic in \z\ < 1 with Rep(z) > 0 and p(z) = 1 + b(1)z + b(2)z(2) +...+ b(n-1)z(n-1) + d(n)z(n) +..., where b(1),b(2),...,b(n-1) are fixed real constants and the coefficients of p(z) are real}. Let T-n(l(1),l(2),...,l(n-1)) = {f(z) : f(z) is analytic in \z\ < 1 and f(z) = z + l(1)z(2) + l(2)z(3) +...+ l(n-1)z(n) + d(n)z(n+1) +...; where l(1),l(2),...,l(n-1) are fixed real constants and the coefficients of f(z) are real}. It is understood that P-n(c(1),c(2),...,c(n-1)), P-R,P-n(b(1),b(2),...,b(n-1)) and T-n(l(1),l(2),...,l(n-1)) are not empty when the constants c(k)(k = 1,...,n-1), b(k)(k = 1,2,...,n-1) and l(k)(k = 1,...,n-1) satisfy certain conditions. This paper obtaines the extreme points of P-n(c(1),...,c(n-1)), P-R,P-n(b(1),...,b(n-1)) and T-n(l(1),...,l(n-1)).

THE SUPPORT POINTS AND EXTREME POINTS OF THE UNIT BALL OF H'_P~ 1

By the author denotes the areal measure on the unit disk . Let H'p = {f(z): f(z) is analytic in D and . Let B H 'p and. This article researches the support points and extreme points of B(H'p).

EXTREME POINTS IN DIAGONAL-DISJOINT IDEALS OF NEST ALGEBRAS

In this paper, the extreme points of the unit ball of diagonal-disjoint ideals L in nest algebras are characterized completely; Furthermore, it is shown that every extreme point of the unit ball of L has essential-norm one.

Extreme points of norm closed nest algebra modules

Suppose that U is a norm closed nest algebra module. Using the characterization of rank one operators in U⊥, a complete description of the extreme points of the unit ball U1 is given.

THREE KINDS OF DENTABILITIES IN BANACH SPACES AND THEIR APPLICATIONS

In this paper,we study some dentabilities in Banach spaces which are closely related to the famous Radon-Nikodym property.We introduce the concepts of the weak^(*)-weak denting point and the weak^(*)-weak^(*)denting point of a set.These are the generalizations of the weak^(*)denting point of a set in a dual Banach space.By use of the weak^(*)-weak denting point,we characterize the very smooth space,the point of weak^(*)-weak continuity,and the extreme point of a unit ball in a dual Banach space.Meanwhile,we also characterize an approximatively weak compact Chebyshev set in dual Banach spaces.Moreover,we define the nearly weak dentability in Banach spaces,which is a generalization of near dentability.We prove the necessary and sufficient conditions of the reflexivity by nearly weak dentability.We also obtain that nearly weak dentability is equivalent to both the approximatively weak compactness of Banach spaces and the w-strong proximinality of every closed convex subset of Banach spaces.

A new algorithm for computing the convex hull of a planar point set

When the edges of a convex polygon are traversed along one direction,the interior of the convex polygon is always on the same side of the edges. Based on this characteristic of convex polygons,a new algorithm for computing the convex hull of a simple polygon is proposed in this paper,which is then extended to a new algorithm for computing the convex hull of a planar point set. First,the extreme points of the planar point set are found,and the subsets of point candidate for vertex of the convex hull between extreme points are obtained. Then,the ordered convex hull point sequences between extreme points are constructed separately and concatenated by removing redundant extreme points to get the convex hull. The time complexity of the new planar convex hull algorithm is O(nlogh) ,which is equal to the time complexity of the best output-sensitive planar convex hull algorithms. Compared with the algorithm having the same complexity,the new algorithm is much faster.

Multi-point Contact of the High-speed Vehicle-turnout System Dynamics

The wheel-rail contact problems, such as the number, location and the track of contact patches, are very important for optimizing the spatial structure of the rails and lowering the vehicle-turnout system dynamics. However, the above problems are not well solved currently because of having the difficulties in how to determine the multi-contact, to preciously present the changeable profiles of the rails and to establish an accurate spatial turnout system dynamics model. Based on a high-speed vehicle-turnout coupled model in which the track is modeled as flexible with rails and sleepers represented by beams, the line tracing extreme point method is introduced to investigate the wheel-rail multiple contact conditions and the key sections of the blade rail, longer nose rail, shorter rail in the switch and nose rail area are discretized to represent the varying profiles of rails in the turnout. The dynamic interaction between the vehicle and turnout is simulated for cases of the vehicle divergently passing the turnout and the multi-point contact is obtained. The tracks of the contact patches on the top of the rails are presented and the wheel-rail impact forces are offered in comparison with the contact patches transference on the rails. The numerical simulation results indicate that the length of two-point contact occurrence of a worn wheel profile and rails is longer than that of the new wheel profile and rails; The two-point contact definitely occurs in the switch and crossing area. Generally, three-point contact doesn’t occur for the new rail profile, which is testified by the wheel-rails interpolation distance and the first order derivative function of the tracing line extreme points. The presented research is not only helpful to optimize the structure of the turnout, but also useful to lower the dynamics of the high speed vehicle-turnout system.

q–differ-integral operator on p–valent functions associated with operator on Hilbert space

Making use of multivalent functions with negative coefficients of the type f (z)=z^(p)-~(∑)_(k=p+1)^(∞)a_(k)z^(k),which are analytic in the open unit disk and applying the q-derivative a q–differintegral operator is considered.Furthermore by using the familiar Riesz-Dunford integral,a linear operator on Hilbert space H is introduced.A new subclass of p-valent functions related to an operator on H is defined.Coefficient estimate,distortion bound and extreme points are obtained.The convolution-preserving property is also investigated.

A Solution to the 3x + 1 Problem

The author makes the claim to have a solution for the famous 3x + 1 problem. The key to its solution is a special proof that the term (3 + ε)r is a non-integer, as well as the use of properties of extremal points of a Collatz sequence.

Extreme Points and Strongly Extreme Points of Musielak-Orlicz Sequences Spaces

Criteria for extreme points and strongly extreme points in Musielak-Orliczsequence spaces, equipped with both the Luxemburg norm and the Orlicz norm, are given.

Rotundity of Orlicz-Bochner Space with Luxemburg Norm

In this paper, by using the generating function Φ and X the criteria of the rotundity for the Orlicz-Bochner function and sequence spaces endowed with the Luxemburg norm were obtained. And the sufficient and necessary conditions were given for the extreme point of Orlicz-Bochner sequence space and the conditions in part for Orlicz-Bochner function space.

A New Class of Harmonic Univalent Functions by the Generalized Salagean Operator

A complex-valued harmonic functions that are univalent and sense preserving in the unit disk U can be written in the form f = h ＋ g^-, where h and g are analytic in U. We define and investigate a new class SHPλ（α,β）by generalized Salagean operator of harmonic univalent functions. We give sufficient coefficient conditions for normalized harmonic functions in the class SHPλ（α,β） These conditions are also shown to be necessary when the coefficients are negative. This leads to distortion bounds and extreme points.

A New Class of Harmonic Univalent Functions

A complex-valued harmonic function that is univalent and sense preserving in the unit disk U can be written in the form of f = h + g,where h and g are analytic in U.We define and investigate a new class LH_λ(α,β) by generalized Salagean operator of harmonic univalent functions.We give sufficient coefficient conditions for normalized harmonic functions in the class LH_λ(α,β).These conditions are also shown to be necessary when the coefficients are negative.This leads to distortion bounds and extreme points.

THE ISOMETRIC EXTENSION OF AN INTO MAPPING FROM THE UNIT SPHERE S[e(Γ)] TO THE UNIT SPHERE S(E)

This is such a article to consider an ＂into＂ isometric mapping between two unit spheres of two infinite dimensional spaces of different types. In this article, we find a useful condition （using the Krein-Milman property） under which an into-isometric mapping from the unit sphere of e（Γ） to the unit sphere of a normed space E can be linearly isometric extended.

SOME PROPERTIES FOR CERTAIN CLASSES OF UNIVALENT FUNCTIONS DEFINED BY DIFFERENTIAL INEQUALITIES

Let A be the space of functions analytic in the unit disk D = {z：｜z｜ 〈 1}.Let U denote the set of all functions f ∈ A satisfying the conditions f（0） = f＇（0）-1 = 0 and｜f＇（z）（z/f（z））2-1｜〈1（｜z｜〈1）.Also,let Ω denote the set of all functions f ∈ A satisfying the conditions f（0） = f＇（0）-1 = 0and｜zf＇（z）-f（z）｜〈1/2（｜z｜〈1）.In this article,we discuss the properties of U and Ω.

Some Properties for New Subclass of Analytic Functions

A new subclass A(β1 , β2 , ··· , βk ; λ) of analytic functions f(z) in the open unit disk U is introduced and studied. We provide coefficient inequalities, distortion theorems, extreme points and radius of close-to-convexity, starlikeness and convexity of this class.

Automatic DEXP method derived from Euler’s Homogeneity equation

The depth from extreme points(DEXP)method can be used for estimating source depths and providing a rough image as a starting model for inversion.However,the application of the DEXP method is limited by the lack of prior information regarding the structural index.Herein,we describe an automatic DEXP method derived from Euler’s Homogeneity equation,and we call it the Euler–DEXP method.We prove that its scaling field is independent of structural indices,and the scaling exponent is a constant for any potential field or its derivative.Therefore,we can simultaneously estimate source depths with diff erent geometries in one DEXP image.The implementation of the Euler–DEXP method is fully automatic.The structural index can be subsequently determined by utilizing the estimated depth.This method has been tested using synthetic cases with single and multiple sources.All estimated solutions are in accordance with theoretical source parameters.We demonstrate the practicability of the Euler–DEXP method with the gravity field data of the Hastings Salt Dome.The results ultimately represent a better understanding of the geometry and depth of the salt dome.