Showing papers on "Extreme point published in 2007"

Null controllability of degenerate parabolic operators with drift

Abstract: We give null controllability results for some degenerate parabolic equations in non divergence form with a drift term in one space dimension. In particular, the coefficient of the second order term may degenerate at the extreme points of the space domain. For this purpose, we obtain an observability inequality for the adjoint problem using suitable Carleman estimates.

Extreme points of the set of density matrices with positive partial transpose

Abstract: We present a necessary and sufficient condition for a finite-dimensional density matrix to be an extreme point of the convex set of density matrices with positive partial transpose with respect to a subsystem. We also give an algorithm for finding such extreme points and illustrate this by some examples.

Characterizations of Local and Global Error Bounds for Convex Inequalities in Banach Spaces

Abstract: This paper studies local and global error bounds for a convex inequality defined by a proper convex function in a Banach space. The concept of weak basic constraint qualification (weak BCQ) is introduced to control the normal directions at a boundary point of the solution set. Local and global error bounds are characterized by a direction-length decomposition condition, which provides a way to independently verify the weak BCQ and the length control of the subdifferential. To further characterize global error bounds, the segment extension property is proposed and studied. It is shown that the verification of the direction-length condition for global error bounds can be sufficiently carried out on any subset having the segment extension property instead of the entire boundary. This leads to a simple formula for the smallest global error bound. In the Euclidean space, the verification of the condition and the computation of the smallest global error bound can be carried out on the set of extreme points.

On the Hardness and Smoothed Complexity of Quasi-Concave Minimization

Abstract: In this paper, we resolve, the smoothed and approximative complexity of low-rank quasi-concave minimization, providing both upper and lower bounds. As an upper bound, we provide the first smoothed analysis of quasi-concave, minimization. The analysis is based on a smoothed bound for the number of extreme points of the projection of the feasible polytope onto a k-dimensional subspace. where k is the rank (informally, the dimension of nonconvexity)ofthe quasi-concave function. Our smoothed bound is polynomial in the original dimension of the problem n and the perturbation size p. and it is exponential in the rank of the function k. From this, we obtain the first randomized fully polynomial-time approximation scheme for low-rank quasi-concave minimization under broad conditions. In contrast with this, we prove log n-hardness of approximation for general quasi-concave minimization. This shows that our smoothed bound is essentially tight, in that no polynomial smoothed bound is possible for quasi-concave functions of general rank k. The tools that we introduce for the smoothed analysis may be of independent interest. All previous smoothed analyses of polytopes analyzed projections onto two-dimensional subspaces and studied them using trigonometry to examine the angles between vectors and 2-planes in Ropf". In this paper, we provide what is, to our knowledge, the first smoothed analysis of the projection of polytopes onto higher-dimensional subspaces. To do this, we replace the trigonometry with tools from random matrix theory and differential geometry on the Grassmannian. Our hardness reduction is based on entirely different proofs that may also be of independent interest; we show that the stochastic 2-stage minimum spanning tree problem has a supermodular objective and that supermodular minimization is hard to approximate.

On the L ∞ -norm of extreme points for crossing supermodular directed network LPs

Abstract: We discuss extensions of Jain’s framework for network design [8] that go beyond undirected graphs. The main problem is approximating a minimum cost set of directed edges that covers a crossing supermodular function. We show that iterated rounding gives a factor 3 approximation, where factor 4 was previously known and factor 2 was conjectured. Our bound is tight for the simplest interpretation of iterated rounding. We also show that (the simplest version of) iterated rounding has unbounded approximation ratio when the problem is extended to mixed graphs.

Accelerating convergence of cutting plane algorithms for disjoint bilinear programming

Abstract: This paper presents two linear cutting plane algorithms that refine existing methods for solving disjoint bilinear programs. The main idea is to avoid constructing (expensive) disjunctive facial cuts and to accelerate convergence through a tighter bounding scheme. These linear programming based cutting plane methods search the extreme points and cut off each one found until an exhaustive process concludes that the global minimizer is in hand. In this paper, a lower bounding step is proposed that serves to effectively fathom the remaining feasible region as not containing a global solution, thereby accelerating convergence. This is accomplished by minimizing the convex envelope of the bilinear objective over the feasible region remaining after introduction of cuts. Computational experiments demonstrate that augmenting existing methods by this simple linear programming step is surprisingly effective at identifying global solutions early by recognizing that the remaining region cannot contain an optimal solution. Numerical results for test problems from both the literature and an application area are reported.

Network design arc set with variable upper bounds

Abstract: In this paper we study the network design arc set with variable upper bounds. This set appears as a common substructure of many network design problems and is a relaxation of several fundamental mixed-integer sets studied earlier independently. In particular, the splittable flow arc set, the unsplittable flow arc set, the single node fixed-charge flow set, and the binary knapsack set are facial restrictions of the network design arc set with variable upper bounds. Here we describe families of strong valid inequalities that cut off all fractional extreme points of the continuous relaxation of the network design arc set with variable upper bounds. Interestingly, some of these inequalities are also new even for the aforementioned restrictions studied earlier. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(1), 17–28 2007

Average of two extreme points in JBW*-triples

Abstract: H.Choda proved that every element in the closed unit ball of a von Neumann algebra is average of two extreme points of the ball. Here, we prove the strict generalisation of Choda’s result to arbitrary $JBW^{*}$-triples.

Stochastic equations on compact groups in discrete negative time

Abstract: In this paper a stochastic equation on compact groups in discrete negative time is studied. The diagonal group action on the extreme points of solutions is proved to be transitive by means of the coupling method. This result is applied to generalize Yor's work which is closely related to Tsirelson's stochastic differential equation and to give criteria for existence of a strong solution and for uniqueness in law.

A modification of Graham's algorithm for determining the convex hull of a finite planar set.

Abstract: In this paper, in our modification of Graham scan for determining the convex hull of a finite planar set, we show a restricted area of the examination of points and its advantage. The actual run times of our scan and Graham scan on the set of random points shows that our modified algorithm runs significantly faster than Graham’s one.

On a subclass of β-uniformly convex functions defined by Dziok-Srivastava linear operator

Abstract: In this paper a new subclass of uniformly convex functions with negative coefficients defined by Dziok-Srivastava Linear operator is introduced. Characterization properties exhibited by certain fractional derivative operators of functions and the result of modified Hadmard product are discussed for this class. Further class preserving ntegral operator, extreme points and other interesting properties for this class are also indicated. 2000mathematics Subj. Classification: 30C45, 26A33.

Extreme Points and Strong U-Points in Musielak–Orlicz Sequence Spaces Equipped with the Orlicz Norm

Abstract: We give some criteria for extreme points and strong U-points in Musielak{ Orlicz sequence spaces equipped with the Orlicz norm. It follows from these results that the notion of the strong U-point is essentially stronger than the notion of the extreme point in these spaces. Keywords. Musielak{Orlicz sequence space, extreme point, strong U-point, Orlicz norm. Mathematics Subject Classiflcation (2000). Primary 46B20, 46E30, secondary 46A45

Using Feasible Direction to Find All Alternative Extreme Optimal Points for Linear Programming Problem

Abstract: We presented a feasible direction method to find all optimal extreme points for the linear programming problem. Our method depends on the conjugate gradient projection method starting with an initial point we generate a sequence of feasible directions towards all alternative extremes.

Probabilistic algorithms for extreme point identification

Abstract: For a finite set of distinct points S = {pi, i e I} , in ℝ d there exists I ⊆ I such that all points in Ŝ = {pi , i e I are extreme points and conv(Ŝ) = conv(S). Since a point pk is extreme if and ...

Ambiguous chance constrained programs: algorithms and applications

Abstract: Chance constrained problems are optimization problems where one or more constraints ensure that the probability of one or more events occurring is less than a prescribed threshold. Although it is typically assumed that the distribution defining the chance constraints are known perfectly; in practice this assumption is unwarranted. We study chance constrained problems where the underlying distributions are not completely specified and are assumed to belong to an uncertainty set Q . We call such problems "ambiguous chance constrained problems." We focus primarily on the special case where the uncertainty set Q of the distributions is of the form Q=Q:rp Q,Q0 ≤b , where ρp denotes the Prohorov metric. We study single and two stage ambiguous chance constrained programs. The single stage ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure Q0 . We show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with a high probability. This result is established using the Strassen-Dudley Representation Theorem. We also show that the robust sampled problem can be solved efficiently both in theory and in practice. Nemirovski and Shapiro [61] formulated two-stage convex chance constrained programs and proposed an ellipsoid-like iterative algorithm for the special case where the impact function f(x, h) is bi-affine. We show that this algorithm extends to bi-convex f(x, h ) in a fairly straightforward fashion. The complexity of the solution algorithm as well as the quality of its output are functions of the radius r of the largest Euclidean ball that can be inscribed in the polytope defined by a random set of linear inequalities generated by the algorithm [61]. In this dissertation we provide some guidance for selecting r. We develop an approximation algorithm to two-stage ambiguous chance constrained programs when the impact function f(x, h) is bi-affine and the extreme points of a certain "dual" polytope are known explicitly.

Stability testing of matrix polytopes

Abstract: Motivated by questions in robust control and switched linear dynamical systems, we consider the problem checking whether every element of a polytope of n×n matrices A is stable. We show that this can be done in polynomial-time in n when the number of extreme points of A is constant, but becomes NP-Hard when the number of extreme points grows as Θ(n). This result has two useful corollaries: (i) for the case when A is a line, we give a stability-testing algorithm considerably faster than the best currently known algorithms (ii) we show that verifying the absolute asymptotic stability of a continuous-time switched linear system with n − 1 n × n matrices A i satisfying 0 ≽ A i + A i T is NP-hard.

Extreme points of the convex set of joint probability distributions with fixed marginals

Abstract: By using a quantum probabilistic approach we obtain a description of the extreme points of the convex set of all joint probability distributions on the product of two standard Borel spaces with fixed marginal distributions.

Properties of Harmonic Functions which are Convex of Order β with Respect to Conjugate Points

Abstract: Let H denote the class of functions f which are harmonic and univalent in the open unit disc D = {z : |z| < 1}. This paper defines and investigates a family of complex-valued harmonic functions that are orientation preserving and univalent in D and are related to the functions convex of order β(0 ≤ β< 1), with respect to conjugate points. We obtain coefficient conditions, growth result, extreme points, convolution and convex combinations for the above harmonic functions.

Sharp inequalities for positive functions

Abstract: In this article we determine the extreme points for the families of positive and positive real functions. We prove a number of new sharp inequalities in addition to quantifying and generalizing some well-known results. We also investigate sharp bounds on |p'(z)| and |p′′(z)| for positive real functions, unknown since 1932. †Dedicated to Professor Peter L. Duren on the occasion of his 70th birthday.

Extreme and nice operators on certain function spaces

Abstract: Let X be a Banach space, and L(X) be the space of bounded linear operators from X into X B1(X) denotes the closed unit ball of X, and S1(X )i s unit sphere of X An element T ∈ S1(L(X)) is called extreme operator if there is no A ∈ L(X) such thatT ± A �≤ 1 The set of extreme points of S1(X) will be denoted by ext(S1(X)) T ∈ S1(L(X)) is called nice if T ∗ (ext(S1(L(X ∗ ))) ⊆ ext(S1(L(X ∗ ))) The object of this paper is to give simpler proofs of old results and present new results on extreme operators of S1(L(� p )) We introduce the concept of k-extreme points Further, we characterize the nice operators on most of the classical function and sequence spaces Nice compact operators onp −spaces are characterized I Introduction Let X be a Banach space The closed unit ball of X will be denoted by B1(X) , and the unit sphere by S1(X) An element x ∈ S1(X) is called an extreme point of B1(X) if whenever x = 1 (y + z), with y and z in S1(X), then x = y = z The space of bounded linear operators on X will be denoted by L(X), and the compact ones by K(X) Extreme elements of S1(L(X)) are called extreme operators An operator T ∈ L(X )i s called nice if the set of extreme points of B1(X ∗ ) is an invariant set for T ∗ L p (I), 1 ≤ p< ∞, denotes the Banach space of p-Bochner integrable functions (equiv- alence classes) defined on the unit interval, with the usual classical norm Similarly, � p denotes the Banach space of p-summable sequences, with the usual classical norm The space of continuous functions on a compact set Ω, with the uniform norm is denoted by C(Ω)

An Efficient Algorithm for Capacitated Multifacility Location Problems

Abstract: In this paper, a squared-Euclidean distance multifacility location problem with inseparable demands under balanced transportation constraints is analyzed. Using calculus to project the problem onto the space of allocation variables, the problem becomes minimizing concave quadratic integer programming problem. The algorithm based on extreme point ranking method combining with logical techniques is developed. The numerical experiments are randomly generated to test efficiency of the proposed algorithm compared with a linearization algorithm. The results show that the proposed algorithm provides a better solution on average with less processing time for all various sizes of problems.

A Convex Analysis Based Criterion for Blind Separation of Non-Negative Sources

Abstract: In this paper, we apply convex analysis to the problem of blind source separation (BSS) of non-negative signals. Under realistic assumptions applicable to many real-world problems such as multichannel biomedical imaging, we formulate a new BSS criterion that does not require statistical source independence, a fundamental assumption to many existing BSS approaches. The new criterion guarantees perfect separation (in the absence of noise), by constructing a convex set from the observations and then finding the extreme points of the convex set. Some experimental results are provided to demonstrate the efficacy of the proposed method.

Extreme points and support points of a family of analytic functions with complex coefficients

Abstract: Let Ω={f(z):f(z) is analytic in |z|1 and f(z)=z+sum from n=2 to +∞(an+ibn)zn,an,bn being real numbers,sum from n=2 to +∞ n (a2n+bn2)~(1/2) ≤1}.In this article,the extreme points and support points of Ω are obtained.

New Algorithms for Computing the Convex Hulls of a Simple Polygon and a Planar Point Set

Abstract: Based on one of the characteristics of convex polygons, i.e. 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 these edges, a new algorithm for computing the convex hull of a simple polygon is proposed first, which is then extended to a new algorithm for computing the convex hull of a planar point set. To compute the convex hull of a planar point set, first to find the extreme points of the planar point set and get the sub collections of points candidate for vertices of the convex hull between them. Then the sorted convex hull point arrays 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 best output-sensitive planar convex hull algorithms. Compared with the same time complexity algorithms, the new algorithm is simple and easy to be implemented, and owing to its sequentially computing the vertices on the convex hull, it is easier to get a space-efficient planar convex hull algorithm based on the new planar convex hull algorithm than others.

Index for Evaluating Heart Rate Variability based on the Extreme Point of Time Series

Abstract: In this study, it aimed at the proposal of real time and continuous estimating method of physiological states using biological signals. The proposal method will be one of the basic technology to develop physiological state monitor. The proposal index is defined in the ratio of the number of extreme points of the heart rate time series and the number of heart rate. This index is called NEP. NEP is defined as equ.(1).In this equation, n is the number of total heart beats, and R ( i ) is the i -th R-R interval. The characteristic of the NEP was shown by simulation analysis. The NEP was compared with classical indices of heart rate variability by the analysis of measured heart rate time series. NEP decreased significantly (p<0.05) when posture changed from supine to standing. In this case, the autonomic nervous activity balance changes from the parasympathetic to the sympathetic nerve. Moreover, the NEP was correlated with the respiratory frequency in supine position (p<0.01). It is not necessary to consider individual variation of NEP in the physiological state evaluation. Standardization process or relative value is not necessary to compare individual persons. This parameter is applicable to evaluate physiological state at real time and continuously.