Showing papers on "Extreme point published in 1995"

Some geometric results in semidefinite programming

Abstract: The purpose of this paper is to develop certain geometric results concerning the feasible regions of Semidefinite Programs, called hereSpectrahedra. We first develop a characterization for the faces of spectrahedra. More specifically, given a pointx in a spectrahedron, we derive an expression for the minimal face containingx. Among other things, this is shown to yield characterizations for extreme points and extreme rays of spectrahedra. We then introduce the notion of an algebraic polar of a spectrahedron, and present its relation to the usual geometric polar.

Solving bicriteria solid transportation problem with fuzzy numbers by a genetic algorithm

Abstract: The proposed Genetic Algorithms (GAs) are incorporated with problem-specific knowledge and the decision maker's degree of optimism, and the criteria space approach for bicriteria linear program conductive to find out the nondominated extreme points in the criteria space.

Smooth norms that depend locally on finitely many coordinates

Abstract: We characterize separable normed spaces that admit equivalent COO-smooth norms depending locally on finitely many coordinates. It follows, in particular, that such norms exist on any normed space with countable algebraic basis. We use the method of Talagrand operators developed by Haydon in [8] and certain integral convolution techniques [2], [13] to characterize separable normed spaces that admit C??-smooth norms depending locally on finitely many coordinates as spaces that admit norms with countable boundaries. As corollaries we obtain improvements on some results of Fonf [3] and Vanderwerff [14] and a new simple proof of a result of Haydon on Cw-smooth renormings of spaces of continuous functions on countable compact sets. Our results should be compared to the result of Godun, Lin, and Troyanski in [5] where it is shown that every separable Banach space can be given an equivalent norm, the unit ball of which contains countably many strongly extreme points. From this result and Theorem 1 below it follows that strongly extreme points need not necessarily form a boundary of Bx* . We will use the notation standard in Banach space theory. In particular, Bx and Sx will denote respectively the unit ball and the unit sphere of a Banach space X. We say that 11 11 depends locally on finitely many coordinates if for each x E Sx there exist an open neighbourhood 0 of x, a finite set {x, ...,Xk*} C X*, and a function f: Rk -+ R such that IIYII =f(X1*(y), ...,xk*(y)) fory EO. If (X, 11) is a normed space, the set B c Sx* is called a boundary of (X, * 11) if for each x E Sx there exists x* E B such that x*(x) = 1. Theorem 1. Let (X, 11 * 11) be a normed space. TFAE: (i) X admits an equivalent norm having a countable boundary. (ii) X admits an equivalent norm with a boundary B, such that there is a sequence {Kn}nEN of norm compact sets in X* satisfing B C UnEN KnX Received by the editors October 13, 1993 and, in revised form, June 15, 1994. 1991 Mathematics Subject Classification. Primary 46B03, 46B 10, 46B20. () 1995 American Mathematical Society

No Quadrangulation is Extremely Odd

Abstract: Given a set S of n points in the plane, a quadrangulation of S is a planar subdivision whose vertices are the points of S, whose outer face is the convex hull of S, and every face of the subdivision (except possibly the outer face) is a quadrilateral We show that S admits a quadrangulation if and only if S does not have an odd number of extreme points If S admits a quadrangulation, we present an algorithm that computes a quadrangulation of S in O(n log n) time even in the presence of collinear points If S does not admit a quadrangulation, then our algorithm can quadrangulate S with the addition of one extra point, which is optimal We also provide an \Omega(n \log n) time lower bound for the problem Finally, our results imply that a k-angulation of a set of points can be achieved with the addition of at most k-3 extra points within the same time bound

A geometrical analysis of the efficient outcome set in multiple objective convex programs with linear criterion functions

Abstract: This article performs a geometrical analysis of the efficient outcome setY E of a multiple objective convex program (MLC) with linear criterion functions. The analysis elucidates the facial structure ofY E and of its pre-image, the efficient decision setX E . The results show thatY E often has a significantly-simpler structure thanX E . For instance, although both sets are generally nonconvex and their maximal efficient faces are always in one-to-one correspondence, large numbers of extreme points and faces inX E can map into non-facial subsets of faces inY E , but not vice versa. Simple tests for the efficiency of faces in the decision and outcome sets are derived, and certain types of faces in the decision set are studied that are immune to a common phenomenon called “collapsing”. The results seem to indicate that significant computational benefits may potentially be derived if algorithms for problem (MLC) were to work directly with the outcome set of the problem to find points and faces ofY E , rather than with the decision set.

On the Master Field in Two Dimensions

Abstract: If a master field exists, for large N gauge theories, it is an extreme point of the set of positive linear central functionals on the group algebra of a group of loops. In two dimensions master fields exist. We exhibit it explicitly for the plane.

Extreme points in spaces of continuous functions

Abstract: We study the A-property for the space <£( T, X) of continuous and bounded functions from a topological space T into a strictly convex Banach space X. We prove that the A-property for

Convexity in complex analysis

Abstract: Convexity is the key concept of functional analysis, but apart from some notable exceptions, it has played a relatively minor role in several complex variables theory. The work of Lempert has focused attention on convex domains, and in these two lectures I will present examples involving invariant metrics in complex analysis where convexity, whether realized as in the case of bounded symmetric domains or assumed as in the case of Blp , is essential. Functional analysis brings to problems a variety of developed concepts such as complex extreme points, complex uniform convexity and an approach which is often coordinate and dimension free. I hope to illustrate these points in my lectures. Throughout this article X will denote a Banach space over the complex numbers C, BX will denote the open unit ball in X and BX its closure. We denote by ∆ the open unit disc in C. Our first example relates the maximum modulus theorem of complex analysis with the functional analytic concept of complex extreme point. A point x in X, (‖x‖ = 1), is a complex extreme point (of the unit ball) if ‖x + λy‖ ≤ 1 for all λ ∈ ∆ implies y = 0. The strong maximum modulus theorem, due to Thorp and Whitely in 1965, states the following:

The construction of extreme compositions

Abstract: If a geochemical compositional datasetX (n×p)is a realization of a physical mixing process, then each of its sample (row) vectors will approximately be a convex combination (mixture) of a fixed set of (l×p)extreme compositions termed endmembers. The kpoints in p-space corresponding to a specified set of k (k

Segments in enumerating faces

Abstract: We introduce the concept of a segment of a degenerate convex polytope specified by a system of linear constraints, and explain its importance in developing algorithms for enumerating the faces. Using segments, we describe an algorithm that enumerates all the faces, in time polynomial in their number. The role of segments in the unsolved problem of enumerating the extreme points of a convex polytope specified by a degenerate system of linear constraints, in time polynomial in the number of extreme points, is discussed.

On g-efficiency calculation for polynomial models

Abstract: We study properties of the variance function of the least squares estimator for the response surface. For polynomial models, we identify a class of approximate designs for which their variance functions are maximized at the extreme points of the design space. As an application, we examine robustness properties of D-optimal designs and $D_{n-r}$-optimal designs under various polynomial model assumptions. Analytic formulas for the G-efficiencies of these designs are derived, along with their D-efficiencies.

A Convex Analytic Framework for Ergodic Control of Semi-Markov Processes

Abstract: The ergodic control problem for semi-Markov processes is reformulated as an optimization problem over the set of suitably defined 'ergodic occupation measures.' This set is shown to be closed and convex, with its extreme points corresponding to stationary strategies. This leads to the existence of optimal stationary strategies under additional hypotheses. A pathwise analysis of the joint empirical occupation measures of the state and control processes shows that this optimality is in the strong i.e., almost sure sense.

Restricted simplicial decomposition with side constraints

Abstract: Restricted Simplicial Decomposition with Side Constraints is a price decomposition method designed for large-scale nonlinear problems with a set of linear constraints with special structure and an additional set of linear side constraints. The resultant algorithm iterates by solving a linear subproblem subject to the set of structured constraints and a small nonlinear master problem whose feasible region is defined by the intersection of a simplex and the set of side constraints. The number of variables of the master problem is controlled by a parameter named r. The conditions required for the parameter r and the rules for dropping extreme points of the simplex are established for global convergence of the algorithm. Computational results are presented for nonlinear network problems with side constraints.

Linear programming with positive semi definite matrices

Abstract: We consider the general linear programming problem over the cone of positive semi-definite matrices. We first provide a simple sufficient condition for existence of optimal solutions and absence of a duality gap without requiring existence of a strictly feasible solution. We then simply characterize the analogues of the standard concepts of linear programming, i.e., extreme points, basis, reduced cost, degeneracy, pivoting step as well as a simplex-like algorithm.

On compactness and extreme points of some sets of quasi-measures and measures. IV

Abstract: We give an affine-topological description of the convex sets E (μ) of all quasi-measure extensions of a quasi-measure μ, defined on an algebra of sets, to a larger algebra of sets in the strongly compact case. We also describe that situation by means of some topological properties of E (μ) as well as in measure-theoretic terms.

A class of multilinear uncertain polynomials to which the edge theorem is applicable

Abstract: This paper gives a class of multilinear uncertain polynomials to which the edge theorem is applicable. In the paper, a notion of "symmetric uncertainty structure" is introduced. The main result is as follows: if a multilinear uncertain polynomial has a symmetric uncertainty structure and the uncertain parameters are nonoverlapping, then the corresponding value sets will be convex polygons. This means that the use of the zero exclusion condition to test the robust stability of such uncertain polynomials is feasible since we need only to calculate the extreme points of the convex polygons. Furthermore, a version of the edge theorem can be used to test the robust stability of such uncertain polynomials.

Experimental investigations in combining primal dual interior point method and simplex based LP solvers

Abstract: The use of a primal dual interior point method (PD) based optimizer as a robust linear programming (LP) solver is now well established. Instead of replacing the sparse simplex algorithm (SSX), the PD is increasingly seen as complementing it. The progress of PD iterations is not hindered by the degeneracy or stalling problem of SSX, indeed it reaches the “near optimum” solution very quickly. The SSX algorithm, in contrast, is not affected by the numeral instabilities which slow down the convergence of the PD near the optimal face. If the solution to the LP problem is non-unique, the PD algorithm converges to an interior point of the solution set while the SSX algorithm finds an extreme point solution. To take advantage of the attractive properties of both the PD and the SSX, we have designed a hybrid framework whereby crossover from PD to SSX can take place at any stage of the PD optimization run. The crossover to SSX involves the partition of the PD solution set to active and dormant variables. In this paper we examine the practical difficulties in partitioning the solution set, we discuss the reliability of predicting the solution set partition before optimality is reached and report the results of combining exact and inexact prediction with SSX basis recovery.

The stability of the lexicographic maximin problem of the flow distribution in multiproduct networks

Abstract: The stability oflexicographic flow problems with piecewise-linear criteria and linear constraints is investigated. For polyhedra of general form defined by equality and inequality constraints, sufficient conditions for the continuous dependence of the set of extreme points of a polyhedron on perturbations of the constraint matrix and the vector of the right-hand sides which conserve its non-emptiness are obtained. Allowing for these conditions, semi-continuity from above of the multivalued mappings defined by sets of extreme points which are optimal solutions of these problems is proved for mutually dual linear programming problems. The properties thus proved are used to investigate the flow problem. Using them it is established that the problem is stable with respect to the criteria, and with respect to the solution, and is also stable in relation to computing errors.

On discrepancy theorems with applications to approximation theory

Abstract: We give an overview on discrepancy theorems based on bounds of the logarithmic potential of signed measures. The results generalize well-known results of P. Erdős and P. Turan on the distribution of zeros of polynomials. Besides of new estimates for the zeros of orthogonal polynomials, we give further applications to approximation theory concerning the distribution of Fekete points, extreme points and zeros of polynomials of best uniform approximation.

Minimum permanent of the polytopes determined by a vector majorization

Abstract: Let $\Omega_n$ denote the set of all $n \times n$ doubly stochatic matrices. Then it is well known that $\Omega_n$ forms convex polytope of dimension $(n-1)^2$ with n! extreme points in the $n^2$-dimensional Euclidean space.

Theory and Methodology A bilevel bottleneck programming problem

Abstract: This paper studies a bilevel programming problem with linear constraints, and in which the objective functions at both levels are concave bottleneck functions which are to be minimized. The problem is a non-convex programming problem. It is shown that an optimal solution to the problem is attainable at an extreme point of the underlying region. The outer level objective function values are ranked in increasing order until a value is reached, one of the solutions corresponding to which is feasible for the problem. This solution is then the required global optimal solution.