Strong convergence theorems obtained by a generalized projections hybrid method for families of mappings in Banach spaces

Let C be a nonempty, closed and convex subset of a uniformly convex and smooth Banach space and let {T_n} be a family of mappings of C into itself such that the set of all common fixed points of {T_n} is nonempty. We consider a sequence {x_n} generated by the hybrid method by generalized projection in mathematical programming. We give conditions on {T_n} under which {x_n} converges strongly to a common fixed point of {T_n} and generalize the results given in [12], [14], [13] and [11].     

Farkas-type results for fractional programming problems

Considering a constrained fractional programming problem, within the present paper we present some necessary and sufficient conditions, which ensure that the optimal objective value of the considered problem is greater than or equal to a given real constant. The desired results are obtained using the Fenchel–Lagrange duality approach applied to an optimization problem with convex or difference of convex (DC) objective functions and finitely many convex constraints. These are obtained from the initial fractional programming problem using an idea due to Dinkelbach. We also show that our general results encompass as special cases some recently obtained Farkas-type results.     

Path-closed sets

Given a digraphG = (V, E), call a node setT ⊆V path-closed ifv, v′ εT andw εV is on a path fromv tov′ impliesw εT. IfG is the comparability graph of a posetP, the path-closed sets ofG are the convex sets ofP. We characterize the convex hull of (the incidence vectors of) all path-closed sets ofG and its antiblocking polyhedron inR ^v , using lattice polyhedra, and give a minmax theorem on partitioning a given subset ofV into path-closed sets. We then derive good algorithms for the linear programs associated to the convex hull, solving the problem of finding a path-closed set of maximum weight sum, and prove another min-max result closely resembling Dilworth’s theorem.     

Fixed point optimization algorithm and its application to network bandwidth allocation

A convex optimization problem for a strictly convex objective function over the fixed point set of a nonexpansive mapping includes a network bandwidth allocation problem, which is one of the central issues in modern communication networks. We devised an iterative algorithm, called a fixed point optimization algorithm, for solving the convex optimization problem and conducted a convergence analysis on the algorithm. The analysis guarantees that the algorithm, with slowly diminishing step-size sequences, weakly converges to a unique solution to the problem. Moreover, we apply the proposed algorithm to a network bandwidth allocation problem and show its effectiveness.     

On The Span Of A Random Channel Assignment Problem

In the radio channel assignment problems considered here, we must assign a ‘channel’ from the set 1,2,... of positive integers to each of n transmitters, and we wish to minimise the span of channels used, subject to the assignment leading to an acceptable level of interference. A standard form of this problem is the ‘constraint matrix’ model. The simplest case of this model (the 0, 1 case) is essentially graph colouring. We consider here a random model for the next simplest case (with lengths 0, 1 or 2), and determine the asymptotic behaviour of the span of channels needed as n→∞. We find that there is a ‘phase change’ in this behaviour, depending on the probabilities for the different lengths.     

The polytope of win vectors

Imagine a graph as representing a fixture list with vertices corresponding to teams, and the number of edges joiningu andv as representing the number of games in whichu andv have to play each other. Each game ends in a win, loss, or tie and we say a vector =(w ₁,...,w _n) is awin vector if it represents the possible outcomes of the games, withw _i denoting the total number of games won by teami. We study combinatorial and geometric properties of the set of win vectors and, in particular, we consider the problem of counting them. We construct a fully polynomial randomized approximation scheme for their number in dense graphs. To do this we prove that the convex hull of the set of win vectors ofG forms an integral polymatroid and then use volume approximation techniques. Supported by the “DAAD Doktorandenstipendium des zweiten Hochschulsonderprogrammes HSPII/AUFE”. Partially supported by RAND-REC EC US030.     

Relations between stochastic and parametric programming for decision problems with a random objective function

In this paper we study the relation between the general concept for an optimal solution for stochastic programming problems with a random objective function-the concept of an £-efficient solution-and the associated parametric problem, We show that it is possible under certain assumptions to obtain some or even all £-efficient solutions of the stochastic problem by solving the parametric problem with respect to a certain parameter set.     

On the symmetry function of a convex set

We attempt a broad exploration of properties and connections between the symmetry function of a convex set S ${S \subset\mathbb{R}^n}We attempt a broad exploration of properties and connections between the symmetry function of a convex set S and other arenas of convexity including convex functions, convex geometry, probability theory on convex sets, and computational complexity. Given a point , let sym(x,S) denote the symmetry value of x in S: , which essentially measures how symmetric S is about the point x, and define x ^* is called a symmetry point of S if x ^* achieves the above maximum. The set S is a symmetric set if sym (S)=1. There are many important properties of symmetric convex sets; herein we explore how these properties extend as a function of sym (S) and/or sym (x,S). By accounting for the role of the symmetry function, we reduce the dependence of many mathematical results on the strong assumption that S is symmetric, and we are able to capture and otherwise quantify many of the ways that the symmetry function influences properties of convex sets and functions. The results in this paper include functional properties of sym (x,S), relations with several convex geometry quantities such as volume, distance, and cross-ratio distance, as well as set approximation results, including a refinement of the L?wner-John rounding theorems, and applications of symmetry to probability theory on convex sets. We provide a characterization of symmetry points x ^* for general convex sets. Finally, in the polyhedral case, we show how to efficiently compute sym(S) and a symmetry point x ^* using linear programming. The paper also contains discussions of open questions as well as unproved conjectures regarding the symmetry function and its connection to other areas of convexity theory. Dedicated to Clovis Gonzaga on the occasion of his 60th birthday.     

Concavity and efficient points of discrete distributions in probabilistic programming

We consider stochastic programming problems with probabilistic constraints involving integer-valued random variables. The concept of a p-efficient point of a probability distribution is used to derive various equivalent problem formulations. Next we introduce the concept of r-concave discrete probability distributions and analyse its relevance for problems under consideration. These notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables. The results are illustrated with numerical examples. Received: October 1998 / Accepted: June 2000?Published online October 18, 2000     

A smoothing Newton method for a type of inverse semi-definite quadratic programming problem

We consider an inverse problem arising from the semi-definite quadratic programming (SDQP) problem. We represent this problem as a cone-constrained minimization problem and its dual (denoted ISDQD) is a semismoothly differentiable (SC¹${\text{SC}}^1$) convex programming problem with fewer variables than the original one. The Karush–Kuhn–Tucker conditions of the dual problem (ISDQD) can be formulated as a system of semismooth equations which involves the projection onto the cone of positive semi-definite matrices. A smoothing Newton method is given for getting a Karush–Kuhn–Tucker point of ISDQD. The proposed method needs to compute the directional derivative of the smoothing projector at the corresponding point and to solve one linear system per iteration. The quadratic convergence of the smoothing Newton method is proved under a suitable condition. Numerical experiments are reported to show that the smoothing Newton method is very effective for solving this type of inverse quadratic programming problems.     

The Number Of Unique-Sink Orientations of the Hypercube*

Let Q_d denote the graph of the d-dimensional cube. A unique-sink orientation (USO) is an orientation of Q_d such that every face of Q_n has exactly one sink (vertex of out degree 0); it does not have to be acyclic. USO have been studied as an abstract model for many geometric optimization problems, such as linear programming, finding the smallest enclosing ball of a given point set, certain classes of convex programming, and certain linear complementarity problems. It is shown that the number of USO is . * This research was partially supported by ETH Zürichan d done in part during the workshop “Towards the Peak” in La Claustra, Switzerland and during a visit to ETH Zürich.     

The signless Laplacian spectral radius of tricyclic graphs and trees with k pendant vertices

In this paper, we consider the following problem: of all tricyclic graphs or trees of order n with k pendant vertices (n,k fixed), which achieves the maximal signless Laplacian spectral radius?We determine the graph with the largest signless Laplacian spectral radius among all tricyclic graphs with n vertices and k pendant vertices. Then we show that the maximal signless Laplacian spectral radius among all trees of order n with k pendant vertices is obtained uniquely at T_n,k, where T_n,k is a tree obtained from a star K_1,k and k paths of almost equal lengths by joining each pendant vertex to one end-vertex of one path. We also discuss the signless Laplacian spectral radius of T_n,k and give some results.     

Fcfs-scheduling in a hard real-time environment under rush-hour conditions

We investigate some real-time behaviour of a (discrete time) single server system with FCFS (first come first serve) task scheduling under rush-hour conditions. The main result deals with the probability distribution of a random variable SRD(T), which describes the time the system operates without violating a fixed task service time deadlineT.Relying on a simple general probability model, asymptotic formulas concerning the mean and the variance of SRD(T) are determined; for instance, if the average arrival rate is larger than the departure rate, the expectation of SRD(T) is proved to fulfilE[SRD(T)]=c ₁+O(T ^–3) forT, wherec ₁ denotes some constant. If the arrival rate equals the departure rate, we findE[SRD(T)]c ₂ T ⁱ for somei2.     

Solving linear programs from sign patterns

This paper is an attempt to provide a connection between qualitative matrix theory and linear programming. A linear program is said to be sign-solvable if the set of sign patterns of the optimal solutions is uniquely determined by the sign patterns of A, b, and c. It turns out to be NP-hard to decide whether a given linear program is sign-solvable or not. We then introduce a class of sign-solvable linear programs in terms of totally sign-nonsingular matrices, which can be recognized in polynomial time. For a linear program in this class, we devise an efficient combinatorial algorithm to obtain the sign pattern of an optimal solution from the sign patterns of A, b, and c. The algorithm runs in O(mγ) time, where m is the number of rows of A and γ is the number of all nonzero entries in A, b, and c.     

A characterization of Delsarte’s linear programming bound as a ratio bound

It is well known that the ratio bound is an upper bound on the stability number α(G) of a regular graph G. In this note it is proved that, if G is a graph whose edge is a union of classes of a symmetric association scheme, the Delsarte’s linear programming bound can alternatively be stated as the minimum of a set of ratio bounds. This result follows from a recently established relationship between a set of convex quadratic bounds on α(G) and the number ?′(G), a well known variant of the Lovász theta number, which was introduced independently by Schrijver [A. Schrijver, A comparison of the Delsarte and Lovász bounds, IEEE Trans. Inform. Theory 25 (1979) 425-429] and McEliece et al. [R.J. McEliece, E.R. Rodemich, H.C. Rumsey Jr, The Lovász bound and some generalizations, J. Combin. Inform. System Sci. 3 (1978) 134-152].     

Partially finite convex programming,Part II: Explicit lattice models

In Part I of this work we derived a duality theorem for partially finite convex programs, problems for which the standard Slater condition fails almost invariably. Our result depended on a constraint qualification involving the notion ofquasi relative interior. The derivation of the primal solution from a dual solution depended on the differentiability of the dual objective function: the differentiability of various convex functions in lattices was considered at the end of Part I. In Part II we shall apply our results to a number of more concrete problems, including variants of semi-infinite linear programming,L ¹ approximation, constrained approximation and interpolation, spectral estimation, semi-infinite transportation problems and the generalized market area problem of Lowe and Hurter (1976). As in Part I, we shall use lattice notation extensively, but, as we illustrated there, in concrete examples lattice-theoretic ideas can be avoided, if preferred, by direct calculation.     

Farkas-type Results for Max-functions and Applications

We present some Farkas-type results for inequality systems involving finitely many convex constraints as well as convex max-functions. Therefore we use the dual of a minmax optimization problem. The main theorem and its consequences allows us to establish, as particular instances, some set containment characterizations and to rediscover two famous theorems of the alternative.     

Convex solution of a permutation problem

In this paper, we show that a problem of finding a permuted version of k vectors from R^N such that they belong to a prescribed rank r subset, can be solved by convex optimization. We prove that under certain generic conditions, the wanted permutation matrix is unique in the convex set of doubly-stochastic matrices. In particular, this implies a solution of the classical correspondence problem of finding a permutation that transforms one collection of points in R^k into the another one. Solutions to these problems have a wide set of applications in Engineering and Computer Science.