E.J. Anderson and P. Nash. Linear programming in infinite-dimensional spaces. Wiley, New York, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....projection # of the Mather measure onto . # 3. Linear programming: continuum case. In this section we discuss analogues for Mather s problem (1.2) 1.5) of the foregoing discrete calculations. These involve infinite dimensional linear programming issues, for which the book of Anderson and Nash [A N] provides a good introduction. Our viewpoint is not that infinite dimensional linear programming immediately provides us with useful theorems: For instance there are subtle problems in finding the proper abstract spaces. Rather we take on the more modest task of pointing out that certain known ....

E. J. Anderson and P. Nash, Linear Prog ramming in Infinite Dimensional Spaces, Wiley, 1987.

....the ideal filters using FIR filters and a min max reconstruction error criterion. We formulate the design problem as a semi infinite linear program. Semi infinite formulations have been successfully applied to other multirate filter design problems [32, 33] and solved using standard techniques [34]. Our FIR filter design formulation is fairly general and can be used to design the interpolation filters for those generalized sampling schemes discussed above. The paper is organized as follows. Section II, contains some basic notation and definitions. In Section III we present discrete time ....

....parts. Finding the dual of this real program, and reconverting to the complex form produces the following dual program: s.t. 60) where is a real and positive measure on . The dual problem can be solved using a simplex type algorithm for semi infinite programs [34]. Recall that whenever the technical conditions in Theorem 1 are satisfied, the set X 8 F perfect reconstruction is achieved. is nonempty. However need not be a singleton set, because the perfect reconstruction filter matrices are not necessarily unique. The optimization always produces ....

E. J. Anderson and P. Nash, Linear Programming in Infinite-Dimensional Spaces, John Wiley & Sons, Chichester, 1987.

....is accurate to within a factor of of the correct answer. Hence, the normalized error is bounded by , and the approximation is quite good for moderately large . This is essentially like approximating circles by sided polygons. The optimization (21) can also be solved using semi infinite programming [24]. D. Lower Bounds on , and The choice of sampling pattern that minimizes the optimal constants , and can be a difficult problem. It is therefore useful to know, even before attempting such a design, how small these constants can be made. In this section, we present lower bounds on these error ....

E. J. Anderson and P. Nash, Linear Programming in Infinite-Dimensional Spaces. New York: Wiley, 1987.

....to verifying whether the constraints are feasible or not, i.e. to a feasibility problem. In the remainder of this section we give a brief discussion of various methods for handling the semi infinite constraints. General semi infinite convex optimization is a well developed field; see for example [8, 44, 7, 27, 26, 41, 47]. While the theory and methods for general semi infinite convex optimization can be technical and complex, our feeling is that the very special form of the semi infinite constraints appearing in filter design problems allows them to be solved with no great theoretical or practical difficulty. Let ....

E. J. Anderson and P. Nash. Linear Programming in Infinite- Dimensional Spaces: Theory and Applications. John Wiley Sons, 1987.

....Much of the work on problems of flowover time reduces the problem to a classical flow problem by using an exponentially sized, time expanded graph [4, 18] In this paper, we discuss a special case of flows over time: we assume all transit times are zero. This special case has been considered in [2, 5, 6, 12, 15, 22, 14], among others. Flows over time with zero transit times capture some time related issues: they can be used to model instances when network capacities restrict the quantity of flow that can be sent at any one time. Solving these problems efficiently may help in finding a more efficient exact or ....

....describes an algorithm that finds the minimum average convex cost circulation that repeats indefinitely in a network with general transit times. Our paper and the above mentioned results all look at discrete time problems. Flows over time have also been considered in the continuous time setting [2, 3, 17, 19]. Most of the work in this area has examined networks with time varying edge capacities, storage capacities, or costs. The focus of this research is on proving the existence of optimal solutions for classes of time varying functions, and proving the convergence of algorithms that eventually find ....

E. J. Anderson and P. Nash. Linear Programming in Infinite-Dimensional Spaces. John Wiley & Sons, 1987.

.... val(LDD) # val(LDR) val(LPR) val(LDP ) which together with the assumption val(LDD) val(LDP ) imply that val(LDD) val(LDR) In the presence of a duality gap we can show that val(LDD) val(LDR) val(LPR) val(LDP ) In this case val(LDR) is a subvalue for (LDD) see Anderson and Nash [3]) Moreover the duality gap can be as large as possible. See Anderson and Nash [3] Glasho# and Gustafson [16] and Bonnans and Shapiro [7] for examples of duality gaps in linear semi infinite programming. We now establish a bound on the number of constraints m in (LDR) Theorem 3 Suppose (LDD) ....

....= val(LDP ) imply that val(LDD) val(LDR) In the presence of a duality gap we can show that val(LDD) val(LDR) val(LPR) val(LDP ) In this case val(LDR) is a subvalue for (LDD) see Anderson and Nash [3] Moreover the duality gap can be as large as possible. See Anderson and Nash [3], Glasho# and Gustafson [16] and Bonnans and Shapiro [7] for examples of duality gaps in linear semi infinite programming. We now establish a bound on the number of constraints m in (LDR) Theorem 3 Suppose (LDD) is consistent and that there exists a finite discretization of (LDD) with the same ....

E.J. Anderson and P. Nash, Linear Programming in InfiniteDimensional Spaces, Wiley, New York, 1987.

....objective function for the fluid job shop is Z 1 0 I X i=1 J i X k=1 w i;k x i;k (t)dt: The problem of whether a polynomial time algorithm exists for the fluid control problem is still open. However, based on several structural properties for this class of problems (see Anderson and Nash [1]) Luo Bertsimas [14] based on earlier work by Pullan [18] propose provably convergent discretization based methods that are able to solve large scale instances in practice fast. The algorithm of Luo Bertsimas [14] is used in our computational study. A key property of the fluid job shop ....

....methods that are able to solve large scale instances in practice fast. The algorithm of Luo Bertsimas [14] is used in our computational study. A key property of the fluid job shop problem that we shall make use of extensively is stated as Proposition 1; its proof can be found in Anderson Nash [1]. Proposition 1 There exists an optimal solution for the fluid job shop scheduling problem such that x(t) is piecewise linear with a finite number of pieces. Note that by Proposition 1, there is always an optimal fluid solution such that T i;k (t) is piecewise linear, and has a finite number of ....

E. J. Anderson and P. Nash. Linear Programming in Infinite-Dimensional Spaces. John Wiley & Sons, New York, 1987.

....minimize (c T x) 2 d T x subject to Ax b 0 (4) 1 Thus x y denotes componentwise inequality when x and y are vectors, and matrix inequality when x and y are (symmetric) matrices. In this paper, the context will always make it clear which is meant. 2 See however Anderson and Nash [AN87] for simplex like methods in semi infinite linear progamming, and Pataki [Pat95] and Lasserre [Las95] for extensions of the simplex method to semidefinite programming. 2 where we assume that d T x 0 whenever Ax b 0. We start with the standard trick of introducing an auxiliary variable t ....

E. J. Anderson and P. Nash. Linear Programming in Infinite-Dimensional Spaces: Theory and Applications. John Wiley & Sons, 1987.

.... Section 31.5 of [Deza and Laurent 1997] The proof can also be obtained via complementarity conditions for semidefinite programming (cf. Barvinok 1995] and [Alizadeh, Haeberly and Overton 1997] which constitute a particular case of the complementary conditions for general linear programs, see [Anderson and Nash 1987]. Thus Theorem 1.1 is somewhat analogous to the statement that the intersection of an affine subspace of codimension r with a non negative orthant in R n , if non empty, will contain a point where some n Gamma r coordinates are zero (see, for example, Pataki 1998] for a discussion of this ....

E. Anderson and P. Nash, Linear Programming in Infinite-Dimensional Spaces. Theory and Applications, Wiley-Interscience, Chichester, 1987.

....I discuss a special case of dynamic network flow problems: I assume all transit times are zero. Dynamic network flow models with zero transit times lie between general dynamic flow models and traditional networks. They have been considered before, for instance, in the book of Anderson and Nash [1]. They capture some time related issues: dynamic networks without transit times can be used to model instances when network capacities restrict the quantity of flow that can be sent at any one time. In this paper, I describe algorithms to solve the quickest transshipment problem minimize the ....

....network. Unfortunately, this graph may be very large, and is thus not practical to work with for large discretizations. Hoppe and Tardos [8] describe the only polynomial time algorithm to solve the 2 discrete problem. Dynamic network flows have also been considered in the continuous time setting [1, 2, 12, 14]. Here, capacities are upper bounds on the rate of flow through the arcs. Most of the work in this area has examined networks with time varying edge capacities, storage capacities, or costs. The focus of this research is on proving the existance of optimal solutions for classes of time varying ....

E. J. Anderson and P. Nash. Linear Programming in Infinite-Dimensional Spaces. John Wiley & Sons, 1987.

....#f(x) #x i #f(x) G(x, y) n X i=1 #f(x) #x i y i ,x#R n ,y= y 1 , y n)#B. Since g # g we have C g (R n ) #C g (R n ) thus the operator L : W # Y is well defined for all f # W . Having a dual pair (X,Y ) we will always consider the weak topology #(X,Y )onX(see [1] or [26] The same applies for (Z, W ) 612 MICHAEL I. TAKSAR For each M = M,N) # X consider the following linear functional on W : T M (f) # M,Lf## #M,Lf# #N, #f, #. 4.5) It follows from (2.1) 2.2) and the definition of C 2 g (R n ) that L is a bounded linear operator ....

....defined as the infimum of #x, c# (respectively, supremum of #b, w#) over all feasible x (respectively, w) and is denoted by inf(P) respectively, sup(P # ) PROPOSITION 6.1. If (P) and (P # ) are both consistent, then sup (P # ) # inf (P) The proof of this proposition can be found in [1]. DEFINITION. If both (P) and (P # ) are consistent with finite values and sup (P) inf (P # ) then it is said that there is no duality gap. Conditions ensuring absence of a duality gap are given by the following theorem (see [1, Theorems 3.10 and 3.22] THEOREM 6.1. Let D be the subset of Z ....

E.J. ANDERSON AND P. NASH, Linear Programming in Infinite-Dimensional Spaces, John Wiley, Chichester, 1989.

.... To study the uniqueness of an equilibrium solution x, we introduce the infinite dimensional linear program LP obtained by setting X to its current value: LP : min y2 Omega hF (X) ffG; yi: This problem is a parametric continuous linear program of the bottleneck type (see Anderson and Nash [1]) Let Y i , i = 1; N be the extreme points of the compact polyhedron b Omega Gamma and set: F i (X) hF (X) Y i i 2 G i = hG; Y i i 2 ; i = 1; N: 8 P. MARCOTTE AND D. L. ZHU Definition 2.5. An extreme point Y i of b Omega is dominated at X if F i (X) ffG ....

Anderson E. J., and Nash, P., Linear Programming in Infinite Dimensional Spaces, John Wiley and Sons, New York, 1987.

....max Q (k 1 k 2 k 3 ) minf1; fi 1 fi 2 g. The proof is similar to that of case iii) This completes the proof of the theorem. 2 We define two operators Omega and Phi that combine intervals according to Theorem 1. Definition 7 Let [ff 1 ; fi 1 ] and [ff 2 ; fi 2 ] be sub intervals of [0,1]. Define: 1) the operator Omega where [ff 1 ; fi 1 ] Omega [ff 2 ; fi 2 ] maxf0; ff 1 ff 2 Gamma 1g; minffi 1 ; fi 2 g] and 2) the operator Phi where [ff 1 ; fi 1 ] Phi [ff 2 ; fi 2 ] maxfff 1 ; ff 2 g; minf1; fi 1 fi 2 g] 2 The following lemma shows a few properties of Omega and ....

....1 ; fi 1 ] Phi [ff 2 ; fi 2 ] maxfff 1 ; ff 2 g; minf1; fi 1 fi 2 g] 2 The following lemma shows a few properties of Omega and Phi that will be used in later proofs. Lemma 1 Let [ff 1 ; fi 1 ] ff 2 ; fi 2 ] ff 3 ; fi 3 ] ffi 1 ; fl 1 ] and [ffi 2 ; fl 2 ] all be sub intervals of [0,1]. 1) Omega and Phi are commutative, e.g. ff 1 ; fi 1 ] Omega [ff 2 ; fi 2 ] ff 2 ; fi 2 ] Omega [ff 1 ; fi 1 ] 2) Omega and Phi are associative, e.g. ff 1 ; fi 1 ] Omega [ff 2 ; fi 2 ] Omega [ff 3 ; fi 3 ] ff 1 ; fi 1 ] Omega ( ff 2 ; fi 2 ] Omega [ff 3 ; fi 3 ] 3) Omega and ....

[Article contains additional citation context not shown here]

E.J. Anderson and P. Nash. (1987) Linear Programming in Infinite-Dimensional Spaces: Theory and Applications, John Wiley & Sons Ltd.

....dual problem (DLSIP ) Hettich and Kortanek [10] provided a general survey of semi infinite programming in 1993. A more focused review on the numerical methods for semi infinite programming can be found in the recent work of Rememtsen and Gorner [18] Survey shows that the cutting plane method [1, 5, 7] is one of the major techniques used for solving linear, quadratic, and convex semi infinite programming problems. When the cutting plane method is applied to solve a quadratic semi infinite programming problem, basically, it solves a sequence of finite dimensional quadratic programs and shows ....

E. J. Anderson and P. Nash. Linear Programming in Infinite--dimensional Spaces. Wiley, Chichester, 1987.

....minfae 1 ; ae n g for some fixed 0 c 1, and 4) f 4 (ae 1 ; ae 2 ) ae 1 =ae 2 , for ae 2 6= 0 and ae 1 ae 2 . 2 We further assume that all annotation functions f are computable in the sense that there is a fixed procedure P f such that if f is n ary, and if ae 1 ; ae n , all in [0,1], are given as inputs to P f , then f(ae 1 ; ae n ) is computed by P f in a finite amount of time. Apart from the interpreted annotation function symbols, L also contains infinitely many variable symbols. Furthermore, we assume that this set of variable symbols are partitioned into two ....

....The other set consists of annotation variable symbols. Annotation variables can only range between 0 and 1. Annotation variable symbols can only appear in annotation terms, a concept defined as follows. Definition 2 1) ae is called an annotation item if it is one of the following: i) a constant in [0,1], or ii) an annotation variable in L, or iii) of the form f(ffi 1 ; ffi n ) where f is an annotation function of arity n and ffi 1 ; ffi n are annotation items. 2) For c; d such that 0 c; d 1, let the closed interval [c; d] be the set fx j c x d g. 3) The closed interval [ae ....

[Article contains additional citation context not shown here]

E.J. Anderson and P. Nash. (1987) Linear Programming in Infinite-Dimensional Spaces: Theory and Applications, John Wiley & Sons Ltd.

....the problem of finding a maximum flow in a network with zero transit times, but with capacities and node storage that vary over time, and develop a duality theory for continuous network flows. In their book, Anderson and Nash present this result as well as other work on continuous linear programs [1]. Since then, there has been much more work on continuous dynamic flow problems in networks with time varying edge capacities, storage capacities, or costs. For example, see [3, 19, 18, 22, 23, 20] One focus of this research is to extend the class of these time varying functions for which optimal ....

....j( represent node j at time , then a generalized cut C in the network N is a set valued function of , defined by a set of cut points fff j g; 1 j jV j in [0; T ] so that j( 2 C; 8 ff j , with ff s = 0 and ff t = T . This definition is actually more restrictive than the cut defined in [1], but we will show that there is a continuous dynamic flow that saturates such a cut. The capacity of generalized cut C is the sum over all edges of the amount of flow that can cross an edge while the end points are on different sides of the cut. Formally, this is X jk2E Z ff k Gamma jk ff ....

E. J. Anderson and P. Nash. Linear Programming in Infinite-Dimensional Spaces. John Wiley & Sons, 1987.

No context found.

E.J. Anderson and P. Nash. Linear programming in infinite-dimensional spaces. Wiley, New York, 1987.

No context found.

E. J. Anderson and P. Nash. Linear Programming in Infinite-dimensional Spaces. Discrete Mathematics and Computation. Wiley-Interscience, 1987.

No context found.

E. J. Anderson and P. Nash. Linear programming in infinite-dimensional spaces. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons Ltd., 1987. Theory and applications, A WileyInterscience Publication.

No context found.

E. J. Anderson and P. Nash. Linear Programming in Infinite-Dimensional Spaces: Theory and Applications. John Wiley & Sons, 1987.

No context found.

Anderson, E. J. and Nash, P., Linear Programming in Infinite-Dimensional Spaces. Theory and Applications, Wiley, Chichester, 1987.

No context found.

E.J. Anderson and P. Nash, (1987). Linear Programming in InfiniteDimensional Spaces, John Wiley & Sons, New York.

No context found.

E. Anderson and P. Nash. Linear programming in infinite dimensional spaces. John Wiley, New York, 1987.

No context found.

E. Anderson and P. Nash, Linear programming in infinite dimensional spaces, John Wiley, New York, 1987.

No context found.

Anderson and Nash, Linear programming in infinite-dimensional spaces. Great Britain, 1987.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC