Convex Sequence Research Articles

Stability of the geometric Ekeland variational principle: Convex case

In geometric terms, the Ekeland variational principle says that a lower-bounded proper lower-semicontinuous functionf defined on a Banach spaceX has a point (x 0,f(x 0)) in its graph that is maximal in the epigraph off with respect to the cone order determined by the convex coneK ? = {(x, ?) ?X × ?:? ?x? ≤ ? ?}, where ? is a fixed positive scalar. In this case, we write (x 0,f(x 0))??-extf. Here, we investigate the following question: if (x 0,f(x 0))??-extf, wheref is a convex function, and if ?f n ? is a sequence of convex functions convergent tof in some sense, can (x 0,f(x 0)) be recovered as a limit of a sequence of points taken from ?-extf n ? The convergence notions that we consider are the bounded Hausdorff convergence, Mosco convergence, and slice convergence, a new convergence notion that agrees with the Mosco convergence in the reflexive setting, but which, unlike the Mosco convergence, behaves well without reflexivity.

Turbulent boundary layers subjected to multiple curvatures and pressure gradients

The effects of abruptly applied cycles of curvatures and pressure gradients on turbulent boundary layers are examined experimentally. Two two-dimensional curved test surfaces are considered: one has a sequence of concave and convex longitudinal surface curvatures and the other has a sequence of convex and concave curvatures. The choice of the curvature sequences were motivated by a desire to study the asymmetric response of turbulent boundary layers to convex and concave curvatures. The relaxation of a boundary layer from the effects of these two opposite sequences has been compared. The effect of the accompanying sequences of pressure gradient has also been examined but the effect of curvature dominates. The growth of internal layers at the curvature junctions have been studied. Measurements of the Górtler and corner vortex systems have been made. The boundary layer recovering from the sequence of concave to convex curvature has a sustained lower skin friction level than in that recovering from the sequence of convex to concave curvature. The amplification and suppression of turbulence due to the curvature sequences have also been studied.

A hybrid optimal synthesis method for truss structures considering shape, material and sizing variables

AbstractA generalized hybrid optimal synthesis method is developed for truss structures in which the design variables are the co‐ordinates of the panel points, cross‐sectional areas and discrete material kinds of member elements. The primary design problem is formulated in terms of the shape, material and sizing variables and approximated to a sequence of convex and separable subproblems by using mixed direct/reciprocal design variables. Each subproblem is solved by a dual method in which the continuous shape and sizing variables are optimized by a Newton‐type algorithm and the discrete material and continuous sizing variables are improved by a discrete sensitivity analysis. Using the proposed method, discrete material kinds, shape and sizing variables can be systematically improved to obtain an optimum solution. The rigourness, reliability and efficiency of the method are demonstrated by applying it to the minimum cost design of trusses subjected to stress and displacement constraints.

An extension of Attouch’s theorem and its application to second-order epi-differentiation of convexly composite functions

In 1977, Hedy Attouch established that a sequence of (closed proper) convex functions epi-converges to a convex function if and only if the graphs of the subdifferentials converge (in the Mosco sense) to the subdifferential of the limiting function and (roughly speaking) there is a condition that fixes the constant of integration. We show that the theorem is valid if instead one considers functions that are the composition of a closed proper convex function with a twice continuously differentiable mapping (in addition a constraint qualification is imposed). Using Attouch’s Theorem, Rockafellar showed that second-order epi-differentiation of a convex function and proto-differentiability of the subdifferential set-valued mapping are equivalent, moreover the subdifferential of one-half the second-order epi-derivative is the proto-derivative of the subdifferential mapping; we will extend this result to the convexly composite setting.

Conjugate convex functions and the epi-distance topology

Let $\Gamma (X)$ denote the proper, lower semicontinuous, convex functions on a normed linear space, and let ${\Gamma ^ * }({X^ * })$ denote the proper, weak*-lower semicontinuous, convex functions on the dual ${X^ * }$ of $X$. It is well-known that the Young-Fenchel transform (conjugate operator) is bicontinuous when $X$ is reflexive and both $\Gamma (X)$ and ${\Gamma ^ * }({X^ * })$ are equipped with the topology of Mosco convergence. We show that without reflexivity, the transform is bicontinuous, provided we equip both $\Gamma (X)$ and ${\Gamma ^ * }({X^ * })$ with the (metrizable) epi-distance topology of Attouch and Wets. Convergence of a sequence of convex functions $\left \langle {{f_n}} \right \rangle$ to $f$ in this topology means uniform convergence on bounded subsets of the associated sequence of distance functional $\left \langle {d( \cdot ,{\text {epi}}{f_n})} \right \rangle$ to $d( \cdot ,{\text {epi}}f)$.

Convexity properties of the overflow in an ordered-entry system with heterogeneous servers

The overflow probability in an Erlang loss system is known to be decreasing convex in the number of servers. Here we consider the GI/M/m loss system with ordered entry and heterogeneous servers. We show that adding a sequence of servers with non-increasing (non-decreasing) service rates will yield a decreasing convex (log-concave) sequence of overflow probabilities. An optimal server allocation problem is solved as a direct application of these results.

Degree theoretic methods in optimal control

Publisher Summary This chapter focuses on degree theoretic methods in optimal control and presents some problems of optimal control and presents an analysis of a coincidence degree for (L, N). Though the notion of coincidence degree has been extended to multivalued functions by Pruszko and Tarafdar and Suat Khoh Teo, the ill-definition ∂l creates some difficulties because it is usually required that N(x, p) be nonempty for all (x, p) in some open bounded set. However, the definition of coincidence degree may be extended to this case by approximating l by a sequence of well-behaved convex functions {ln}. The chapter also focuses on the existence theorem.

Absolute a- summability factors

In thîs paper we prove the following theorem: k THEOREM; Let (> J be a convex sequence sucb tb^t is eonvengent n If there exist a [A > 0 such tbat sequence (nj^**^* is JB, logn,«p, Ijp-bounded then l

A note on the maximum probability property of MLE's in multiparameter exponential families

For multiparameter exponential models a short and direct proof is given that the maximum likelihood estimator is a maximum probability estimator with respect to a certain sequence of convex and bounded sets inR(k) that are symmetric about the origin; asymptotically these sets are allowed to be unbounded.

Limit distributions for the maxima of stationary Gaussian processes

Let { X n } be a stationary Gaussian sequence with E{ X 0} = 0, { X 2 0} = 1 and E{ X 0 X n } = r n n Let c n = (2ln n) built1 2 , b n = c n − 1 2 c -1 n ln(4π ln n), and set M n = max 0 ⩽ k⩽ n X k . A classical result for independent normal random variables is that P[c n(M n−b n)⩽x]→exp[-e -x] as n → ∞ for all x. Berman has shown that (1) applies as well to dependent sequences provided r n ln n = o(1). Suppose now that { r n } is a convex correlation sequence satisfying r n = o(1), ( r n ln n) -1 is monotone for large n and o(1). Then P[r n -1 2 (M n − (1−r n) 1 2 b n)⩽x] → Ф(x) for all x, where Ф is the normal distribution function. While the normal can thus be viewed as a second natural limit distribution for { M n }, there are others. In particular, the limit distribution is given below when r n is (sufficiently close to) γ/ln n. We further exhibit a collection of limit distributions which can arise when r n decays to zero in a nonsmooth manner. Continuous parameter Gaussian processes are also considered. A modified version of (1) has been given by Pickands for some continuous processes which possess sufficient asymptotic independence properties. Under a weaker form of asymptotic independence, we obtain a version of (2).

Variation norm convergence of function sequences

We prove that a pointwise convergent sequence of convex functions with a continuous limit converges with respect to the total variation norm. This yields a theorem on convexity-preserving operators which has as a corollary the result that a complex function f f is absolutely continuous on [ 0 , 1 ] [0,1] if and only if the sequence B . ( f ) B.(f) of Bernstein polynomials of f f converges to f f with respect to the total variation norm.

The serial correlation coefficients of waiting times in a stationary single server queue

SummaryIn a stationaryGI/G/1 queueing system in which the waiting time variance is finite, it can be shown that the serial correlation coefficients {ρn} of a (stationary) sequence of waiting times are non-negative and decrease monotonically to zero. By means of renewal theory we find a representation for Σ∞0ρnfrom which necessary and sufficient condition for its finiteness can be found. InM/G/1 rather more can be said: {ρn} is convex sequence, the asymptotic form of ρncan be given in a nearly saturated queue, and a simple explicit expression for Σ∞0ρnexists. For the stationaryM/M/1queue we find the ρn's explicitly, illustrate them numerically, and derive a representation which shows that {ρn} is completely monotonic.

Remarks on Fourier series

We prove the following propositions. An even integrable function whose Fourier coefficients form a convex sequence is absolutely continuous if and only if its Fourier series converges absolutely. If the function f(t)is convex on [0, π],f(t)=f(π—t), then for odd n while for even n, b0=0.

Matrix summability of convex sequences

3 3, . In this paper we obtain three conditions which are necessary and sufficient for a matrix to sum every convergent convex sequence. The first two conditions are (1) and (2) above, while the third condition is a weakened version of (3)'. Matrices considered here have complex elements even though convex sequences are necessarily real sequences. The following lemma embodies well-known properties of convex sequences which we will need. LEMMA. If {,up } is a bounded convex sequence, then

Absolute Q-Summability Factors

In thîs paper we prove the following theorem: k THEOREM; Let (> J be a convex sequence sucb tb^t is eonvengent n If there exist a [A > 0 such tbat sequence (nj^**^* is JB, logn,«p, Ijp-bounded then l

On the absolute Cesàro summability factors of infinite series

The author has established that if {λn| is a convex sequence such that the series\(\sum {\frac{{\lambda _n }}{n}} \) is convergent and if Σan is bounded [R, logn, 1] with indexk, then\(\sum {a_n \lambda _n } \) is summable |C, 1|k fork>1. The casek=1 of the theorem is due to Pati [3].

On the summability factors of infinite series

The author has established that if [λn] is a convex sequence such that the series Σn -1λn is convergent and the sequence {K n} satisfies the condition |K n|=O[log(n+1)]k(C, 1),k⩾0, whereK n denotes the (R, logn, 1) mean of the sequence {n log (n+1)a n}, then the series Σlog(n+1)1-kλn a n is summable |R, logn, 1|. The result obtained for the particular casek=0 generalises a previous result of the author [1].