Non-decreasing Convex Function Research Articles

Non-clairvoyantly Scheduling to Minimize Convex Functions

The paper considers scheduling jobs online to minimize the objective $$\sum _{i \in [n]}w_ig(C_i-r_i)$$ , where $$w_i$$ is the weight of job i, $$r_i$$ is its release time, $$C_i$$ is its completion time and g is any non-decreasing convex function. It is known that the clairvoyant algorithm Highest-Density-First (HDF) is $$(2+\epsilon )$$ -speed O(1)-competitive for this objective on a single machine for any fixed $$ 0< \epsilon < 1$$ (Im et al., in: ACM-SIAM symposium on discrete algorithms, pp 1254–1265, 2012). In this paper, we give the first non-trivial results for this problem when g is a non-decreasing convex function and the algorithm must be non-clairvoyant. More specifically, our results include: The paper gives the first non-trivial upper-bound on multiple machines even if the algorithm is allowed to be clairvoyant. All performance guarantees above hold for all non-decreasing convex functions gsimultaneously. The positive results are supplemented by almost matching lower bounds. We show that any algorithm that is oblivious to g is not O(1)-competitive with speed augmentation less than 2 on a single machine. Further, any non-clairvoyent algorithm that knows the function g cannot be O(1)-competitive with speed augmentation less than $$\sqrt{2}$$ on a single machine or $$(2-\frac{1}{m})$$ on m identical machines.

Dynamic job assignment: A column generation approach with an application to surgery allocation

We consider the assignment of jobs to heterogeneous agents in a dynamic system with a rolling time horizon. An example is a hospital operating theatre where the jobs are surgeries and the agents are the surgeons. The paper is presented in the context of surgery allocation and the system is characterized as follows: Patients are grouped into categories and they arrive continually following a stochastic process. Patients in each group have specific time limits within which they need treatment and if it cannot be accommodated then the patients are outsourced. The service level is the percentage of patients in each group treated within the time limit. Surgery durations are stochastic and depend on the surgeon conducting the surgeries. Each surgeon has limited time available and expected overtime is penalized by a non-decreasing convex function. We develop a column generation approach for the assignment of already arrived patients and tentative future patients to surgeons on specific days. It balances the conflicting objectives of including as many arrived patients as possible within their time limits, maximizing the service level of future patients, and minimizing the expected overtime of surgeons. A computational study is conducted with the model embedded in a rolling time horizon frame. The study indicates that the assignment of patients based on our model increases system performance in terms of service level and reduced overtime compared to a First-Come-First-Served (FCFS) policy when the arrival rates of patients are medium to high compared to the capacity of the system.

An isoperimetric inequality in the plane with a log-convex density

Given a positive lower semi-continuous density f on mathbb {R}^2 the weighted volume V_f:=fmathscr {L}^2 is defined on the mathscr {L}^2-measurable sets in mathbb {R}^2. The f-weighted perimeter of a set of finite perimeter E in mathbb {R}^2 is written P_f(E). We study minimisers for the weighted isoperimetric problem If(v):=inf{Pf(E):Eis a set of finite perimeter inR2andVf(E)=v}\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} I_f(v):=\\inf \\Big \\{ P_f(E):E\\text { is a set of finite perimeter in }\\mathbb {R}^2\\text { and }V_f(E)=v\\Big \\} \\end{aligned}$$\\end{document}for v>0. Suppose f takes the form f:mathbb {R}^2rightarrow (0,+infty );xmapsto e^{h(|x|)} where h:[0,+infty )rightarrow mathbb {R} is a non-decreasing convex function. Let v>0 and B a centred ball in mathbb {R}^2 with V_f(B)=v. We show that B is a minimiser for the above variational problem and obtain a uniqueness result.

Generalizations of the Aluthge transform of operators

Let A be an operator with the polar decomposition A = U|A|. The Aluthge transform of the operator A, denoted by ?, is defined as ? = |A|1/2U |A|1/2. In this paper, first we generalize the definition of Aluthge transformfor non-negative continuous functions f,g such that f(x)g(x) = x (x ? 0). Then, by using this definition, we get some numerical radius inequalities. Among other inequalities, it is shown that if A is bounded linear operator on a complex Hilbert space H, then h (w(A)) ? 1/4||h(g2 (|A|)) + h(f2(|A|)|| + 1/2h (w(? f,g)), where f,g are non-negative continuous functions such that f(x)g(x) = x (x ? 0), h is a non-negative and non-decreasing convex function on [0,?) and ? f,g = f (|A|)Ug(|A|).

Sherman, Hermite-Hadamard and Fejér like inequalities for convex sequences and nondecreasing convex functions

In this paper, we prove Sherman like inequalities for convex sequences and nondecreasing convex functions. Thus we develop some results by S. Wu and L. Debnath [19]. In consequence, we derive discrete versions for convex sequences of Petrovic and Giaccardi?s inequalities. As applications, we establish some generalizatons of Fej?r inequality for convex sequences. We also study inequalities of Hermite-Hadamard type. Thus we extend some recent results of Latreuch and Bela?di [8]. In our considerations we use some matrix methods based on column stochastic and doubly stochastic matrices.

Inequalities for convex sequences and nondecreasing convex functions

In this paper, by using Wu–Debnath’s method, we establish inequalities of Hammer–Bullen type for convex sequences and nondecreasing convex functions. In particular, we give an extension of a recent Farissi’s inequality. By assuming the symmetry of an involved sequence, we derive an inequality of Fejer–Hammer–Bullen type for convex sequences. In addition, we establish some companions of an Hermite–Hadamard like inequality by Latreuch and Belaidi. In our approach we use matrix methods based on column stochastic matrices.

Scheduling with non-decreasing deterioration jobs and variable maintenance activities on a single machine

ABSTRACTThere is a situation found in many manufacturing systems, such as steel rolling mills, fire fighting or single-server cycle-queues, where a job that is processed later consumes more time than that same job when processed earlier. The research finds that machine maintenance can improve the worsening of processing conditions. After maintenance activity, the machine will be restored. The maintenance duration is a positive and non-decreasing differentiable convex function of the total processing times of the jobs between maintenance activities. Motivated by this observation, the makespan and the total completion time minimization problems in the scheduling of jobs with non-decreasing rates of job processing time on a single machine are considered in this article. It is shown that both the makespan and the total completion time minimization problems are NP-hard in the strong sense when the number of maintenance activities is arbitrary, while the makespan minimization problem is NP-hard in the ordinary sense when the number of maintenance activities is fixed. If the deterioration rates of the jobs are identical and the maintenance duration is a linear function of the total processing times of the jobs between maintenance activities, then this article shows that the group balance principle is satisfied for the makespan minimization problem. Furthermore, two polynomial-time algorithms are presented for solving the makespan problem and the total completion time problem under identical deterioration rates, respectively.

Estimates for the quenching time of a MEMS equation with fringing field

In this paper we consider a zero initial–boundary value parabolic problem in MEMS with fringing field, ut−Δu=λg(u)(1+δ|∇u|2), in a bounded domain Ω of RN. Here λ>0 is a parameter related to the applied voltage, δ>0, g is a positive nondecreasing convex function, diverging as u→1. Firstly, we show that the solvability of the stationary problem −Δw=λg(w)(1+δ|∇w|2) with Dirichlet boundary condition is characterized by a parameter λδ∗. Meanwhile it is shown that for λ>λδ∗, any solution to the parabolic equation will quench (u→1) at a finite time. Secondly, we focus on estimating the quenching time Tδ∗ in terms of λ, λδ∗, i.e. the quenching time Tδ∗=O((λ−λδ∗)−12), as λ→(λδ∗)+.

Optimal arrival rate and service rate control of multi-server queues

We consider the problem of optimal control of a multi-server queue with controllable arrival and service rates. This study is motivated by its potential application to the design and control of data centers. The cost structure includes customer holding cost which is a non-decreasing convex function of the number of customers in the system, server operating cost which is a non-decreasing convex function of the chosen service rate, and system operating reward which is a non-decreasing concave function of the chosen arrival rate. We formulate the problem as a continuous-time Markov decision process and derive structural properties of the optimal control policies under both discounted cost and average cost criterions.

Regularity of the extremal solution for some elliptic problems with singular nonlinearity and advection

In this note, we investigate the regularity of the extremal solution u ⁎ for the semilinear elliptic equation − △ u + c ( x ) ⋅ ∇ u = λ f ( u ) on a bounded smooth domain of R n with Dirichlet boundary condition. Here f is a positive nondecreasing convex function, exploding at a finite value a ∈ ( 0 , ∞ ) . We show that the extremal solution is regular in the low-dimensional case. In particular, we prove that for the radial case, all extremal solutions are regular in dimension two.

Isoperimetry for spherically symmetric log-concave probability measures

We prove an isoperimetric inequality for probability measures $\\mu$ on $\\mathbb{R}^n$ with density proportional to $\\exp(-\\phi(\\lambda |x|))$, where $|x|$ is the euclidean norm on $\\mathbb{R}^n$ and $\\phi$ is a non-decreasing convex function. It applies in particular when $\\phi(x)=x^\\alpha$ with $\\alpha \\ge 1$. Under mild assumptions on $\\phi$, the inequality is dimension-free if $\\lambda$ is chosen such that the covariance of $\\mu$ is the identity.

On a general family of nonautonomous elliptic and parabolic equations

We consider the parabolic equation u t − Δu = λ f (x)g(u) as well as the corresponding elliptic problem, with a nonnegative profile f and a positive nondecreasing convex function g verifying $${\lim_{u \to 1^-} g(u) = \infty}$$ . Our study is motivated by a simplified Micro-Electromechanical Systems (MEMS) device model. We extend or improve many qualitative and quantitative results for the MEMS modeling to this very general setting, which help us to understand more about the influence of f on the pull-in voltage λ* and the quenching phenomenon. Especially, we show some new estimates for λ* and the quenching time T.

Composition operators on weighted Bergman-Orlicz spaces

In this paper, composition operators acting on Bergman-Orlicz spaces are studied, where ψ is a non-constant, non-decreasing convex function defined on (-∞, ∞) which satisfies the growth condition . In fact, under a mild condition on ∞, we show that every holomorphic-self map ∞ of induces a bounded composition operator on and C∞ is compact on if and only if it is compact on .

’n Algoritme vir die minimum van die konkawe knapsakprobleem

This paper addresses the problem of resource allocation among activities where the cost of each is described by a concave function. There is a single linear constraint (limited resource) and each activity has an upper and lower bound (maximum and minimum resource allocations). The objective is to minimise the sum of the functions. The problem with convex functions is well-studied and since a local minimum is also global, this problem was tamed early by Luss and Gupta [Luss H & Gupta SK, 1975, Allocation of effort resources among competing activities, Operations Research, 23, pp. 360–366], and by Bitran and Hax [Bitran GR & Hax AC, 1979, On the solution of convex knapsack problems with bounded variables, pp. 357–367 in Prekopa A (Ed.), Survey of Mathematical Programming]. In contrast the minimisation of a sum of concave functions has received less attention and then the emphasis has often been on an objective function which is nonseparable quadratic, and on the complexity of finding a true local minimiser (e.g. More and Vavasis [More JJ & Vavasis SS, 1991, On the solution of concave knapsack problems, Mathematical Programming, 49, pp. 397–411]). We are concerned with the computational problem of finding the global optimum for a (separable) sum of nondecreasing general concave functions and the approach is via the Kuhn-Tucker necessary conditions. These are improved by using the result that a minimiser must be an extreme point, which means that all but one variable (at most) is at an upper or lower bound. (Mor´e and Vavasis base their CKP algorithm on this property.) The improved necessary conditions form the basis of the method of greatest differences (GVA), our algorithm to improve a feasible solution. A greedy heuristic to produce a first feasible solution is also proposed. Using four groups of 10 instances, each from the four classes of concave functions of Luss and Gupta, one thousand different runs for incremental values of the resource were made for both the CKP and our GVA. While the CKP often found a globally suboptimal answer for functions which intersect, the GVA found the correct answer in all 16 000 runs. This is no guarantee that a method based on necessary conditions will always find the global maximum but the GVA is numerically promising and it masters a class of problems that the CKP does not. The greedy first feasible solution was found to be optimal more frequently (64% to 49%, and 73% to 40%) than that proposed by More and Vavasis. The GVA does not depend on the kind of function, requiring only that the functions be nondecreasing and concave. With minor alterations it can be used for the maximisation of the sum of nondecreasing convex functions. It is much faster than dynamic programming on problems with up to 10 functions and should be superior when tested on large problems.

Majorization of the Critical Points of a Polynomial by Its Zeros

Let z1, …, zn be the zeros of a polynomial f(z) and let \(\zeta _{1},\cdots,\zeta_{n}\) be those of zf′(z). Suppose that for both polynomials the zeros are labelled in order of non-increasing modulus. We show that $$\sum_{v=1}^{k}|\zeta_{v}|\leq \sum_{v=1}^{k}|z_{v}|,\qquad k = 1\cdots,n$$ , which means that the moduli of the zeros of f(z) weakly majorize those of zf′(z). This refines the Gauss-Lucas Theorem. Moreover, this weak majorization is preserved if we replace \(|\zeta_{v}|\ {\rm by}\ \psi (|\zeta_{v}|)\ {\rm and}\ |z_{v}|\ {\rm by}\ \psi (|z_{v}|) \) forv = 1,…, n, where ψ o exp is any non-decreasing convex function on ℝ. Actually, we establish more general results which hold for a polynomial f and a certain multiplicative composition which may be interpreted as a Hadamard product of f with a polynomial from a certain class.

Constructing Nowhere Differentiable Functions From Convex Functions

We find an easy way to construct a continuous nowhere differentiable function from any nondecreasing convex function mapping the unit interval onto itself. We give a number of examples of nowhere differentiable functions constructed this way.

On the structure of optimal entropy-constrained scalar quantizers

The nearest neighbor condition implies that when searching for a mean-square optimal fixed-rate quantizer it is enough to consider the class of regular quantizers, i.e., quantizers having convex cells and codepoints which lie inside the associated cells. In contrast, quantizer regularity can preclude optimality in entropy-constrained quantization. This can be seen by exhibiting a simple discrete scalar source for which the mean-square optimal entropy-constrained scalar quantizer (ECSQ) has disconnected (and hence nonconvex) cells at certain rates. In this work, new results concerning the structure and existence of optimal ECSQs are presented. One main result shows that for continuous sources and distortion measures of the form d(x,y)=/spl rho/(|x-y|), where /spl rho/ is a nondecreasing convex function, any finite-level ECSQ can be "regularized" so that the resulting regular quantizer has the same entropy and equal or less distortion. Regarding the existence of optimal ECSQs, we prove that under rather general conditions there exists an "almost regular" optimal ECSQ for any entropy constraint. For the squared error distortion measure and sources with piecewise-monotone and continuous densities, the existence of a regular optimal ECSQ is shown.

Some best possible prophet inequalities for convex functions of sums of independent variates and unordered martingale difference sequences

Let $\Phi (\cdot)$ be a nondecreasing convex function on $[0, \infty)$. We show that for any integer $n \geq 1$ and real $a$, $$E \Phi ((M_n - a)^+) \leq 2E \Phi ((S_n - a)^+) - \Phi (0)$$ and $$E(M_n \vee \mathrm{med} S_n) \leq E|S_n - \mathrm{med} S_n|.$$ where $X_1, X_2, \dots$ are any independent mean zero random variables with partial sums $S_0 = 0, S_k = X_1 + \dots + X_k$ and partial sum maxima $M_n = \max_{0 \leq k \leq n} S_k$. There are various instances in which these inequalities are best possible for fixed $n$ and/or as $n \to \infty$. These inequalities remain valid if $\{X_k\}$ is a martingale difference sequence such that $E(X_k \{X_i; i \not= k\}) = 0$ a.s. for each $k \geq 1$. Modified versions of these inequalities hold if the variates have arbitrary means but are independent.

Constrained Maximum Likelihood Exemplified by Isotonic Convex Logistic Regression

Abstract Maximum likelihood estimates in problems in which the likelihood is smooth and the parameter space is defined by linear or smooth nonlinear inequality constraints can be obtained using available nonlinear optimization packages, such as NPSOL. This class of models includes generalized linear models with order restrictions, convexity, or smoothness constraints on the parameters (smoothness constraints being in the form of bounds on finite differences of the regression function). Methods for such models are demonstrated by an analysis of data from four published studies of the incidence of Down's syndrome in single-year maternal age intervals. Constrained logistic regression of the incidence of Down's syndrome on maternal age shows that the data are fitted well by a nondecreasing convex function. P values for the likelihood ratio test of this model against alternatives are obtained by the parametric bootstrap and iterated parametric bootstrap. The iterated bootstrap is used as a diagnostic tool: It demonstrates that the bootstrap works in addition to providing a correction term. These methods provide a general approach applicable to most models specified by constraints on the parameter space.

Constrained Maximum Likelihood Exemplified by Isotonic Convex Logistic Regression

Abstract Maximum likelihood estimates in problems in which the likelihood is smooth and the parameter space is defined by linear or smooth nonlinear inequality constraints can be obtained using available nonlinear optimization packages, such as NPSOL. This class of models includes generalized linear models with order restrictions, convexity, or smoothness constraints on the parameters (smoothness constraints being in the form of bounds on finite differences of the regression function). Methods for such models are demonstrated by an analysis of data from four published studies of the incidence of Down's syndrome in single-year maternal age intervals. Constrained logistic regression of the incidence of Down's syndrome on maternal age shows that the data are fitted well by a nondecreasing convex function. P values for the likelihood ratio test of this model against alternatives are obtained by the parametric bootstrap and iterated parametric bootstrap. The iterated bootstrap is used as a diagnostic tool: It ...