Continuous Convex Function Research Articles

On a problem of T. Szostok concerning the Hermite–Hadamard inequalities

In the present paper we solve a problem posed by Tomasz Szostok who asked about the solutions f and F to the system of inequalities f ( x + y 2 ) ≤ F ( y ) − F ( x ) y − x ≤ f ( x ) + f ( y ) 2 . We show that f and F are the solutions to the above system of inequalities if and only if f is a continuous convex function and F is primitive function of f . This result can be interpreted as a regularity phenomenon-the solutions to the system of functional inequalities turn out to be regular without any additional assumptions.

Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law

AbstractIn this paper, we consider the definition of “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law associated with the real utility distribution to give the results for majorizatioQn inequalities by using monotonic sequences. We obtain the equivalent statements between continuous convex functions and Green functions via majorization inequalities, “useful” Csiszár functional and “useful” Zipf-Mandelbrot law. By considering “useful” Csiszár divergence in the integral case, we give the results for integral majorization inequality. Towards the end, some applications are given.

The TV advertisements scheduling problem

A TV channel has a single advertisement break of duration h and a convex continuous function $$f{:}\;[0,h] \rightarrow \mathbb {R}^+$$ representing the TV rating points within the advertisement break. Given n TV advertisements of different durations $$p_j$$ that sum up to h, and willingness to pay coefficients $$w_j$$ , the objective is to schedule them on the TV break in order to maximize the total revenue of the TV channel $$\sum _j w_j \int _{c_j-p_j}^{c_j} f(t) dt,$$ where $$[c_j-p_j,c_j)$$ is the broadcast time interval of TV advertisement j. We show that this problem is NP-hard and propose a fully polynomial time approximation scheme, using a special dominance property of an optimal schedule and the technique of K-approximation sets and functions introduced by Halman et al. (Math Oper Res 34:674–685, 2009).

Multifractal properties of typical convex functions

We study the singularity (multifractal) spectrum of continuous convex functions defined on $$[0,1]^{d}$$ . Let $$E_f({h}) $$ be the set of points at which f has a pointwise exponent equal to h. We first obtain general upper bounds for the Hausdorff dimension of these sets $$E_f(h)$$ , for all convex functions f and all $$h\ge 0$$ . We prove that for typical/generic (in the sense of Baire) continuous convex functions $$f:[0,1]^{d}\rightarrow { \mathbb {R}}$$ , one has $$\dim E_f(h) =d-2+h$$ for all $$h\in [1,2],$$ and in addition, we obtain that the set $$ E_f({h} )$$ is empty if $$h\in (0,1)\cup (1,+\infty )$$ . Also, when f is typical, the boundary of $$[0,1]^{d}$$ belongs to $$E_{f}({0})$$ .

Modified basic projection methods for a class of equilibrium problems

Projection methods are a popular class of methods for solving equilibrium problems. In this paper, we propose approximate one projection methods for solving a class of equilibrium problems, where the cost bifunctions are paramonotone, the feasible sets are defined by a continuous convex function inequality and not necessarily differentiable in the Euclidean space \(\mathcal R^{s}\). At each main iteration step in our algorithms, the usual projections onto the feasible set are replaced by computing inexact subgradients and one projection onto the intersection of two halfspaces containing the solution set of the equilibrium problems. Then, by choosing suitable parameters, we prove convergence of the whole generated sequence to a solution of the problems, under only the assumptions of continuity and paramonotonicity of the bifunctions. Finally, we present some computational examples to illustrate the assumptions of the proposed algorithms.

Resource constrained scheduling problems with general truncated sum-of-processing time dependent effect under single machine and unrelated parallel machines

This paper studies some scheduling problems with general truncated sum-of-processing time dependent effect under single machine and unrelated parallel machines. Under the single machine, we consider that the actual processing time of a job is bivariate, continuous and non-increasing convex function of the total processing time of the processed jobs, its position, the amount of resource allocation and control parameter. SLK due date assignment models are involved, in which the length of due window is the same. We present the polynomial time algorithms to find the optimal job sequence, the due date and resource allocations that minimizing including the slack due date, earliness cost, tardiness cost, resource consumption costs and makespan. Under the unrelated machines, we only consider no resource locations scheduling problems. Minimizing the total the machine load and minimizing the total completion time can also be solved in a polynomial time when the number of machines is fixed, respectively.

Superiorization with level control

The convex feasibility problem is to find a common point of a finite family of closed convex subsets. In many applications one requires something more, namely finding a common point of closed convex subsets which minimizes a continuous convex function. The latter requirement leads to an application of the superiorization methodology which is actually settled between methods for convex feasibility problem and the convex constrained minimization. Inspired by the superiorization idea we introduce a method which sequentially applies a long-step algorithm for a sequence of convex feasibility problems; the method employs quasi-nonexpansive operators as well as subgradient projections with level control and does not require evaluation of the metric projection. We replace a perturbation of the iterations (applied in the superiorization methodology) by a perturbation of the current level in minimizing the objective function. We consider the method in the Euclidean space in order to guarantee the strong convergence, although the method is well defined in a Hilbert space.

Hölder’s means and absolute normalized norms on R2

Recently, Cui and Lu have introduced the constant Hp(X) (p  R) of a Banach space X by using Hölder’s means. In this paper, we determine and estimate the new constant under the absolute normalized norms on R2 by means of their corresponding continuous convex functions on [0,1]. Furthermore, the exact values of the constant are calculated in some concrete Banach spaces. In particular, we calculate the precise values of the constants A2(X) and T(X) and the Gao constant E(X) in these concrete spaces.

Hausdorff dimension of the nondifferentiability set of a convex function

We find an upper bound for the Hausdorff dimension of the nondifferentiability set of a continuous convex function defined on a Riemannian manifold. As an application, we show that the boundary of a convex open subset of Rn, n ? 2, has Hausdorff dimension at most n - 2.

A Generalization on Weighted Means and Convex Functions with respect to the Non-Newtonian Calculus

This paper is devoted to investigating some characteristic features of weighted means and convex functions in terms of the non-Newtonian calculus which is a self-contained system independent of any other system of calculus. It is shown that there are infinitely many such useful types of weighted means and convex functions depending on the choice of generating functions. Moreover, some relations between classical weighted mean and its non-Newtonian version are compared and discussed in a table. Also, some geometric interpretations of convex functions are presented with respect to the non-Newtonian slope. Finally, using multiplicative continuous convex functions we give an application.

On operator inequalities of Jensen type for convex functions

Let f be a continuous convex function on an interval J and be Hilbert spaces. Let be self-adjoint operators with spectra in [m, M] for some scalars and be self-adjoint operators with spectra in . If is a k-tuple and is a n-tuple of positive linear mappings from into with , and , then we prove the inequalitiesAs applications, we prove Jensen’s inequality for operators without operator convexity and obtain some operator inequalities of Jensen type.

Schäffer-type constant and uniform normal structure in Banach spaces

The exact value of the Schaffer-type constants are investigated under the absolute normalized norms on R2 by means of their corresponding continuous convex functions on [0,1]. Moreover, some sufficient conditions which imply uniform normal structure are presented. These results improve some known results.

Convergence of functions of self-adjoint operators and applications

The main result (roughly) is that if $(H_i)$ converges weakly to $H$ and if also $f(H_i)$ converges weakly to $f(H)$, for a single strictly convex continuous function $f$, then $(H_i)$ must converge strongly to $H$. One application is that if $f({\operatorname{pr}}(H))={\operatorname{pr}} (f(H))$, where ${\operatorname{pr}}$ denotes compression to a closed subspace $M$, then $M$ must be invariant for $H$. A consequence of this is the verification of a conjecture of Arveson, that Theorem 9.4 of [Arv] remains true in the infinite dimensional case. And there are two applications to operator algebras. If $h$ and $f(h)$ are both quasimultipliers, then $h$ must be a multiplier. Also (still roughly stated), if $h$ and $f(h)$ are both in $pA_{\operatorname{sa}} p$, for a closed projection $p$, then $h$ must be strongly $q$-continuous on $p$.

A separating hyperplane theorem, the fundamental theorem of asset pricing, and Markov's principle

We prove constructively that every uniformly continuous convex function f:X→R+ has positive infimum, where X is the convex hull of finitely many vectors. Using this result, we prove that a separating hyperplane theorem, the fundamental theorem of asset pricing, and Markov's principle are constructively equivalent. This is the first time that important theorems are classified into Markov's principle within constructive reverse mathematics.

On a convexity property

In this article we proved an interesting property of the class of continuous convex functions. This leads to the form of pre-Hermite-Hadamard inequality which in turn admits a generalization of the famous Hermite-Hadamard inequality. Some further discussion is also given.

A relaxed-projection splitting algorithm for variational inequalities in Hilbert spaces

We introduce a relaxed-projection splitting algorithm for solving variational inequalities in Hilbert spaces for the sum of nonsmooth maximal monotone operators, where the feasible set is defined by a nonlinear and nonsmooth continuous convex function inequality. In our scheme, the orthogonal projections onto the feasible set are replaced by projections onto separating hyperplanes. Furthermore, each iteration of the proposed method consists of simple subgradient-like steps, which does not demand the solution of a nontrivial subproblem, using only individual operators, which exploits the structure of the problem. Assuming monotonicity of the individual operators and the existence of solutions, we prove that the generated sequence converges weakly to a solution.

KdV Hamiltonian as a Function of Actions

We prove that the non-linear part of the Hamiltonian of the KdV equation on the circle, written as a function of the actions, defines a continuous convex function on the l2 space and derive for it lower and upper bounds in terms of some functions of the l2-norm. The proof is based on a new representation of the Hamiltonian in terms of the quasimomentum, obtained via the conformal mapping theory.

Well-Posedness of Approximation and Optimization Problems for Weakly Convex Sets and Functions

We consider the class of weakly convex sets with respect to a quasiball in a Banach space. This class generalizes the classes of sets with positive reach, proximal smooth sets and prox-regular sets. We prove the well-posedness of the closest points problem of two sets, one of which is weakly convex with respect to a quasiball M, and the other one is a summand of the quasiball −rM, where r ∈ (0, 1). We show that if a quasiball B is a summand of a quasiball M, then a set that is weakly convex with respect to the quasiball M is also weakly convex with respect to the quasiball B. We consider the class of weakly convex functions with respect to a given convex continuous function γ that consists of functions whose epigraphs are weakly convex sets with respect to the epigraph of γ. We obtain a sufficient condition for the well-posedness of the infimal convolution problem, and also a sufficient condition for the existence, uniqueness, and continuous dependence on parameters of the minimizer.

On the convex formulation of area for slicing floorplans

In this paper, it is shown that the area optimization problem of a compact slicing floorplan may be formulated as a convex optimization problem when the areas of the analog components are modeled with continuous convex functions of the width (height). It is proved that the area of a compact slicing floorplan is a convex function of its width (height). The convexity is shown for the cases with and without dead (empty) space. This feature can be exploited to efficiently optimize the dimensions of layout components with multiple variants, without enumerating all possible combinations. Layout of a voltage-doubler circuit is used to quantitatively verify the proof.

A criterion of \varGamma -nullness and differentiability of convex and quasiconvex functions

We introduce a criterion for a set to be $\varGamma $-null. Using it we give a shorter proof of the result that the set of points where a continuous convex function on a separable Asplund space is not Fréchet differentiable is $\varGamma $-null. Our crite