Continuous Convex Research Articles

A library for continuous convex separable quadratic knapsack problems

The Continuous Convex Separable Quadratic Knapsack problem (CQKnP) is an easy but useful model that has very many different applications. Although the problem can be solved quickly, it must typically be solved very many times within approaches to (much) more difficult models; hence an efficient solution approach is required. We present and discuss a small open-source library for its solution that we have recently developed and distributed.

A note on extreme norms on $\mathbb R$2

We denote by AN2 the set of all absolute normalized norms on $\mathbb R$2. It is known that the set AN2 and the set of all continuous convex functions ψ on [0,1] with max{1−t,t} ≤ ψ(t) ≤ 1 for t ∈ [0,1] (denoted by Ψ2) are in a one to one correspondence under the equation ψ(t) = ||(1−t,t)||. Recently, we characterized extreme points of AN2 by considering Ψ2. In this paper we give another proof of this result.

On differentiability of convex operators

The main known results on differentiability of continuous convex operators f from a Banach space X to an ordered Banach space Y are due to J.M. Borwein and N.K. Kirov. Our aim is to prove some “supergeneric” results, i.e., to show that, sometimes, the set of Gâteaux or Fréchet nondifferentiability points is not only a first-category set, but also smaller in a stronger sense. For example, we prove that if Y is countably Daniell and the space L(X,Y) of bounded linear operators is separable, then each continuous convex operator f:X→Y is Fréchet differentiable except for a Γ-null angle-small set. Some applications of such supergeneric results are shown.

On average values of convex functions

The aim of this paper is to give an extension of an inequality proved by Wulbert (Math Comput Model 37:1383–1391, 2003, Lemma 2.5) and to define Stolarsky type means as an application of this inequality. Further, we discuss some properties of averages of a continuous convex function, some consequences of a double inequality given by Wulbert (Math Comput Model 37:1383–1391, 2003, Theorem 3.3) and obtain improvement results of Wulbert (Math Comput Model 37:1383–1391, 2003, Corollary 4.3).

New converses of the Jessen and Lah-Ribarič inequalities II

New converses of the Jessen and Lah-Ribariˇ c inequalities for continuous convex func- tions are studied. Applications are given for generalized and arithmetic means, Holder's inequal- ity, Hadamard's inequality, and the inequalities of Giaccardi and Petrovic.

Refined Jensen's operator inequality with condition on spectra

We give a refinement of Jensen's inequality for n- tuples of self-adjoint operators, unital n-tuples of positive linear mappings and real valued continuous convex functions with condition on the spectra of the operators. The refined Jensen's inequality is used to obtain a refinement of inequalities among quasi-arithmetic means under similar conditions. As an application of these results we give a refinement of inequalities among power means.

Measuring Voting Power in Convex Policy Spaces

Classical power index analysis considers the individual's ability to influence the aggregated group decision by changing its own vote, where all decisions and votes are assumed to be binary. In many practical applications we have more options than either yes or no. Here we generalize three important power indices to continuous convex policy spaces. This allows the analysis of a collection of economic problems like e.g. tax rates or spending that otherwise would not be covered in binary models.

An operator inequality and its consequences

Let f be a continuous convex function on an interval J, let A, B, C, D be self-adjoint operators acting on a Hilbert space with spectra contained in J such that A+D=B+C and A⩽m⩽B,C⩽M⩽D for two real numbers m<M, and let Φ be a unital positive linear map on B(H). We prove the inequalityf(Φ(B))+f(Φ(C))⩽Φ(f(A))+Φ(f(D))and apply it to obtain several inequalities such as the Jensen–Mercer operator inequality and the Petrović operator inequality.

PROJECTIONS OF LOG-CONCAVE FUNCTIONS

Recently, it has been proven in [V. Milman, A. Segal and B. Slomka, A characterization of duality through section/projection correspondence in the finite dimensional setting, J. Funct. Anal. 261(11) (2011) 3366–3389] that the well-known duality mapping on the class of closed convex sets in ℝncontaining the origin is the only operation, up to obvious linear modifications, that interchanges linear sections with projections. In this paper, we extend this result to the class of geometric log-concave functions (attaining 1 at the origin). As the notions of polarity and the support function were recently uniquely extended to this class by Artstein-Avidan and Milman, a natural notion of projection arises. This notion of projection is justified by our result. As a consequence of our main result, we prove that, on the class of lower semi continuous non-negative convex functions attaining 0 at the origin, the polarity operation is the only operation interchanging addition with geometric inf-convolution and the support function is the only operation interchanging addition with inf-convolution.

Braess's paradox in expanders

AbstractExpander graphs are known to facilitate effective routing and most real‐world networks have expansion properties. At the other extreme, it has been shown that in some special graphs, removing certain edges can lead to moreefficient routing. This phenomenon is known as Braess's paradox and is usually regarded as a rare event. In contrast to what one might expect, we show that Braess's paradox is ubiquitous in expander graphs. Specifically, we prove that Braess's paradox occurs in a large class of expander graphs with continuous convex latency functions. Our results extend previous work which held only when the graph was both denser and random and for random linear latency functions. We identify deterministic sufficient conditions for a graph with as few as a linear number of edges, such that Braess's Paradox almost always occurs, with respect to a general family of random latency functions. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012

Efficient Algorithm for Level Set Method Preserving Distance Function

The level set method is a popular technique for tracking moving interfaces in several disciplines, including computer vision and fluid dynamics. However, despite its high flexibility, the original level set method is limited by two important numerical issues. First, the level set method does not implicitly preserve the level set function as a distance function, which is necessary to estimate accurately geometric features, s.a. the curvature or the contour normal. Second, the level set algorithm is slow because the time step is limited by the standard Courant-Friedrichs-Lewy (CFL) condition, which is also essential to the numerical stability of the iterative scheme. Recent advances with graph cut methods and continuous convex relaxation methods provide powerful alternatives to the level set method for image processing problems because they are fast, accurate, and guaranteed to find the global minimizer independently to the initialization. These recent techniques use binary functions to represent the contour rather than distance functions, which are usually considered for the level set method. However, the binary function cannot provide the distance information, which can be essential for some applications, s.a. the surface reconstruction problem from scattered points and the cortex segmentation problem in medical imaging. In this paper, we propose a fast algorithm to preserve distance functions in level set methods. Our algorithm is inspired by recent efficient l(1) optimization techniques, which will provide an efficient and easy to implement algorithm. It is interesting to note that our algorithm is not limited by the CFL condition and it naturally preserves the level set function as a distance function during the evolution, which avoids the classical re-distancing problem in level set methods. We apply the proposed algorithm to carry out image segmentation, where our methods prove to be 5-6 times faster than standard distance preserving level set techniques. We also present two applications where preserving a distance function is essential. Nonetheless, our method stays generic and can be applied to any level set methods that require the distance information.

The Ptolemy constant of absolute normalized norms on ℝ2

We determine and estimate the Ptolemy constant of absolute normalized norms on ℝ2 by means of their corresponding continuous convex functions on [0, 1]. Moreover, the exact values were calculated in some concrete Banach spaces.

Continuous convex MS-differentiable function need not Be HL-differentiable

We construct here an example of a continuous convex function on a locally convex space, which is MS-differentiable at a point, but is not HL-differentiable at this point.

A construction of convex functions

We describe a construction of convex functions on infinite-dimensional spaces and apply this construction to give an illustration to a theorem of Borwein-Fabian from [1]. Namely, we give a simple explicit example of a continuous convex function on lp, p ≥ 1, which is everywhere compactly differentiable, but not Frechet differentiable at zero.

Invariant valuations on quaternionic vector spaces

AbstractThe spaces of Sp(n)-, Sp(n) · U(1)- and Sp(n) · Sp(1)-invariant, translation-invariant, continuous convex valuations on the quaternionic vector space ℍn are studied. Combinatorial dimension formulae involving Young diagrams and Schur polynomials are proved.

The problem of optimizing the torsional rigidity of a prismatic body about a cross section

Some extremal problems, related to the torsional rigidity of a homogeneous body, are investigated. The problem of optimizing the torsional rigidity of a cylindrical body about a cross section is solved by determining the variation of the region when using the one-to-one correspondence between bounded convex regions and continuous positive-homogeneous convex functions. Using this approach a formula is obtained for the torsional rigidity, and conditions are also obtained characterizing the optimum region and the maximum value of the functional.

Some better bounds in converses of the Jensen operator inequality

In this paper we study converses of a generalized Jensen's inequality for a continuous field of self-adjoint operators, a unital field of positive linear mappings and real values continuous convex functions. We obtain some better bounds than the ones calculated in a series of papers in which these inequalities are studied. As an application, we provide a refined calculation of bounds in the case of power functions.

Extension of the refined Jensen's operator inequality with condition on spectra

‎We give an extension of the refined Jensen's operator inequality for‎ ‎$n-$tuples of self-adjoint operators‎, ‎unital $n-$tuples of positive‎ ‎linear mappings and real valued continuous convex functions with‎ ‎conditions on the spectra of the operators‎. ‎We also study the order‎ ‎among quasi-arithmetic means under similar conditions‎.

Convergence of quantum random walks with decoherence

In this paper, we study the discrete-time quantum random walks on a line subject to decoherence. The convergence of the rescaled position probability distribution $p(x,t)$ depends mainly on the spectrum of the superoperator $\mathcal{L}_{kk}$. We show that if 1 is an eigenvalue of the superoperator with multiplicity one and there is no other eigenvalue whose modulus equals to 1, then $\hat {P}(\frac{\nu} {\sqrt t},t)$ converges to a convex combination of normal distributions. In terms of position space, the rescaled probability mass function $p_t (x, t) \equiv p(\sqrt t x, t)$, $ x \in Z/\sqrt t$, converges in distribution to a continuous convex combination of normal distributions. We give an necessary and sufficient condition for a U(2) decoherent quantum walk that satisfies the eigenvalue conditions. We also give a complete description of the behavior of quantum walks whose eigenvalues do not satisfy these assumptions. Specific examples such as the Hadamard walk, walks under real and complex rotations are illustrated. For the O(2) quantum random walks, an explicit formula is provided for the scaling limit of $p(x,t)$ and their moments. We also obtain exact critical exponents for their moments at the critical point and show universality classes with respect to these critical exponents.

Fuzzy extensions of bargaining sets and their existence in cooperative fuzzy games

Recently, the concept of classical bargaining set given by Aumann and Maschler in 1964 has been extended to fuzzy bargaining set. In this paper, we give a modification to correct some weakness of this extension. We also extend the concept of the Mas-Colell's bargaining set (the other major type of bargaining sets) to its corresponding fuzzy bargaining set. Our main effort is to prove existence theorems for these two types of fuzzy bargaining sets. We will also give necessary and sufficient conditions for these bargaining sets to coincide with the Aubin Core in a continuous superadditive cooperative fuzzy game which has a crisp maximal coalition of maximum excess at each payoff vector. We show that both Aumann–Maschler and Mas-Colell fuzzy bargaining sets of a continuous convex cooperative fuzzy game coincide with its Aubin core.