New existence results of solutions for vector optimization programs with multifunctions

This research work follows the main aim to present new existence conditions for the solutions of vector optimization programs with set-valued objective maps taking values especially in separated and ordered locally convex spaces     

Probability boxes on totally preordered spaces for multivariate modelling

A pair of lower and upper cumulative distribution functions, also called probability box or p-box, is among the most popular models used in imprecise probability theory. They arise naturally in expert elicitation, for instance in cases where bounds are specified on the quantiles of a random variable, or when quantiles are specified only at a finite number of points. Many practical and formal results concerning p-boxes already exist in the literature. In this paper, we provide new efficient tools to construct multivariate p-boxes and develop algorithms to draw inferences from them. For this purpose, we formalise and extend the theory of p-boxes using Walley’s behavioural theory of imprecise probabilities, and heavily rely on its notion of natural extension and existing results about independence modeling. In particular, we allow p-boxes to be defined on arbitrary totally preordered spaces, hence thereby also admitting multivariate p-boxes via probability bounds over any collection of nested sets. We focus on the cases of independence (using the factorization property), and of unknown dependence (using the Fréchet bounds), and we show that our approach extends the probabilistic arithmetic of Williamson and Downs. Two design problems—a damped oscillator, and a river dike—demonstrate the practical feasibility of our results.     

Multifractal analysis based on the Choquet capacity: Application to solar magnetograms

We explore the multiscale properties of the line-of-sight component of Solar magnetic fields using magnetograms of the full disc obtained from the Michelson Doppler Imager (MDI) on board the Solar and Heliospheric Observatory (SOHO). Multifractal spectra are estimated by different methods, based on the Choquet capacity, instead of the traditional Borel measure. We have extracted spectra corresponding to active regions (AR) as well as those from quiet regions of the Sun. The shapes of spectra of active regions and those of quiet regions of the Sun are different, displaying different lengths of left-hand and right-hand branches. We indicate that multifractal scaling of magnetograms can be produced by a set of statistically similar elements in digital high resolution images. The same features are found in images of many terrestrial scenes.     

The Wijsman hyperspace of a metric hereditarily Baire space is Baire

In this paper, we show that the Wijsman hyperspace of a metric hereditarily Baire space is Baire. This solves a recent question posed by Zsilinszky.     

模糊积分变换与模糊Choquet积分的一致连续性

在一般非负单调函数空间 m[0 ,a]上引入模糊积分变换与距离的概念 ,证明了这种模糊积分变换与模糊 Choquet积分在 m[0 ,a]上关于这种距离是一致连续的 ,从而说明当 m[0 ,a]上两个函数变化不大时 ,不会使相应的模糊积分变换与模糊 Choquet积分产生较大的变化 .     

Sturdy Harmonic Functions and their Integral Representations

Sturdy harmonic functions constitute all but the least tractable of the positive harmonic functions in potential-theoretic settings. They are the uniform limits on compact sets of positive, bounded harmonic functions and are also produced by a simple integral representation on the boundary of a natural compactification of the space on which they are defined. The boundary of that compactification is metrizable, and more regular for the Dirichlet problem, in general, than is the Martin boundary if that boundary is even defined in the setting.     

Existence and uniqueness of solutions of differential equations with respect to non-additive measures

By taking Sugeno-derivative into account, first, we investigate the existence of solutions to the initial value problems (IVP) of first-order differential equations with respect to non-additive measure (more precisely, distorted Lebesgue measure). It particularly occurs in the mathematical modeling of biology. We begin by expressing the differential equation in terms of ordinary derivative and the derivative with respect to the distorted Lebesgue measure. Then, by using the fixed point theorem on cones, we construct an operator and prove the existence of positive non-decreasing solutions on cones in semi-order Banach spaces. In addition, we also use Picard–Lindelöf theorem to prove the existence and uniqueness of the solution of the equation. Second, we investigate the existence of a solution to the boundary value problem (BVP) on cones with integral boundary conditions of a mix-order differential equation with respect to non-additive measures. Moreover, the Krasnoselskii fixed point theorem is also applied to both BVP and IVP and obtains at least one positive non-decreasing solution. Examples with graphs are provided to validate the results.     

Integral representation of belief measures on compact spaces

An integral representation theorem for outer continuous and inner regular belief measures on compact topological spaces is elaborated under the condition that compact sets are countable intersections of open sets (e.g. metric compact spaces). Extreme points of this set of belief measures are identified with unanimity games with compact support. Then, the Choquet integral of a real valued continuous function can be expressed as a minimum of means over the sigma-core and also as a mean of minima over the compact subsets. Similarly, for bounded measurable functions, the Choquet integral is expressed as min of means over the core, we prove in addition that it is a mean of infima over the compact subsets. Then, we obtain Choquet–Revuz' measure representation theorem and introduce the Möbius transform of a belief measure. An extension to locally compact and sigma-compact topological spaces is provided.     

Stochastic multiobjective acceptability analysis for the Choquet integral preference model and the scale construction problem

The Choquet integral preference model is adopted in Multiple Criteria Decision Aiding (MCDA) to deal with interactions between criteria, while the Stochastic Multiobjective Acceptability Analysis (SMAA) is an MCDA methodology considered to take into account uncertainty or imprecision on the considered data and preference parameters. In this paper, we propose to combine the Choquet integral preference model with the SMAA methodology in order to get robust recommendations taking into account all parameters compatible with the preference information provided by the Decision Maker (DM). In case the criteria are on a common scale, one has to elicit only a set of non-additive weights, technically a capacity, compatible with the DM’s preference information. Instead, if the criteria are on different scales, besides the capacity, one has to elicit also a common scale compatible with the preferences given by the DM. Our approach permits to explore the whole space of capacities and common scales compatible with the DM’s preference information.