Abstract homogeneity of interval-valued functions
In this paper we develop the idea of abstract homogeneity in the context of interval-valued (IV) functions endowed with admissible orders and investigate some of its properties.
cs.LO↗
arXiv subjects
Radko Mesiar
Publications and source records attributed to Radko Mesiar.
On Monotonicities of Interval Valued Functions
In this paper, we develop the notions of weak/directional monotonicity (developed by Sesma-Sara et al. in terms of the \emph{Kulisch-Miranker order}) and the notion of $ G $-monotonicity (introduced by Santiago et al. for $ [0,\!1]$) for intervals endowed with admissible orders.
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Generalized-Hukuhara Subgradient Method for Optimization Problem with Interval-valued Functions and its Application in Lasso Problem
In this study, a \emph{$gH$-subgradient technique} is developed to obtain efficient solutions to the optimization problems with nonsmooth nonlinear convex interval-valued functions. The algorithmic implementation of the developed $gH$-subgradient technique is illustrated. As an application of the proposed \emph{$gH$-subgradient technique}, an $\ell_1$ penalized linear regression problem, known as a \emph{lasso problem}, with interval-valued features is solved.
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Generalized-Hukuhara Subdifferential Analysis and Its Application in Nonconvex Composite Optimization Problems with Interval-valued Functions
In this article, we study $gH$-subdifferential calculus of convex interval-valued functions (IVFs) and apply it in a nonconvex composite model of interval optimization problems (IOPs). It is found that the $gH$-directional derivative of maximum of finitely many comparable IVFs is the maximum of their $gH$-directional derivative. Proposed concepts of $gH$-subdifferential are observed to be useful to derive Fritz-John-type and KKT-type efficiency conditions for weak efficient solutions of IOPs. Further, we extract a necessary and sufficient condition to characterize the weak efficient solutions of nonconvex composite IOPs by applying the proposed concepts. To derive the results on $gH$-subdifferentials, the concepts of limit supremum and limit infimum with certain properties for IVFs are defined in the sequel. The whole analysis is supported by appropriate expository examples.
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Towards interval uncertainty propagation control in bivariate aggregation processes and the introduction of width-limited interval-valued overlap functions
Overlap functions are a class of aggregation functions that measure the overlapping degree between two values. Interval-valued overlap functions were defined as an extension to express the overlapping of interval-valued data, and they have been usually applied when there is uncertainty regarding the assignment of membership degrees. The choice of a total order for intervals can be significant, which motivated the recent developments on interval-valued aggregation functions and interval-valued overlap functions that are increasing to a given admissible order, that is, a total order that refines the usual partial order for intervals. Also, width preservation has been considered on these recent works, in an intent to avoid the uncertainty increase and guarantee the information quality, but no deeper study was made regarding the relation between the widths of the input intervals and the output interval, when applying interval-valued functions, or how one can control such uncertainty propagation based on this relation. Thus, in this paper we: (i) introduce and develop the concepts of width-limited interval-valued functions and width limiting functions, presenting a theoretical approach to analyze the relation between the widths of the input and output intervals of bivariate interval-valued functions, with special attention to interval-valued aggregation functions; (ii) introduce the concept of $(a,b)$-ultramodular aggregation functions, a less restrictive extension of one-dimension convexity for bivariate aggregation functions, which have an important predictable behaviour with respect to the width when extended to the interval-valued context; (iii) define width-limited interval-valued overlap functions, taking into account a function that controls the width of the output interval; (iv) present and compare three construction methods for these width-limited interval-valued overlap functions.
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Generalized-Hukuhara Subgradient and its Application in Optimization Problem with Interval-valued Functions
In this article, the concepts of gH-subgradients and gH-subdifferentials of interval-valued functions are illustrated. Several important characteristics of the gH-subdifferential of a convex interval-valued function, e.g., closeness, boundedness, chain rule, etc. are studied. Alongside, we prove that gH-subdifferential of a gH-differentiable convex interval-valued function only contains gH-gradient of that interval-valued function. It is observed that the gH-directional derivative of a convex interval-valued function in each direction is maximum of all the products of gH-subgradients and the direction. Importantly, we show that a convex interval-valued function is gH-Lipschitz continuous if it has gH-subgradients at each point in its domain. Furthermore, the relations between efficient solutions of an optimization problem with interval-valued function and its gH-subgradients are derived.
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On generating of idempotent aggregation functions on finite lattices
In a recent paper we proposed the study of aggregation functions on lattices via clone theory approach. Observing that aggregation functions on lattices just correspond to $0,1$-monotone clones, we have shown that all aggregation functions on a finite lattice $L$ can be obtained as usual composition of lattice operations $\wedge,\vee$, and certain unary and binary aggregation functions. The aim of this paper is to present a generating set for the class of intermediate (or, equivalently, idempotent) aggregation functions. This set consists of lattice operations and certain ternary idempotent aggregation functions.
math.RA↗
Generalized comonotonicity and new axiomatizations of Sugeno integrals on bounded distributive lattices
Two new generalizations of the relation of comonotonicity of lattice-valued vectors are introduced and discussed. These new relations coincide on distributive lattices and they share several properties with the comonotonicity for the real-valued vectors (which need not hold for $L$-valued vectors comonotonicity, in general). Based on these newly introduced generalized types of comonotonicity of $L$-valued vectors, several new axiomatizations of $L$-valued Sugeno integrals are introduced. One of them brings a substantial decrease of computational complexity when checking an aggregation function to be a Sugeno integral.
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A new characterization of the discrete Sugeno integral
We introduce a new property of the discrete Sugeno integrals which can be seen as their characterization, too. This property, compatibility with respect to congruences on $[0,1]$, stresses the importance of the Sugeno integrals in multicriteria decision support as well.
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Congruences and the discrete Sugeno integrals on bounded distributive lattices
We study compatible aggregation functions on a general bounded distributive lattice $L$, where the compatibility is related to the congruences on $L$. As a by-product, a new proof of an earlier result of G. Gr\"atzer is obtained. Moreover, our results yield a new characterization of discrete Sugeno integrals on bounded distributive lattices.
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Description of sup- and inf-preserving aggregation functions via families of clusters in data tables
Connection between the theory of aggregation functions and formal concept analysis is discussed and studied, thus filling a gap in the literature by building a bridge between these two theories, one of them living in the world of data fusion, the second one in the area of data mining. We show how Galois connections can be used to describe an important class of aggregation functions preserving suprema, and, by duality, to describe aggregation functions preserving infima. Our discovered method gives an elegant and complete description of these classes. Also possible applications of our results within certain biclustering fuzzy FCA-based methods are discussed.
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Generators of aggregation functions and fuzzy connectives
We show that the class of all aggregation functions on $[0,1]$ can be generated as a composition of infinitary sup-operation $\bigvee$ acting on sets with cardinality not exceeding $\mathfrak{c}$, $b$-medians $\mathsf{Med}_b$, $b\in[0,1[$, and unary aggregation functions $1_{]0,1]}$ and $1_{[a,1]}$, $a\in ]0,1]$. Moreover, we show that we cannot relax the cardinality of argument sets for suprema to be countable, thus showing a kind of minimality of the introduced generating set. As a by product, generating sets for fuzzy connectives, such as fuzzy unions, fuzzy intersections and fuzzy implications are obtained, too.
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A special class of fuzzy measures: Choquet integral and applications
Core of an economy and Walras equilibria are considered for a product space $X \times [0,1]$ using the Choquet integral with respect to a fuzzy measure.
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On linearity of pan-integral and pan-integrable functions space
$L\sp{p}$ space is a crucial aspect of classical measure theory. For nonadditive measure, it is known that $L\sp{p}$ space theory holds for the Choquet integral whenever the monotone measure $μ$ is submodular and continuous from below. The main purpose of this paper is to generalize $L\sp{p}$ space theory to $+,\cdot$-based pan-integral. Let $(X, {\cal A}, μ)$ be a monotone measure space. We prove that the $+,\cdot$-based pan-integral is additive with respect to integrands if $μ$ is subadditive. Then we introduce the pan-integral for real-valued functions(not necessarily nonnegative), and prove that this integral possesses linearity if $μ$ is subadditive. By using the linearity of pan-integral, we finally show that all of the pan-integrable functions form a Banach space. Since the $+,\cdot$-based pan-integral coincides with the concave integral for subadditive measure, the results obtained in this paper remain valid for the concave integral. Noticing that an outer measure is subadditive, we can define a Lebesgue-like integral(possesses linearity) from an outer measure, and the $L\sp{p}$ theory holds for this integral. {\it Keywords:} Monotone measure; Subadditivity; Pan-integral; Linearity; Pan-integrable space; Completeness
math.FA↗
Decomposition approaches to integration without a measure
Extending the idea of Even and Lehrer [3], we discuss a general approach to integration based on a given decomposition system equipped with a weighting function, and a decomposition of the integrated function. We distinguish two type of decompositions: sub-decomposition based integrals (in economics linked with optimization problems to maximize the possible profit) and super-decomposition based integrals (linked with costs minimization). We provide several examples (both theoretical and realistic) to stress that our approach generalizes that of Even and Lehrer [3] and also covers problems of linear programming and combinatorial optimization. Finally, we introduce some new types of integrals related to optimization tasks.
math.FA↗
On distance distribution functions-valued submeasures related to aggregation functions
Probabilistic submeasures generalizing the classical (numerical) submeasures are introduced and discussed in connection with some classes of aggregation functions. A special attention is paid to triangular norm-based probabilistic submeasures and semi-copula-based probabilistic submeasures. Some algebraic properties of classes of such submeasures are also studied.
math.CA↗
Meaningful aggregation functions mapping ordinal scales into an ordinal scale: a state of the art
We present an overview of the meaningful aggregation functions mapping ordinal scales into an ordinal scale. Three main classes are discussed, namely order invariant functions, comparison meaningful functions on a single ordinal scale, and comparison meaningful functions on independent ordinal scales. It appears that the most prominent meaningful aggregation functions are lattice polynomial functions, that is, functions built only on projections and minimum and maximum operations.
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