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Miguel Couceiro - One of the best experts on this subject based on the ideXlab platform.
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Pseudo-Polynomial Functions over finite distributive lattices
Fuzzy Sets and Systems, 2014Co-Authors: Miguel Couceiro, Tamas WaldhauserAbstract:In this paper we extend the authors’ previous works by considering a multi-attribute aggregation model based on a composition of a Polynomial Function over a finite distributive lattice with local utility Functions; these are referred to as pseudo-Polynomial Functions. We present an axiomatization for this class of pseudo-Polynomial Functions which differs from the previous ones both in flavour and nature, and develop general tools which are then used to obtain all possible such factorizations of a given pseudo-Polynomial Function.
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Pseudo-Polynomial Functions over finite distributive lattices
arXiv: Rings and Algebras, 2011Co-Authors: Miguel Couceiro, Tamas WaldhauserAbstract:In this paper we consider an aggregation model f: X1 x ... x Xn --> Y for arbitrary sets X1, ..., Xn and a finite distributive lattice Y, factorizable as f(x1, ..., xn) = p(u1(x1), ..., un(xn)), where p is an n-variable lattice Polynomial Function over Y, and each uk is a map from Xk to Y. The resulting Functions are referred to as pseudo-Polynomial Functions. We present an axiomatization for this class of pseudo-Polynomial Functions which differs from the previous ones both in flavour and nature, and develop general tools which are then used to obtain all possible such factorizations of a given pseudo-Polynomial Function.
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Self-commuting lattice Polynomial Functions on chains
Aequationes mathematicae, 2010Co-Authors: Miguel Couceiro, Erkko LehtonenAbstract:We provide sufficient conditions for a lattice Polynomial Function to be self-commuting. We explicitly describe self-commuting Polynomial Functions over chains.Comment: 10 page
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Quasi-Polynomial Functions over bounded distributive lattices
Aequationes mathematicae, 2010Co-Authors: Miguel Couceiro, Jean-luc MarichalAbstract:In [6] the authors introduced the notion of quasi-Polynomial Function as being a mapping f : X n → X defined and valued on a bounded chain X and which can be factorized as \({f(x_1,\ldots,x_n)=p(\varphi(x_1),\ldots,\varphi(x_n))}\) , where p is a Polynomial Function (i.e., a combination of variables and constants using the chain operations \({\wedge}\) and \({\vee}\)) and \({\varphi}\) is an order-preserving map. In the current paper we study this notion in the more general setting where the underlying domain and codomain sets are, possibly different, bounded distributive lattices, and where the inner Function is not necessarily order-preserving. These Functions appear naturally within the scope of decision making under uncertainty since, as shown in this paper, they subsume overall preference Functionals associated with Sugeno integrals whose variables are transformed by a given utility Function. To axiomatize the class of quasi-Polynomial Functions, we propose several generalizations of well-established properties in aggregation theory, as well as show that some of the characterizations given in [6] still hold in this general setting. Moreover, we investigate the so-called transformed Polynomial Functions (essentially, compositions of unary mappings with Polynomial Functions) and show that, under certain conditions, they reduce to quasi-Polynomial Functions.
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IFSA/EUSFLAT Conf. - Quasi-Polynomial Functions on bounded chains
2009Co-Authors: Miguel Couceiro, Jean-luc MarichalAbstract:Two emergent properties in aggregation theory are in- vestigated, namely horizontal maxitivity and comonotonic maxitivity (as well as their dual counterparts) which are commonly defined by means of certain Functional equations. We present complete descrip- tions of the Function classes axiomatized by each of these proper- ties, up to weak versions of monotonicity, in the cases of horizontal maxitivity and minitivity. While studying the classes axiomatized by combinations of these properties, we introduce the concept of quasi- Polynomial Function which appears as a natural extension of the well-established notion of Polynomial Function. We present further axiomatizations for this class both in terms of Functional equations and natural relaxations of homogeneity and median decomposabil- ity. As noteworthy particular cases, we investigate those subclasses of quasi-term Functions and quasi-weighted maximum and minimum Functions, and present characterizations accordingly. Keywords— Discrete Sugeno integral, quasi-Polynomial func- tion, horizontal maxitivity and minitivity, comonotonic maxitivity and minitivity, Functional equation.
Tamas Waldhauser - One of the best experts on this subject based on the ideXlab platform.
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Pseudo-Polynomial Functions over finite distributive lattices
Fuzzy Sets and Systems, 2014Co-Authors: Miguel Couceiro, Tamas WaldhauserAbstract:In this paper we extend the authors’ previous works by considering a multi-attribute aggregation model based on a composition of a Polynomial Function over a finite distributive lattice with local utility Functions; these are referred to as pseudo-Polynomial Functions. We present an axiomatization for this class of pseudo-Polynomial Functions which differs from the previous ones both in flavour and nature, and develop general tools which are then used to obtain all possible such factorizations of a given pseudo-Polynomial Function.
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Pseudo-Polynomial Functions over finite distributive lattices
arXiv: Rings and Algebras, 2011Co-Authors: Miguel Couceiro, Tamas WaldhauserAbstract:In this paper we consider an aggregation model f: X1 x ... x Xn --> Y for arbitrary sets X1, ..., Xn and a finite distributive lattice Y, factorizable as f(x1, ..., xn) = p(u1(x1), ..., un(xn)), where p is an n-variable lattice Polynomial Function over Y, and each uk is a map from Xk to Y. The resulting Functions are referred to as pseudo-Polynomial Functions. We present an axiomatization for this class of pseudo-Polynomial Functions which differs from the previous ones both in flavour and nature, and develop general tools which are then used to obtain all possible such factorizations of a given pseudo-Polynomial Function.
Javad Lavaei - One of the best experts on this subject based on the ideXlab platform.
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CDC - Inverse Function theorem for Polynomial equations using semidefinite programming
2015 54th IEEE Conference on Decision and Control (CDC), 2015Co-Authors: Morteza Ashraphijuo, Ramtin Madani, Javad LavaeiAbstract:This paper is concerned with obtaining the inverse of Polynomial Functions using semidefinite programming (SDP). Given a Polynomial Function and a nominal point at which the Jacobian of the Function is invertible, the inverse Function theorem states that the inverse of the Polynomial Function exists at a neighborhood of the nominal point. In this work, we show that this inverse Function can be found locally using convex optimization. More precisely, we propose infinitely many SDPs, each of which finds the inverse Function at a neighborhood of the nominal point. We also design a convex optimization to check the existence of an SDP problem that finds the inverse of the Polynomial Function at multiple nominal points and a neighborhood around each point. This makes it possible to identify an SDP problem (if any) that finds the inverse Function over a large region. As an application, any system of Polynomial equations can be solved by means of the proposed SDP problem whenever an approximate solution is available. The method developed in this work is numerically compared with Newton's method and the nuclear-norm technique.
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Inverse Function theorem for Polynomial equations using semidefinite programming
2015 54th IEEE Conference on Decision and Control (CDC), 2015Co-Authors: Morteza Ashraphijuo, Ramtin Madani, Javad LavaeiAbstract:This paper is concerned with obtaining the inverse of Polynomial Functions using semidefinite programming (SDP). Given a Polynomial Function and a nominal point at which the Jacobian of the Function is invertible, the inverse Function theorem states that the inverse of the Polynomial Function exists at a neighborhood of the nominal point. In this work, we show that this inverse Function can be found locally using convex optimization. More precisely, we propose infinitely many SDPs, each of which finds the inverse Function at a neighborhood of the nominal point. We also design a convex optimization to check the existence of an SDP problem that finds the inverse of the Polynomial Function at multiple nominal points and a neighborhood around each point. This makes it possible to identify an SDP problem (if any) that finds the inverse Function over a large region. As an application, any system of Polynomial equations can be solved by means of the proposed SDP problem whenever an approximate solution is available. The method developed in this work is numerically compared with Newton's method and the nuclear-norm technique.
Morteza Ashraphijuo - One of the best experts on this subject based on the ideXlab platform.
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CDC - Inverse Function theorem for Polynomial equations using semidefinite programming
2015 54th IEEE Conference on Decision and Control (CDC), 2015Co-Authors: Morteza Ashraphijuo, Ramtin Madani, Javad LavaeiAbstract:This paper is concerned with obtaining the inverse of Polynomial Functions using semidefinite programming (SDP). Given a Polynomial Function and a nominal point at which the Jacobian of the Function is invertible, the inverse Function theorem states that the inverse of the Polynomial Function exists at a neighborhood of the nominal point. In this work, we show that this inverse Function can be found locally using convex optimization. More precisely, we propose infinitely many SDPs, each of which finds the inverse Function at a neighborhood of the nominal point. We also design a convex optimization to check the existence of an SDP problem that finds the inverse of the Polynomial Function at multiple nominal points and a neighborhood around each point. This makes it possible to identify an SDP problem (if any) that finds the inverse Function over a large region. As an application, any system of Polynomial equations can be solved by means of the proposed SDP problem whenever an approximate solution is available. The method developed in this work is numerically compared with Newton's method and the nuclear-norm technique.
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Inverse Function theorem for Polynomial equations using semidefinite programming
2015 54th IEEE Conference on Decision and Control (CDC), 2015Co-Authors: Morteza Ashraphijuo, Ramtin Madani, Javad LavaeiAbstract:This paper is concerned with obtaining the inverse of Polynomial Functions using semidefinite programming (SDP). Given a Polynomial Function and a nominal point at which the Jacobian of the Function is invertible, the inverse Function theorem states that the inverse of the Polynomial Function exists at a neighborhood of the nominal point. In this work, we show that this inverse Function can be found locally using convex optimization. More precisely, we propose infinitely many SDPs, each of which finds the inverse Function at a neighborhood of the nominal point. We also design a convex optimization to check the existence of an SDP problem that finds the inverse of the Polynomial Function at multiple nominal points and a neighborhood around each point. This makes it possible to identify an SDP problem (if any) that finds the inverse Function over a large region. As an application, any system of Polynomial equations can be solved by means of the proposed SDP problem whenever an approximate solution is available. The method developed in this work is numerically compared with Newton's method and the nuclear-norm technique.
Masato Seki - One of the best experts on this subject based on the ideXlab platform.
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multiplicative update for a class of constrained optimization problems related to nmf and its global convergence
European Signal Processing Conference, 2016Co-Authors: Norikazu Takahashi, Masato SekiAbstract:Multiplicative updates are widely used for nonnegative matrix factorization (NMF) as an efficient computational method. In this paper, we consider a class of constrained optimization problems in which a Polynomial Function of the product of two matrices is minimized subject to the nonnegativity constraints. These problems are closely related to NMF because the Polynomial Function covers many error Function used for NMF. We first derive a multiplicative update rule for those problems by using the unified method developed by Yang and Oja. We next prove that a modified version of the update rule has the global convergence property in the sense of Zangwill under certain conditions. This result can be applied to many existing multiplicative update rules for NMF to guarantee their global convergence.
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EUSIPCO - Multiplicative update for a class of constrained optimization problems related to NMF and its global convergence
2016 24th European Signal Processing Conference (EUSIPCO), 2016Co-Authors: Norikazu Takahashi, Masato SekiAbstract:Multiplicative updates are widely used for nonnegative matrix factorization (NMF) as an efficient computational method. In this paper, we consider a class of constrained optimization problems in which a Polynomial Function of the product of two matrices is minimized subject to the nonnegativity constraints. These problems are closely related to NMF because the Polynomial Function covers many error Function used for NMF. We first derive a multiplicative update rule for those problems by using the unified method developed by Yang and Oja. We next prove that a modified version of the update rule has the global convergence property in the sense of Zangwill under certain conditions. This result can be applied to many existing multiplicative update rules for NMF to guarantee their global convergence.