Showing papers on "Extreme point published in 2011"
TL;DR: Experimental results demonstrate that the proposed method can reduce training set considerably, while the obtained model maintains generalization capability to the level of a model trained on the full training set, but uses less support vectors and exhibits faster training speed.
67 citations
TL;DR: In this article, the authors give a characterization of the minimal tropical half-spaces containing a given tropical polyhedron, from which they derive a counter-example showing that the number of such minimal halfspaces can be infinite, contradicting some statements which appeared in the tropical literature.
Abstract: We give a characterization of the minimal tropical half-spaces containing a given tropical polyhedron, from which we derive a counter-example showing that the number of such minimal half-spaces can be infinite, contradicting some statements which appeared in the tropical literature, and disproving a conjecture of F. Block and J. Yu. We also establish an analogue of the Minkowski---Weyl theorem, showing that a tropical polyhedron can be equivalently represented internally (in terms of extreme points and rays) or externally (in terms of half-spaces containing it). A canonical external representation of a polyhedron turns out to be provided by the extreme elements of its tropical polar. We characterize these extreme elements, showing in particular that they are determined by support vectors.
47 citations
TL;DR: The celebrated upper bound theorem of McMullen determines the maximal number of extreme points of a polyhedron in terms of its dimension and the number of constraints which define it, showing that the maximum is attained by the polar of the cyclic polytope.
42 citations
TL;DR: A distance approach based on extreme points, or predefined ideal and anti-ideal points, is proposed to improve on the TOPSIS method of multiple criteria ranking, and produces results that are generally consistent with the original technique but offers new features such as a clear interpretation of extreme points.
30 citations
TL;DR: In this article, the authors introduced the notion of Volterra quadratic stochastic operators of a two-sex population and constructed several Lyapunov functions for these operators.
Abstract: We introduce the notion of Volterra quadratic stochastic operators of a two-sex population. The description of the fixed points of Volterra quadratic stochastic operators of a two-sex population is reduced to the description of the fixed points of Volterra-type operators. Several Lyapunov functions are constructed for the Volterra quadratic stochastic operators of a two-sex population. By using these functions, we obtain an upper bound for the ω-limit set of trajectories. It is shown that the set of all Volterra quadratic stochastic operators of a two-sex population is a convex compact set, and the extreme points of this set are found. Volterra quadratic stochastic operators of a two-sex population that have a 2-periodic orbit (trajectory) are constructed.
29 citations
TL;DR: Several new characterizations of correlated equilibria in games with continuous utility functions are presented, which have the advantage of being more computationally and analytically tractable than the standard definition in terms of departure functions.
27 citations
TL;DR: This work proposes an alternative clipped extension of classical MDM that results in a simpler algorithm with the same classification accuracy than that of the extensions already mentioned, but also with a much faster numerical convergence.
27 citations
TL;DR: A class of necessary linear constraints on standard imsets are introduced and a conjecture that these constraints characterize the polytope P is formed, which has been confirmed in the case of (at most) 4 variables.
16 citations
Dissertation•
01 Jan 2011
TL;DR: In this article, Voda et al. showed that a Loewner chain with minimal regularity assumptions (Df (0, ·) of local bounded variation) satisfies an associated LoWner equation, and showed that the existence of a bounded solution depends on the real resonance of A, but there always exists a polynomially bounded solution.
Abstract: Loewner Theory in Several Complex Variables and Related Problems Mircea Iulian Voda Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2011 The first part of the thesis deals with aspects of Loewner theory in several complex variables. First we show that a Loewner chain with minimal regularity assumptions (Df (0, ·) of local bounded variation) satisfies an associated Loewner equation. Next we give a way of renormalizing a general Loewner chain so that it corresponds to the same increasing family of domains. To do this we will prove a generalization of the converse of Caratheodory’s kernel convergence theorem. Next we address the problem of finding a Loewner chain solution to a given Loewner chain equation. The main result is a complete solution in the case when the infinitesimal generator satisfies Dh (0, t) = A where inf {Re 〈Az, z〉 : ‖z‖ = 1} > 0. We will see that the existence of a bounded solution depends on the real resonances of A, but there always exists a polynomially bounded solution. Finally we discuss some properties of classes of biholomorphic mappings associated to A-normalized Loewner chains. In particular we give a characterization of the compactness of the class of spirallike mappings in terms of the resonance of A. The second part of the thesis deals with the problem of finding examples of extreme points for some classes of mappings. We see that straightforward generalizations of one dimensional extreme functions give examples of extreme Caratheodory mappings and extreme starlike mappings on the polydisc, but not on the ball. We also find examples of extreme Caratheodory mappings on the ball starting from a known example of extreme Caratheodory function in higher dimensions.
13 citations
Posted Content•
TL;DR: In this article, the authors give a necessary and sufficient condition for a channel to be extreme point of a convex set in terms of a complementary channel, a notion of big importance in quantum information theory.
Abstract: The set of all channels with fixed input and output is convex. We first give a convenient formulation of necessary and sufficient condition for a channel to be extreme point of this set in terms of complementary channel, a notion of big importance in quantum information theory. This formulation is based on the general approach to extremality of completely positive maps in an operator algebra due to Arveson. We then apply this formulation to prove the main result of this note: under certain nondegeneracy conditions, purity of the environment is necessary and sufficient for extremality of Bosonic linear (quasi-free) channel. It follows that Gaussian channel between finite-mode Bosonic systems is extreme if and only if it has minimal noise.
13 citations
Posted Content•
TL;DR: This paper shows how one can extend Marcet and Marimon's recursive saddle point method to a weakly concave Pareto frontier by expanding the state space to include the realizations of an end of period lottery over the extreme points of a flat region of the Pare to frontier.
Abstract: Marcet and Marimon (1994, revised 1998) developed a recursive saddle point method which can be used to solve dynamic contracting problems that include participation, enforcement and incentive constraints. Their method uses a recursive multiplier to capture implicit prior promises to the agent(s) that were made in order to satisfy earlier instances of these constraints. As a result, their method relies on the invertibility of the derivative of the Pareto frontier and cannot be applied to problems for which this frontier is not strictly concave. In this paper we show how one can extend their method to a weakly concave Pareto frontier by expanding the state space to include the realizations of an end of period lottery over the extreme points of a .at region of the Pareto frontier. With this expansion the basic insight of Marcet and Marimon goes through .one can make the problem recursive in the Lagrangian multiplier which yields significant computational advantages over the conventional approach of using utility as the state variable. The case of a weakly concave Pareto frontier arises naturally in applications where the principal’s choice set is not convex but where randomization is possible.
TL;DR: The robust network design problem with (splittable) dynamic routing is polynomially solvable, whereas it is co-NP-hard in the general case, which applies to particular instances of the single-source Hose model.
Posted Content•
01 Jan 2011
TL;DR: In this paper, it was shown that in additional hypotesys (like the linearity) the nature of a stationary point remains the same for the restricted function and for the initial one.
Abstract: Many papers treat the classical problem of the determination function’s extreme points subject to equality type restrictions. The well-known method of Lagrange’s multipliers gives necessary conditions but not sufficient. In this paper, it is shown that in additional hypotesys (like the linearity) the nature of a stationary point remains the same for the restricted function and for the initial one.
TL;DR: In this article, the authors analyzed the mechanisms related to the secondary splitting of zero-gradient points of scalar fields using the two-dimensional case of a scalar extreme point lying in a region of local strain.
Abstract: The mechanisms related to the secondary splitting of zero-gradient points of scalar fields are analyzed using the two-dimensional case of a scalar extreme point lying in a region of local strain. The velocity field is assumed to resemble a stagnation-point flow, cf. Gibson (Phys Fluids 11:2305–2315, 1968), which is approximated using a Taylor expansion up to third order. The temporal evolution of the scalar field in the vicinity of the stagnation point is derived using a series expansion, and it is found that the splitting can only be explained when the third-order terms of the Taylor expansion of the flow field are included. The non-dimensional splitting time turns out to depend on three parameters, namely the local Peclet number Pe
δ
based on the initial size of the extreme point δ and two parameters which are measures of the rate of change of the local strain. For the limiting casePe
δ
→ 0, the splitting time is found to be finite but Peclet-number independent, while for the case of Pe
δ
→ ∞ it increases logarithmically with the Peclet number. The physical implications of the two-dimensional mathematical solution are discussed and compared with the splitting times obtained numerically from a Taylor–Green vortex.
01 Jan 2011
TL;DR: In this paper, a comprehensive family of analytic univalent functions is introduced and studied, which contains various well-known classes of analytic functions as well as many new ones, and coefficient bounds, distortion bounds, and extreme points are derived.
Abstract: We introduce and study a comprehensive family of analytic univalent functions which contains various well-known classes of analytic univalent functions as well as many new ones. In this paper, we obtain coefficient bounds, distortion bounds and extreme points. Convolution conditions and convex combination are also determined for functions in this family.
08 Aug 2011
TL;DR: The output tracking problem for linear systems subject to actuator faults, saturation and disturbances is studied and the conditions for perfecting tracking are derived in the form of linear matrix inequality (LMI).
Abstract: The output tracking problem for linear systems subject to actuator faults, saturation and disturbances is studied in this paper. By incorporating an intelligent integrator and some formulation of the equilibrium points, the zero steady-state error tracking of constant reference command is converted into the convergence of the equilibrium point. Then a saturated state feedback controller and the associated domain of stability are synthesized. However, the equilibrium point of the closed-loop system may change due to actuator faults. To guarantee all the equilibrium points are converged subject to defined faults, two methods are proposed by building the extreme point model and the middle point model for continuous actuator faults. All the conditions for perfecting tracking are derived in the form of linear matrix inequality (LMI). The results of the illustrating example verify the effectiveness of the presented design techniques.
Journal Article•
TL;DR: In this paper, a subclass of harmonic univalent functions defined by convolution and integral convolution is studied, and a class of functions with coefficients, extreme points, distortion, convolution conditions and convex combination are determined.
Abstract: In this paper, we introduce and study a subclass of harmonic univalent functions defined by convolution and integral convolution. Coefficient bounds, extreme points, distortion bounds, convolution conditions and convex combination are determined for functions in this family. Consequently, many of our results are either extensions or new approaches to those corresponding to previously known results.
TL;DR: It is proved that if a given undirected graph is Hamiltonian, then with probability one this random walk algorithm detects its Hamiltonicity in a finite number of iterations.
Abstract: We develop a new, random walk-based, algorithm for the Hamiltonian cycle problem. The random walk is on pairs of extreme points of two suitably constructed polytopes. The latter are derived from geometric properties of the space of discounted occupational measures corresponding to a given graph. The algorithm searches for a measure that corresponds to a common extreme point in these two specially constructed polyhedral subsets of that space. We prove that if a given undirected graph is Hamiltonian, then with probability one this random walk algorithm detects its Hamiltonicity in a finite number of iterations. We support these theoretical results by numerical experiments that demonstrate a surprisingly slow growth in the required number of iterations with the size of the given graph.
Posted Content•
TL;DR: In this paper, the Fiedler vector for the graph Laplacian of a tree has its most extreme values at the vertices which are the farthest apart from the center of the tree.
Abstract: In this paper I present a counter-example to the conjecture: The Fiedler vector for the graph Laplacian of a tree has its most extreme values at the verticies which are the farthest apart. This counter-example looks roughly like a flower and so I have named it the "Fiedler rose".
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TL;DR: The hybrid-LP is a hybrid method that uses a computationally efficient pivot to move in the interior of the feasible region in its first phase, which reduces both the number of iterations and the running time compared to the simplex method.
TL;DR: In this article, the authors represent quantum observables as normalized positive operator valued measures and consider convex sets of observables which are covariant with respect to a unitary representation of a locally compact Abelian symmetry group G. The value space of such observables is a transitive G-space.
Abstract: We represent quantum observables as normalized positive operator valued measures and consider convex sets of observables which are covariant with respect to a unitary representation of a locally compact Abelian symmetry group G. The value space of such observables is a transitive G-space. We characterize the extreme points of covariant observables and also determine the covariant extreme points of the larger set of all quantum observables. The results are applied to position, position difference, and time observables.
01 Mar 2011
TL;DR: Chu and Cohen as mentioned in this paper showed that if there exists an isomorphism T: U c (X) → U c(Y) with ∥T∥·∥T -1 ∥ < 2, then ext X is homeomorphic to ext Y.
Abstract: Let X, Y be compact convex sets such that every extreme point of X and Y is a weak peak point and both ext X and ext Y are Lindelof spaces. We prove that if there exists an isomorphism T: U c (X) → U c (Y) with ∥T∥·∥T -1 ∥ < 2, then ext X is homeomorphic to ext Y. This generalizes results of C. H. Chu and H. B. Cohen.
TL;DR: The aim of the paper is to compute the set of the parameters (corresponding to one coefficient) for which a given extreme point is a weakly efficient solution in multiple objective linear programming (MOLP).
Abstract: This paper, studies the sensitivity analysis of weakly efficient extreme solutions in multiple objective linear programming (MOLP). The aim of the paper is to compute the set of the parameters (corresponding to one coefficient) for which a given extreme point is a weakly efficient solution. We also focus on the properties of the parameters set by proving convexity and closeness of this set. Moreover, we compare the results of the sensitivity analysis of efficiency and of weak efficiency in MOLP.
TL;DR: In this article, it was shown that a state on a quantum structure E like an effect algebra, a pseudo effect algebra E satisfying some kind of the Riesz Decomposition Properties (RDP) or on an MV-algebra, a BL- algebra, and a pseudo BL algebra is an integral through a regular Borel probability measure defined on the Borel σ -algebra of a Choquet simplex.
Abstract: We show conditions when a state on a quantum structure E like an effect algebra, a pseudo effect algebra E satisfying some kind of the Riesz Decomposition Properties (RDP) or on an MV-algebra, a BL-algebra, a pseudo MV-algebra and a pseudo BL-algebra is an integral through a regular Borel probability measure defined on the Borel σ-algebra of a Choquet simplex K. In particular, if E satisfies the strongest type of (RDP), the representing Borel probability measure can be uniquely chosen to have its support in the set of the extreme points of K. The same is true for states on an MV-algebra and a BL-algebra and their noncommutative variants.
TL;DR: In this article, the authors studied the properties of a(G), particularly its spectrum and its dual von Neumann algebra, and showed that G∗ is the set of Dirac measures of the dual group Ĝ, and l(G) can be identified as l(Ĝ).
Abstract: Let G be a locally compact group, G∗ be the set of all extreme points of the set of normalized continuous positive definite functions of G, and a(G) be the closed subalgebra generated by G∗ in B(G). When G is abelian, G∗ is the set of Dirac measures of the dual group Ĝ, and a(G) can be identified as l(Ĝ). We study the properties of a(G), particularly its spectrum and its dual von Neumann algebra.
TL;DR: In this article, it was shown that the transition probability for the particles to pass through windows E ˜ k at times t k, satisfies, in a new set of variables, a non-linear PDE which can be expressed as a near-Wronskian; that is a determinant of a matrix of size p + 1, with each row being a derivative of the previous, except for the last column.
TL;DR: In this article, it was shown that every state on an interval pseudo effect algebra E satisfying an appropriate version of the Riesz Decomposition Property (RDP for short) is an integral through a regular Borel probability measure defined on the Borel σ-algebra of a Choquet simplex K.
Abstract: We show that every state on an interval pseudo effect algebra E satisfying an appropriate version of the Riesz Decomposition Property (RDP for short) is an integral through a regular Borel probability measure defined on the Borel σ-algebra of a Choquet simplex K. In particular, if E satisfies the strongest type of RDP, the representing Borel probability measure can be uniquely chosen to have its support in the set of the extreme points of K.
Dissertation•
01 May 2011
TL;DR: In this article, the variational principle is proved for a large class of functions, and a more restrictive modulus of continuity condition is provided that guarantees a function's Gibbs measures to be a nonempty, weakly compact, convex set of measures that coincides with the sets of measures obeying a form of the DLR equations.
Abstract: Consider a multidimensional shift space with a countably infinite alphabet, which serves in mathematical physics as a classical lattice gas or lattice spin system. A new definition of a Gibbs measure is introduced for suitable real-valued functions of the configuration space, which play the physical role of specific internal energy. The variational principle is proved for a large class of functions, and then a more restrictive modulus of continuity condition is provided that guarantees a function's Gibbs measures to be a nonempty, weakly compact, convex set of measures that coincides with the set of measures obeying a form of the DLR equations (which has been adapted so as to be stated entirely in terms of specific internal energy instead of the Hamiltonians for an interaction potential). The variational equilibrium measures for a such a function are then characterized as the shift invariant Gibbs measures of finite entropy, and a condition is provided to determine if a function's Gibbs measures have infinite entropy or not. Moreover the spatially averaged limiting Gibbs measures, i.e. constructive equilibria, are shown to exist and their weakly closed convex hull is shown to coincide with the set of true variational equilibrium measures. It follows that the "pure thermodynamic phases", which correspond to the extreme points in the convex set of equilibrium measures, must be constructive equilibria. Finally, for an even smoother class of functions a method is presented to construct a compatible interaction potential and it is checked that the two different structures generate the same sets of Gibbs and equilibrium measures, respectively.
Posted Content•
TL;DR: The key point of this approach is that the conditions are much more explicit and can be tested in a more direct manner, removing the need for a reduction algorithm.
Abstract: We study a mixed integer linear program with m integer variables and k nonnegative continuous variables in the form of the relaxation of the corner polyhedron that was introduced by Andersen, Louveaux, Weismantel and Wolsey [Inequalities from two rows of a simplex tableau, Proc. IPCO 2007, LNCS, vol. 4513, Springer, pp. 1{15]. We describe the facets of this mixed integer linear program via the extreme points of a well-dened polyhedron. We then utilize this description to give polynomial time algorithms to derive valid inequalities with optimal lp norm for arbitrary, but xed m. For the case of m = 2, we give a renement and a new proof of a characterization of the facets by Cornu ejols and Margot [ On the facets of mixed integer programs with two integer variables and two constraints, Math. Programming 120 (2009), 429{456]. The key point of our approach is that the conditions are much more explicit and can be tested in a more direct manner, removing the need for a reduction algorithm. These results allow us to show that the relaxed corner polyhedron has only polynomially many facets.
24 Jul 2011
TL;DR: This paper presents an extreme-point-based finite-termination procedure for obtaining a global maximizer of the associated Lagrangian dual problem, which is, in general, a piecewise-affine concave function.
Abstract: Prices in electricity markets are given by the dual variables associated with the supply-demand constraint in the dispatch problem. However, in unit-commitment-based day-ahead markets, these variables are less easy to obtain. A common approach relies on resolving the dispatch problem with the commitment decisions fixed and utilizing the associated dual variables. Yet, this avenue leads to inadequate revenues to generators and necessitates an uplift payment to be made by the market operator. Recently, a convex hull pricing scheme has been proposed to reduce the impact of such payments and requires the global maximization of the associated Lagrangian dual problem, which is, in general, a piecewise-affine concave function. In this paper, we present an extreme-point-based finite-termination procedure for obtaining such a global maximizer. Unlike standard subgradient schemes where an arbitrary subgradient is used, we present a novel technique where the steepest ascent direction is constructed by solving a continuous quadratic program. The scheme initiates a move along this direction with an a priori constant steplength, with the intent of reaching the boundary of the face. A backtracking scheme allows for mitigating the impact of excessively large steps. Termination of the scheme occurs when the set of subgradients contains the zero vector. Preliminary numerical tests are seen to be promising and display the finite-termination property. Furthermore, the scheme is seen to significantly outperform standard subgradient methods.