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1.
P. Sawyer 《Transactions of the American Mathematical Society》1997,349(9):3569-3584
In this note, we obtain a recursive formula for the spherical functions associated with the symmetric cone of a formally real Jordan algebra. We use this formula as an inspiration for a similar recursive formula involving the Jack polynomials.
2.
We show that penalized functions of the Fischer–Burmeister and the natural residual functions defined on symmetric cones are
complementarity functions. Boundedness of the solution set of a symmetric cone complementarity problem, based on the penalized
natural residual function, is proved under monotonicity and strict feasibility. The proof relies on a trace inequality on
Euclidean Jordan algebras. 相似文献
3.
In this paper we show that the so-called commutative class of primal-dual interior point algorithms which were designed by
Monteiro and Zhang for semidefinite programming extends word-for-word to optimization problems over all symmetric cones. The
machinery of Euclidean Jordan algebras is used to carry out this extension. Unlike some non-commutative algorithms such as
the XS+SX method, this class of extensions does not use concepts outside of the Euclidean Jordan algebras. In particular no assumption
is made about representability of the underlying Jordan algebra. As a special case, we prove polynomial iteration complexities
for variants of the short-, semi-long-, and long-step path-following algorithms using the Nesterov-Todd, XS, or SX directions.
Received: April 2000 / Accepted: May 2002
Published online: March 28, 2003
RID="⋆"
ID="⋆" Part of this research was conducted when the first author was a postdoctoral associate at Center for Computational
Optimization at Columbia University.
RID="⋆⋆"
ID="⋆⋆" Research supported in part by the U.S. National Science Foundation grant CCR-9901991 and Office of Naval Research
contract number N00014-96-1-0704. 相似文献
4.
I. A. Drozhzhin 《Mathematical Notes》1989,46(6):920-924
Translated from Matematicheskie Zametki, Vol. 46, No. 6, pp. 46–52, December, 1989. 相似文献
5.
Michael Aristidou 《Bulletin des Sciences Mathématiques》2006,130(3):246
Let Ω be a symmetric cone and V the corresponding simple Euclidean Jordan algebra. In our previous papers (some with G. Zhang) we considered the family of generalized Laguerre functions on Ω that generalize the classical Laguerre functions on R+. This family forms an orthogonal basis for the subspace of L-invariant functions in L2(Ω,dμν), where dμν is a certain measure on the cone and where L is the group of linear transformations on V that leave the cone Ω invariant and fix the identity in Ω. The space L2(Ω,dμν) supports a highest weight representation of the group G of holomorphic diffeomorphisms that act on the tube domain T(Ω)=Ω+iV. In this article we give an explicit formula for the action of the Lie algebra of G and via this action determine second order differential operators which give differential recursion relations for the generalized Laguerre functions generalizing the classical creation, preservation, and annihilation relations for the Laguerre functions on R+. 相似文献
6.
Hongming Ding 《Transactions of the American Mathematical Society》2007,359(7):3239-3250
We obtain the differential equation and recurrence relations satisfied by the Laguerre functions on an arbitrary symmetric cone .
7.
《Communications in Nonlinear Science & Numerical Simulation》2010,15(11):3299-3305
A popular approach to solving the complementarity problem is to reformulate it as an equivalent system of smooth equations via a smoothing complementarity function. In this paper, first we propose a new class of smoothing complementarity functions, which contains the natural residual smoothing function and the Fischer–Burmeister smoothing function for symmetric cone complementarity problems. Then we give some unified formulae of the Fréchet derivatives associated with Jordan product. Finally, the derivative of the new proposed class of smoothing complementarity functions is deduced over symmetric cones. 相似文献
8.
We prove (Theorem 1.1) that if e0>>er>0 are coprime integers, then the Newton functions , i=0,…,r, generate over the field of symmetric rational functions in X1,…,Xr. This generalizes a previous result of us for r=2. This extension requires new methods, including: (i) a study of irreducibility and Galois-theoretic properties of Schur polynomials (Theorem 3.1), and (ii) the study of the dimension of the varieties obtained by intersecting Fermat hypersurfaces (Theorem 4.1). We shall also observe how these results have implications to the study of zeros of linear recurrences over function fields; in particular, we give (Theorem 4.2) a complete classification of the zeros of recurrences of order four with constant coefficients over a function field of dimension 1. 相似文献
9.
Michael Aristidou Mark Davidson Gestur Ólafsson 《Journal of Computational and Applied Mathematics》2007
In this article we derive differential recursion relations for the Laguerre functions on the cone Ω of positive definite real matrices. The highest weight representations of the group Sp(n,R) play a fundamental role. Each such representation acts on a Hilbert space of holomorphic functions on the tube domain Ω+iSym(n,R). We then use the Laplace transform to carry the Lie algebra action over to L2(Ω,dμν). The differential recursion relations result by restricting to a distinguished three-dimensional subalgebra, which is isomorphic to sl(2,R). 相似文献
10.
We prove that for any fixed the generating function of the projection of the set of integer points in a rational -dimensional polytope can be computed in polynomial time. As a corollary, we deduce that various interesting sets of lattice points, notably integer semigroups and (minimal) Hilbert bases of rational cones, have short rational generating functions provided certain parameters (the dimension and the number of generators) are fixed. It follows then that many computational problems for such sets (for example, finding the number of positive integers not representable as a non-negative integer combination of given coprime positive integers ) admit polynomial time algorithms. We also discuss a related problem of computing the Hilbert series of a ring generated by monomials.
11.
I provide a simple example of a quadratic utility function that generates a Giffen demand. The utility function is symmetric, increasing and concave. Interestingly, the Giffen effect arises in the subspace where the utility function is strictly increasing and strictly concave. A full characterization of the parameter conditions under which the Giffen demand arises is provided. 相似文献
12.
《Optimization》2012,61(3):345-377
We consider the extension of primal dual interior point methods for linear programming on symmetric cones, to a wider class of problems that includes approximate necessary optimality conditions for functions expressible as the difference of two convex functions of a special form. Our analysis applies the Jordan-algebraic approach to symmetric cones. As the basic method is local, we apply the idea of the filter method for a globalization strategy. 相似文献
13.
Shaohua Pan 《Operations Research Letters》2010,38(5):372-377
For the symmetric cone complementarity problem, we show that each stationary point of the unconstrained minimization reformulation based on the Fischer-Burmeister merit function is a solution to the problem, provided that the gradient operators of the mappings involved in the problem satisfy column monotonicity or have the Cartesian P0-property. These results answer the open question proposed in the article that appeared in Journal of Mathematical Analysis and Applications 355 (2009) 195-215. 相似文献
14.
Motivated by the similarities between the properties of Z-matrices on $R^{n}_+$ and Lyapunov and Stein transformations on the semidefinite cone $\mathcal {S}^n_+$ , we introduce and study Z-transformations on proper cones. We show that many properties of Z-matrices extend to Z-transformations. We describe the diagonal stability of such a transformation on a symmetric cone by means of quadratic representations. Finally, we study the equivalence of Q and P properties of Z-transformations on symmetric cones. In particular, we prove such an equivalence on the Lorentz cone. 相似文献
15.
Deren Han 《Journal of Mathematical Analysis and Applications》2007,336(1):727-737
One of the popular solution methods for the complementarity problem over symmetric cones is to reformulate it as the global minimization of a certain merit function. An important question to be answered for this class of methods is under what conditions the level sets of the merit function are bounded (the coerciveness of the merit function). In this paper, we introduce the generalized weak-coerciveness of a continuous transformation. Under this condition, we prove the coerciveness of some merit functions, such as the natural residual function, the normal map, and the Fukushima-Yamashita function for complementarity problems over symmetric cones. We note that this is a much milder condition than strong monotonicity, used in the current literature. 相似文献
16.
Yongdo Lim 《Mathematische Zeitschrift》2000,234(2):407-411
17.
《European Journal of Operational Research》2002,141(3):471-479
In this paper, a pair of multiobjective symmetric dual programs over arbitrary cones are formulated for cone-convex functions. Weak, strong, converse and self-duality theorems are proved for these programs. 相似文献
18.
19.
Yongdo Lim 《Mathematische Annalen》2000,316(2):379-389
Let V be a simple Euclidean Jordan algebra with an associative inner product and let be the corresponding symmetric cone. Let be the compact symmetric space of all primitive idempotents of V. We show that the function s(a,b) defined by
is a (the automorphism group of )-invariant complete metric on and it coincides with a natural Finsler distance on We also show that the metric s(a,b) (strictly) contracts any (strict) conformal compression of .
Received: 24 May 1999 / in final form: 15 March 1999 相似文献
20.
A. M. Borodin 《Functional Analysis and Its Applications》2000,34(1):10-23
A method for computing the correlation functions of stochastic point processes arising in the decomposition problem for characters
of the infinite symmetric group is presented.
Supported by the Russian Program for Support of Scientific Schools (grant 96-15-96060).
Institute for Problems in Information Transmission, Russian Academy of Science, and University of Pennsylvania, USA. Translated
from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 34, No. 1, pp. 12–28, January–March, 2000.
Translated by A. M. Borodin. 相似文献