scholarly journals Lipschitz Continuity of the Solution Mapping of Symmetric Cone Complementarity Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Xin-He Miao ◽  
Jein-Shan Chen
Keyword(s):  
Lipschitz Continuity ◽  
Complementarity Problems ◽  
Euclidean Jordan Algebra ◽  
Symmetric Cone ◽  
Monotonicity Property ◽  
Jordan Algebras ◽  
Strong Monotonicity ◽  
Linear Transformations ◽  
Lipschitz Continuous ◽  
Solution Mapping

This paper investigates the Lipschitz continuity of the solution mapping of symmetric cone (linear or nonlinear) complementarity problems (SCLCP or SCCP, resp.) over Euclidean Jordan algebras. We show that if the transformation has uniform CartesianP-property, then the solution mapping of the SCCP is Lipschitz continuous. Moreover, we establish that the monotonicity of mapping and the Lipschitz continuity of solutions of the SCLCP imply ultraP-property, which is a concept recently developed for linear transformations on Euclidean Jordan algebra. For a Lyapunov transformation, we prove that the strong monotonicity property, the ultraP-property, the CartesianP-property, and the Lipschitz continuity of the solutions are all equivalent to each other.

A REPRESENTATION THEOREM FOR LYAPUNOV-LIKE TRANSFORMATIONS ON EUCLIDEAN JORDAN ALGEBRAS

2013 ◽  
Vol 15 (04) ◽  
pp. 1340034 ◽  
Author(s):  
JIYUAN TAO ◽  
M. SEETHARAMA GOWDA
Keyword(s):  
Jordan Algebra ◽  
Linear Transformation ◽  
Representation Theorem ◽  
Elementary Proof ◽  
Euclidean Jordan Algebra ◽  
Symmetric Cone ◽  
Jordan Algebras ◽  
Euclidean Jordan Algebras

A Lyapunov-like (linear) transformation L on a Euclidean Jordan algebra V is defined by the condition [Formula: see text]where K is the symmetric cone of V. In this paper, we give an elementary proof (avoiding Lie algebraic ideas and results) of the fact that Lyapunov-like transformations on V are of the form La + D, where a ∈ V, D is a derivation, and La(x) = a ◦ x for all x ∈ V.

scholarly journals An Arc Search Interior-Point Algorithm for Monotone Linear Complementarity Problems over Symmetric Cones

2018 ◽  
Vol 23 (1) ◽  
pp. 1-16
Author(s):  
Mohammad Pirhaji ◽  
Maryam Zangiabadi ◽  
Hossein Mansouri ◽  
Saman H. Amin
Keyword(s):  
Interior Point ◽  
Complementarity Problems ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Symmetric Cone ◽  
Central Path ◽  
Interior Point Algorithm ◽  
Complexity Bound ◽  
Wide Neighborhood ◽  
Mathematical Problems

An arc search interior-point algorithm for monotone symmetric cone linear complementarity problem is presented. The algorithm estimates the central path by an ellipse and follows an ellipsoidal approximation of the central path to reach an "-approximate solution of the problem in a wide neighborhood of the central path. The convergence analysis of the algorithm is derived. Furthermore, we prove that the algorithm has the complexity bound O ( p rL) using Nesterov-Todd search direction and O (rL) by the xs and sx search directions. The obtained iteration complexities coincide with the best-known ones obtained by any proposed interior- point algorithm for this class of mathematical problems.

Statistical Inference of Second-Order Cone Programming

2019 ◽  
Vol 36 (02) ◽  
pp. 1940003
Author(s):  
Liwei Zhang ◽  
Shengzhe Gao ◽  
Saoyan Guo
Keyword(s):  
Lipschitz Continuity ◽  
Metric Regularity ◽  
Optimal Solution ◽  
Second Order ◽  
Lipschitz Continuous ◽  
Set Valued Mapping ◽  
Second Order Cone ◽  
Degeneracy Condition ◽  
Optimal Solution Set ◽  
The Stability

In this paper, we study the stability of stochastic second-order programming when the probability measure is perturbed. Under the Lipschitz continuity of the objective function and metric regularity of the feasible set-valued mapping, the outer semicontinuity of the optimal solution set and Lipschitz continuity of optimal values are demonstrated. Moreover, we prove that, if the constraint non-degeneracy condition and strong second-order sufficient condition hold at a local minimum point of the original problem, there exists a Lipschitz continuous solution path satisfying the Karush–Kuhn–Tucker conditions.

LIPSCHITZ CONTINUITY FOR H-CONVEX FUNCTIONS IN CARNOT GROUPS

2006 ◽  
Vol 08 (01) ◽  
pp. 1-8 ◽  
Author(s):  
MINGBAO SUN ◽  
XIAOPING YANG
Keyword(s):  
Convex Functions ◽  
Lipschitz Continuity ◽  
Carnot Group ◽  
Carnot Groups ◽  
Lipschitz Continuous ◽  
Locally Lipschitz ◽  
Locally Lipschitz Continuous

For a Carnot group G of step two, we prove that H-convex functions are locally bounded from above. Therefore, H-convex functions on a Carnot group G of step two are locally Lipschitz continuous by using recent results by Magnani.

Some New Results for Linear Transformations on Euclidean Jordan Algebras

2009 ◽  
Author(s):  
Jiyuan Tao ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras
Keyword(s):  
Jordan Algebras ◽  
Linear Transformations ◽  
Euclidean Jordan Algebras

scholarly journals Lipschitz Continuity for the Solutions of Triharmonic Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Zhou Yu ◽  
Xiao Bing
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Dirichlet Boundary ◽  
Lipschitz Continuity ◽  
Boundary Value ◽  
Jordan Domain ◽  
Lipschitz Continuous ◽  
Equivalent Conditions

Let D be the unit disk in the complex plane C and denote T=∂D. Write Hom+T,∂Ω for the class of all sense-preserving homeomorphism of T onto the boundary of a C2 convex Jordan domain Ω. In this paper, five equivalent conditions for the solutions of triharmonic equations ∂z∂z¯3ω=ff∈CD¯ with Dirichlet boundary value conditions ωzz¯zz¯T=γ2∈CT,ωzz¯T=γ1∈CT and ωT=γ0∈Hom+T,∂Ω to be Lipschitz continuous are presented.

Sign in / Sign up

Export Citation Format

Share Document