Three classes of merit functions for the complementarity problem over a closed convex cone

Optimization ◽  
2013 ◽  
Vol 62 (4) ◽  
pp. 545-560 ◽  
Author(s):  
Nan Lu ◽  
Zheng-Hai Huang
Keyword(s):  
Complementarity Problem ◽  
Convex Cone ◽  
Closed Convex Cone ◽  
Merit Functions

scholarly journals A complex nonlinear complementarity problem

1978 ◽  
Vol 19 (3) ◽  
pp. 437-444 ◽  
Author(s):  
Sribatsa Nanda ◽  
Sudarsan Nanda
Keyword(s):  
Continuous Function ◽  
Complementarity Problem ◽  
Convex Cone ◽  
Existence And Uniqueness ◽  
Nonlinear Complementarity Problem ◽  
Closed Convex Cone ◽  
Uniqueness Of Solutions ◽  
Existence Of A Solution ◽  
Polar Cone ◽  
Nonlinear Complementarity

In this paper we study the existence and uniqueness of solutions for the following complex nonlinear complementarity problem: find z ∈ S such that g(z) ∈ S* and re(g(z), z) = 0, where S is a closed convex cone in Cn, S* the polar cone, and g is a continuous function from Cn into itself. We show that the existence of a z ∈ S with g(z) ∈ int S* implies the existence of a solution to the nonlinear complementarity problem if g is monotone on S and the solution is unique if g is strictly monotone. We also show that the above problem has a unique solution if the mapping g is strongly monotone on S.

scholarly journals Global Error Bound Estimation for the Generalized Nonlinear Complementarity Problem over a Closed Convex Cone

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Hongchun Sun ◽  
Yiju Wang
Keyword(s):  
Error Bound ◽  
Complementarity Problem ◽  
Convex Cone ◽  
Nonlinear Complementarity Problem ◽  
Global Error ◽  
Closed Convex Cone ◽  
Global Error Bound ◽  
Nonlinear Complementarity ◽  
Bound Estimation

The global error bound estimation for the generalized nonlinear complementarity problem over a closed convex cone (GNCP) is considered. To obtain a global error bound for the GNCP, we first develop an equivalent reformulation of the problem. Based on this, a global error bound for the GNCP is established. The results obtained in this paper can be taken as an extension of previously known results.

scholarly journals Solvability of Two Classes of Tensor Complementarity Problems

2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Yang Xu ◽  
Weizhe Gu ◽  
He Huang
Keyword(s):  
Complementarity Problem ◽  
Convex Cone ◽  
Complementarity Problems ◽  
Positive Semidefinite ◽  
Closed Convex Cone ◽  
Tensor Complementarity Problem ◽  
Closed Cone ◽  
Strictly Copositive Tensor ◽  
Tensor Complementarity ◽  
Tensor Complementarity Problems

In this paper, we first introduce a class of tensors, called positive semidefinite plus tensors on a closed cone, and discuss its simple properties; and then, we focus on investigating properties of solution sets of two classes of tensor complementarity problems. We study the solvability of a generalized tensor complementarity problem with aD-strictly copositive tensor and a positive semidefinite plus tensor on a closed cone and show that the solution set of such a complementarity problem is bounded. Moreover, we prove that a related conic tensor complementarity problem is solvable if the involved tensor is positive semidefinite on a closed convex cone and is uniquely solvable if the involved tensor is strictly positive semidefinite on a closed convex cone. As an application, we also investigate a static traffic equilibrium problem which is reformulated as a concerned complementarity problem. A specific example is also given.

scholarly journals An existence theorem for the generalized complementarity problem

1978 ◽  
Vol 19 (1) ◽  
pp. 51-58
Author(s):  
J. Parida ◽  
B. Sahoo
Keyword(s):  
Complementarity Problem ◽  
Convex Cone ◽  
Existence Theorems ◽  
Nonlinear Complementarity Problem ◽  
Closed Convex Cone ◽  
Coercivity Conditions ◽  
Existence Of A Solution ◽  
New Class ◽  
Image Position ◽  
Class Of Functions

Given a closed, convex cone S, in Rn, its polar S* and a mapping g from Rn into itself, the generalized nonlinear complementarity problem is to find a z ∈ Rn such thatMany existence theorems for the problem have been established under varying conditions on g. We introduce new mappings, denoted by J(S)-functions, each of which is used to guarantee the existence of a solution to the generalized problem under certain coercivity conditions on itself. A mapping g:S → Rn is a J(S)-function ifimply that z = 0. It is observed that the new class of functions is a broader class than the previously studied ones.

Remarks on CCR and CAR flows over closed convex cones

2022 ◽  
Author(s):  
Anbu Arjunan
Keyword(s):  
Convex Cone ◽  
Convex Cones ◽  
Closed Convex Cone ◽  
Cocycle Conjugacy

For a closed convex cone [Formula: see text] in [Formula: see text] which is spanning and pointed, i.e. [Formula: see text] and [Formula: see text] we consider a family of [Formula: see text]-semigroups over [Formula: see text] consisting of a certain family of CCR flows and CAR flows over [Formula: see text] and classify them up to the cocycle conjugacy.

scholarly journals Best Simultaneous approximation of quasi-continuous functions by monotone functions

1991 ◽  
Vol 50 (3) ◽  
pp. 391-408 ◽  
Author(s):  
Salem M. A. Sahab
Keyword(s):  
Banach Space ◽  
Convex Cone ◽  
Simultaneous Approximation ◽  
Continuous Functions ◽  
Unit Interval ◽  
Closed Convex Cone ◽  
Monotone Functions ◽  
Measurable Functions ◽  
Best Simultaneous Approximation ◽  
Nondecreasing Functions

AbstractLet Q denote the Banach space (under the sup norm) of quasi-continuous functions on the unit interval [0, 1]. Let ℳ denote the closed convex cone comprised of monotone nondecreasing functions on [0, 1]. For f and g in Q and 1 < p < ∞, let hp denote the best Lp-simultaneous approximant of f and g by elements of ℳ. It is shown that hp converges uniformly as p → ∞ to a best L∞-simultaneous approximant of f and g by elements of ℳ. However, this convergence is not true in general for any pair of bounded Lebesgue measurable functions. If f and g are continuous, then each hp is continuous; so is limp→∞ hp = h∞.

Positive and Z-operators on Closed Convex Cones

2018 ◽  
Vol 34 ◽  
pp. 444-458
Author(s):  
Michael Orlitzky
Keyword(s):  
Game Theory ◽  
Hilbert Space ◽  
Dynamical Systems ◽  
Linear Operator ◽  
Positive Operator ◽  
Convex Cone ◽  
Real Hilbert Space ◽  
Closed Convex Cone ◽  
Nonnegative Matrices ◽  
Finite Dimensional

Let $K$ be a closed convex cone with dual $\dual{K}$ in a finite-dimensional real Hilbert space. A \emph{positive operator} on $K$ is a linear operator $L$ such that $L\of{K} \subseteq K$. Positive operators generalize the nonnegative matrices and are essential to the Perron-Frobenius theory. It is said that $L$ is a \emph{\textbf{Z}-operator} on $K$ if % \begin{equation*} \ip{L\of{x}}{s} \le 0 \;\text{ for all } \pair{x}{s} \in \cartprod{K}{\dual{K}} \text{ such that } \ip{x}{s} = 0. \end{equation*} % The \textbf{Z}-operators are generalizations of \textbf{Z}-matrices (whose off-diagonal elements are nonpositive) and they arise in dynamical systems, economics, game theory, and elsewhere. In this paper, the positive and \textbf{Z}-operators are connected. This extends the work of Schneider, Vidyasagar, and Tam on proper cones, and reveals some interesting similarities between the two families.

scholarly journals Further Application ofH-Differentiability to Generalized Complementarity Problems Based on Generalized Fisher-Burmeister Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Wei-Zhe Gu ◽  
Mohamed A. Tawhid
Keyword(s):  
Stationary Point ◽  
Complementarity Problem ◽  
Global Minimum ◽  
Complementarity Problems ◽  
Merit Function ◽  
Merit Functions ◽  
Generalized Complementarity Problem ◽  
Nonlinear Complementarity ◽  
Generalized Complementarity Problems ◽  
The Given

We study nonsmooth generalized complementarity problems based on the generalized Fisher-Burmeister function and its generalizations, denoted by GCP(f,g) wherefandgareH-differentiable. We describeH-differentials of some GCP functions based on the generalized Fisher-Burmeister function and its generalizations, and their merit functions. Under appropriate conditions on theH-differentials offandg, we show that a local/global minimum of a merit function (or a “stationary point” of a merit function) is coincident with the solution of the given generalized complementarity problem. When specializing GCP(f,g)to the nonlinear complementarity problems, our results not only give new results but also extend/unify various similar results proved forC1, semismooth, and locally Lipschitzian.

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