**2010 Workshop on Nonlinear** **Analysis and Optimization**

**Department of Mathematics** **National Taiwan Normal University**

**November 24-26, 20010**

**Sponsored by**

**College of Science, National Taiwan Normal University** **Mathematics Division, National Center for Theoretical Sciences**

**(Taipei Oﬃce)**

**Organized by**

**Mau-Hsiang Shih and Jein-Shan Chen**

1

**Schedule of Programs**

**Place : M210, Mathematics Building**

**Table 1: November 24, Wednesday**

Chair **Speaker** Title

09:10 J-S Chen D. Sun An introduction to a class of matrix cone programming 09:50

09:50 J-S Chen P-W Chen A novel kernel correlation model with the

10:30 correspondence estimation

10:50 J-S Jung L-J Lin Variational relation problems and equivalent forms of 11:30 generalized Fan-Browder ﬁxed point theorems with

applications to Stampacchia equilibrium problems 11:30 J-S Jung C-T Pang Asymptotic stability of interval systems 12:10

*Lunch Break*

14:00 D-S Kim J-S Jung *Some results on a general iterative method for k-strictly*

14:40 pseudo-contractive mappings

14:40 D-S Kim S-N Lee *Large maximal IC-colorings for K** _{1,m,n}*
15:20

15:40 L-J Lin J-L Ho A combinatorial ﬁxed point theorem in Boolean algebra 16:20

16:20 L-J Lin D-S Kim Duality relations for nondiﬀerentiable fractional

17:00 multiobjective programming problems

2

**Table 2: November 25, Thursday**

Chair **Speaker** Title

09:10 J-S Chen W. Takahashi Fixed point and nonlinear ergodic theorems for

09:50 generalized hybrid mappings

09:50 J-S Chen X. Chen Nonsmooth, nonconvex optimization with applications 10:30

10:50 J-S Chen C-B Chua Target-following framework for symmetric cone

11:30 programming

11:30 J-S Chen R-L Sheu Duality and solutions for quadratic programming over

12:10 single non-homogeneous quadratic constraint

*Lunch Break*

14:00 Takahashi H-C Lai Complex minimax programming with complex variables 14:40

14:40 Takahashi S-C Huang A hybrid extragradient method for asymptotically strict 15:20 pseudo-contractions in the intermediate sense and

inverse-strongly monotone mappings 15:40 H-C Lai Y-A Hwang Consistency of the Hirsch-index 16:20

16:20 H-C Lai H-K Xu Stochastic F´ejer-monotonicity and its applications 17:00

3

**Table 3: November 26, Friday**

Chair **Speaker** Title

09:10 J-S Chen Y-L Chang Stationary point conditions for the FB merit function

09:50 associated with symmetric cones

10:00 J-S Chen X. Miao The column-suﬃciency and row-suﬃciency of the linear

10:40 transformation on Hilbert spaces

10:50 J-S Chen H-J Chen Convergence rate analysis on interval-type algorithms for

11:30 generalized fractional programming

4

**Stationary point conditions for the FB merit function associated with**
**symmetric cones**

Yu-Lin Chang

Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan E-mail:ylchang@math.ntnu.edu.tw

**Abstract. For the symmetric cone complementarity problem, we show that each sta-**
tionary point of the unconstrained minimization reformulation based on the Fischer-
Burmeister merit function is a solution to the problem, provided that the gradient oper-
ators of the mappings involved in the problem satisfy column monotonicity or have the
*Cartesian P*0-property. These results answer the open question proposed in the article
appeared in Journal of Mathematical Analysis and Applications, vol. 355, pp. 195–215,
2009.

5

**Convergence rate analysis on interval-type algorithms for generalized**
**fractional programming**

Hui-Ju Chen

Department of Mathematics National Cheng Kung University

Tainan 70101, Taiwan E-mail: vanillascody@gmail.com

**Abstract. The generalized fractional programming is to minimize the largest of n ra-**
tios. Most algorithms for solving the generalized fractional programming are called the

”Dinkelbach-type” which converts the original problem into a sequence of parametric subproblems. Interval-type algorithms diﬀer from Dinkelbach-type in providing ﬂexibil- ity to select iterate parameters within the intervals, but the diﬃculty of estimating the convergence rate is to cope with the oscillating behavior of iterate parameters. In this talk, we will introduce the generic algorithm which can be regarded as a generalized version for interval-type algorithms and it creates a sequence of nested intervals contain- ing the optimal value to the original problem whose lengths decrease to 0. The generic algorithm not only uniﬁes various versions of the Dinkelbach-type algorithms, but give a stronger convergence result and the convergence as well as the convergence rate analysis are carried out through geometric observations and fundamental properties of convex analysis.

6

**A novel kernel correlation model with the correspondence estimation**

Peng-Wen Chen Department of Mathematics

National Taiwan University Taipei 10617, Taiwan

E-mail: pengwen@math.ntu.edu.tw

**Abstract. We present a novel multiple-linked iterative closest point method to estimate**
correspondences and the rigid/non-rigid transformations between point-sets or shapes.

The estimation task is carried out by maximizing a symmetric similarity function, which is the product of the square roots of correspondences and a kernel correlation. The local mean square error analysis and robustness analysis are provided to show our method’s superior performance to the kernel correlation method.

7

**Nonsmooth, nonconvex optimization with applications**

Xiaojun Chen

Department of Applied Mathematics The Hong Kong Polytechnic University

E-mail: maxjchen@polyu.edu.hk

**Abstract. This talk will discuss nonsmooth, nonconvex optimization problems in stochas-**
*tic equilibrium problems and l*_{2}*-l*_{p}*(0 < p < 1) minimization problems, as well as their*
applications in transportation planning, signal reconstruction and variable selection. In
particular, this talk will present our recent results in the following two parts.

(i) We reformulate the stochastic variational inequality problems as expected residual minimization problems using residual functions of variational inequalities. Math- ematical theorems and practical examples of traﬃc assignments show that the re- formulations are robust and reliable in uncertain environments.

(ii) We derive a lower bound theory for nonzero entries in every local minimizer of the
*l*_{2}*-l** _{p}* minimization problems. This theory shows clearly the relationship between
the sparsity of the solution and the choice of parameters in the model. We prove

*global convergence of the l*

_{1}reweighted minimization algorithm and uniqueness of solution under the truncated null space property which is weaker than the restricted isometry property.

8

**Target-following framework for symmetric cone programming**

Chek Beng Chua

Department of Mathematical Sciences Nanyang Technological University

SPMS-03-01, 21 Nanyang Link 637371, Singapore E-mail: CBChua@ntu.edu.sg

**Abstract. The ﬁrst target-following algorithm was given by Shinji Mizuno in 1992 for**
linear complementarity problems, using the notion of delta sequences. The delta sequence
is a sequence of targets in the ‘v-space’ that lead the strictly feasible primal-dual solutions
towards optimality. In this talk, I will present a generalization of the target-following
framework to symmetric cone programming.

9

**A Combinatorial ﬁxed point theorem in Boolean algebra**

Juei-Ling Ho Department of Fiance Tainan University of Technology

Tainan 71002, Taiwan E-mail: t20054@mail.tut.edu.tw

**Abstract. We propose to answer the Jacobian conjecture in boolean algebra.** The
boolean analogue of the Jacobian problem in 0,1? has been proved: if a map from
*{0, 1}? to itself deﬁnes a boolean network has the property that all the boolean eigenval-*
ues of the discrete Jacobian matrix of this map evaluated at each element of *{0, 1}?are*
zero, then it has a unique ﬁxed point. We propose extending this result to any map F
from the product space X of n ﬁnite boolean algebras to itself.

Keywords: Jacobian conjecture; Combinatorial ﬁxed point theorem, Discrete Boolean eigenvalues; Finite boolean algebras.

10

**A hybrid extragradient method for asymptotically strict pseudo-contractions**
**in the intermediate sense and inverse-strongly monotone mappings**

Shue-Chin Huang

Department of Applied Mathematics National Dong Hwa University

Hualien 97401, Taiwan E-mail: shuang@mail.ndhu.edu.tw

**Abstract. This talk is devoted to investigating a new hybrid extragradient method for**
*an asymptotically strict pseudo-contraction in the intermediate sense S and an inverse-*
*strongly monotone mapping A in a Hilbert space. The main purpose is to use this*
iteration method to generate a sequence to approximate a common element of the ﬁxed
*point set of S and the solution set of the variational inequality problem for A. Weak con-*
vergence and strong convergence for the related sequences are established with respective
iteration processes.

11

**Consistency of the Hirsch-index**

Yan-An Hwang

Department of Applied Mathematics National Dong Hwa University

Hualien 97401, Taiwan E-mail: yahwang@mail.ndhu.edu.tw

**Abstract. The Hirsch-index is an index for measuring and comparing the output of**
researchers. Under the condition of monotonicity, Woeginger (2008) provides a charac-
terization of the Hirsch-index by three axioms. Replacing monotonicity by expansion
consistency, we characterize the Hirsch-index by only two of Woeginger’s axioms. Be-
sides, we also introduce an axiom contraction consistency. It is a dual viewpoint of
expansion consistency. Based on contraction consistency, an additional characterization
of the Hirsch-index is reported.

12

**Some results on a general iterative method for k-strictly pseudo-contractive****mappings**

Jong-Soo Jung

Department of Mathematics Dong-A University Busan 604-714, Korea E-mail: jungjs@mail.donga.ac.kr

**Abstract. In this paper, we introduce a new general iterative scheme for a k-strictly***pseudo-contractive mapping related to an operator F which is κ-Lipschizian and η-*
strongly monotone and then prove that under certain diﬀerent control conditions, the
sequence generated by the proposed iterative scheme converges strongly to a ﬁxed point
*of the mapping, which solves a variational inequality related to the operator F . Addi-*
tional results of main results are also obtained. Our results substantially improve and
develop the corresponding ones announced by many authors recently.

*Key words. k-strictly pseudo-contractive mapping; Nonexpansive mapping; Fixed points;*

*Contraction; κ-Lipschizian and η-strongly monotone operator; Hilbert space; Variational*
inequality.

*2000 Mathematics Subject Classification. 47H09, 47H10, 47J20, 47J25, 49M05.*

13

**Duality relations for nondiﬀerentiable fractional multiobjective**
**programming problems**

Do Sang Kim

Division of Mathematical Sciences Pukyong National University Busan 608-737, Republic of Korea

E-mail : dskim@pknu.ac.kr

**Abstract. In this talk, we consider the nondiﬀerentiable multiobjective fractional pro-**
gramming problem involving support functions and cone constraints. For this problem,
Wolfe and Mond-Weir type duals are proposed. We establish weak and strong duality
theorems for a weakly eﬃcient solution by using generalized convexity conditions. In ad-
dition, we introduce a pair of nondiﬀerentiable multiobjective symmetric dual problems
with cone constraints over arbitrary closed convex cones. Weak, strong and converse
duality theorems are established under suitable generalized convexity conditions for a
weakly eﬃcient solution. As special cases of our duality relations are given.

*2000 Mathematics Subject Classification. 90C29; 90C32; 90C46.*

*Key words and phrases. Multiobjective fractional programming, support functions, op-*
timality conditions, duality theorems.

14

**Complex minimax programming with complex variables**

Hang-Chin Lai

Department of Applied Mathematics Chung Yuan Christian University

Chungli 32023, Taiwan E-mail: hclai@cycu.edu.tw

**Abstract. Consider a nondiﬀerentiable minimax fractional programming problems with**
complex variables as the following form:

(FP* _{c}*) min

*ζ**∈X*max

*η**∈Y*

*Re [f (ζ, η) + (z*^{H}*Az)** ^{1/2}*]

*Re [g(ζ, η)− (z*

^{H}*Bz)*

*]*

^{1/2}s.t. *X ={ζ = (z, z) ∈ C*^{2n}*| − h(ζ) ∈ S}*

*where Y is a speciﬁed compact subset of* C^{2m}*, A and B are positive semideﬁnite Hermi-*
tian matrices in C^{n}^{×n}*, S is a polyhedral cone in* C^{p}*, f (·, ·) and g(·, ·) are continuous on*
C^{n}*× C*^{m}*, and for each η* *∈ Y , f(·, η), g(·, η) and h(·) are analytic functions.*

In this talk, the duality models of (FP* _{c}*) are established and the duality theorems related
to problem (FP

*) are proved with nonduality gap under some conditions.*

_{c}**References**

i. H.C. Lai and T.Y. Huang. Optimality conditions for a nondiﬀerentiable minimax
*programming in complex spaces. Nonlinear Analysis, 71: 1205-1212, 2009.*

ii. H.C. Lai and J.C. Liu. Duality for nondiﬀerentiable minimax programming in
*complex spaces. Nonlinear Analysis, 71: e224-e233, 2009.*

iii. H.C. Lai and J.C. Liu. Complex fractional programming involving generalized
*quasi/pseudo convex functions. Z. Angew. Math. Mech., 82(3): 159-166, 2002.*

iv. H.C. Lai and T.Y. Huang. Optimality conditions for nondiﬀerentiable minimax
*fractional programming with complex variables. Journal of Mathematical Analysis*
*and Applications, 359: 229-239, 2009.*

15

**Large maximal IC-colorings for K**_{1,m,n}

Shyh-Nan Lee

Department of Applied Mathematics Chung Yuan Christian University

Chungli 32023, Taiwan E-mail: nan@math.cycu.edu.tw

**Abstract. The IC-index M (K**_{1,m,n}*) of the class K** _{1,m,n}* of all complete tripartite graphs

*and its corresponding maximal colorings are obtained. We prove that M (K*

*) = 3*

_{1,1,n}*·*2

^{n}*+ 1 (n*

*≥ 1) and M(K*

*1,m,n*) = 13

*· 2*

^{m+n}

^{−3}*− 2*

^{m}

^{−2}*+ 2 (n*

*≥ m ≥ 3) and that, up*

*to IC-equivalence, the classes K*

_{1,1,1}*, K*

_{1,1,2}*, K*

_{1,1,n}*(n*

*≥ 3), K*

*1,2,2*

*, K*

_{1,2,n}*(n*

*≥ 3) and*

*K*

_{1,m,n}*(n*

*≥ m ≥ 3) have exactly one, four, two, six, four and one maximal colorings,*respectively.

16

**Variational relation problems and equivalent forms of generalized**
**Fan-Browder ﬁxed point theorems with applications to Stampacchia**

**equilibrium problems**

Lai-Jiu Lin

Department of Mathematics

National Changhua University of Education Changhua 50058, Taiwan

E-mail: maljlin@cc.ncue.edu.tw

**Abstract. In this paper, we study the existence theorems of solution for variational**
relation problems. From the existence theorems of solution for variational relation prob-
lems, we study equivalent forms of generalized Fan-Browder ﬁxed point theorem, exis-
tence theorems of solutions for Stampacchia vector equilibrium problems and generalized
Stampacchia vector equilibrium problems. Our results contains many orginal results and
have many applications in Nonlinear Analysis.

17

**The column-suﬃciency and row-suﬃciency of the linear transformation on**
**Hilbert spaces**

Xinhe Miao

Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan

E-mail: miaoxinhe@abel.math.ntnu.edu.tw

**Abstract. In this talk, ﬁrstly, we introduce the concepts of the column-suﬃciency and**
row-suﬃciency of the linear transformation on Hilbert space. Secondly, we show that
the row-suﬃciency of T is equivalent to the existence of the solution of the linear com-
plementarity problem under an operator commutative condition; moreover, the column-
suﬃciency along with cross commutative property is equivalent to the convexity of the
solution set of the linear complementarity problem. In our analysis, the properties of the
Jordan product and the Lorentz cone in Hilbert space play important roles.

18

**Asymptotic stability of interval systems**

Chin-Tzong Pang

Department of Information Management Yuan Ze University

Chungli 32003, Taiwan

E-mail: imctpong@saturn.yzu.edu.tw

**Abstract. Previous works about the convergence of the powers of interval matrices have**
focused on the iteration of a single interval matrix. But in robust stability analysis of
uncertain systems, there is associated with a set of coupled interval matrices. The most
basic issue is the the asymptotic stability of a set of interval matrices. Here we introduce
the notion of simultaneous Schur stability by linking the concepts of the majorant and
the joint spectral radius, and prove the asymptotic stability of a set of interval matrices
governed by simultaneous Schur stability. The present result may lead to the stability
analysis of discrete dynamical interval systems.

19

**Duality and solutions for quadratic programming over single**
**non-homogeneous quadratic constraint**

Ruey-Lin Sheu

Department of Mathematics National Cheng Kung University

Tainan 70101, Taiwan E-mail: rsheu@mail.ncku.edu.tw

**Abstract. The quadratic programming over one quadratic constraint (QP1QC) was**
mostly studied under certain constraint qualiﬁcations such as the Slater condition. In
this talk, we relax the assumption to cover more general cases when the two matrices
from the objective and the constraint functions can be simultaneously diagonalizable via
congruence. Under such an assumption, the nonconvex (QP1QC) problem can be classi-
ﬁed into three types: (i) a unbounded below problem; or (ii) a unconstrained quadratic
problem; or (iii) one with a feasible dual problem with no duality gap. In other words,
the (QP1QC) problem is a “good” non-convex programming. We can explain by showing
that the (QP1QC) problem is indeed equivalent to a linearly constrained convex problem,
which happens to be dual of the dual of itself.

20

**An introduction to a class of matrix cone programming**

Defeng Sun

Department of Mathematics National University of Singapore

10 Lower Kent Ridge Road 119076, Singapore E-mail: matsundf@nus.edu.sg

**Abstract. In this talk, we shall introduce a class of linear conic programming (which**
we call matrix cone programming or MCP) involving the epigraphs of ﬁve commonly
used matrix norms and the well studied symmetric cone. MCP has recently found many
important applications, for example, in nuclear norm relaxations of aﬃne rank mini-
mization problems. In order to make the deﬁned MCP tractable and meaningful, we
must ﬁrst understand the structure of these epigraphs. So far, only the epigraph of the
Frobenius matrix norm, which can be regarded as a second order cone, has been well
studied. Here, we take an initial step to study several important properties, including
its closed form solution, calm Bouligand-diﬀerentiability and strong semismoothness, of
*the metric projection operator over the epigraph of the l*_{1}*, l** _{∞}*, spectral or operator, and
nuclear matrix norm, respectively. These properties make it possible to apply augmented
Lagrangian methods, which have recently received a great deal of interests due to their
high eﬃciency in solving large scale semideﬁnite programming, to this class of MCP
problems. The work done on MCP is far from comprehensive. Rather it is intended as a
starting point to call for more insightful research on MCP so that it can serve as a basic
tool to solve more challenging convex matrix optimization problems in years to come.

[This is a joint work with Chao DING and Kim Chuan TOH]

21

**Fixed point and nonlinear ergodic theorems for generalized hybrid mappings**

Wataru Takahashi Tokyo Institute of Technology

and Keio University, Japan E-mail: wataru@is.titech.ac.jp

**Abstract. Let H be a real Hilbert space and let C be a nonempty subset of H. Then a***mapping T : C* *→ H is said to be nonexpansive if*

*∥T x − T y∥ ≤ ∥x − y∥*

*for all x, y* *∈ C. The set of ﬁxed points of T is denoted by F (T ). In 1975, Baillon proved*
the following ﬁrst nonlinear ergodic theorem in a Hilbert space.

**Theorem 1. Let C be a nonempty closed convex subset of H and let T : C***→ C be*
*a nonexpansive mapping with F (T )* *̸= ∅. Then, for any x ∈ C, S**n**x =* ^{1}* _{n}*∑

_{n}

_{−1}*k=0**T*^{k}*x*
*converges weakly to an element z∈ F (T ).*

An important example of nonexpansive mappings in a Hilbert space is a ﬁrmly non-
expansive mapping. Recently, Kohsaka and Takahashi, and Takahashi introduced the
following mappings which are deduced from a ﬁrmly nonexpansive mapping in a Hilbert
*space. A mapping T : C* *→ H is called nonspreading if*

2*∥T x − T y∥*^{2} *≤ ∥T x − y∥*^{2}+*∥T y − x∥*^{2}
*for all x, y∈ C. A mapping T : C → H is called hybrid if*

3*∥T x − T y∥*^{2} *≤ ∥x − y∥*^{2}+*∥T x − y∥*^{2}+*∥T y − x∥*^{2}
*for all x, y∈ C.*

*In this talk, we ﬁrst introduce a broad class of mappings T : C* *→ H called generalized*
*hybrid such that for some α, β* *∈ R,*

*α∥T x − T y∥*^{2} + (1*− α)∥x − T y∥*^{2} *≤ β∥T x − y∥*^{2}+ (1*− β)∥x − y∥*^{2}

*for all x, y* *∈ C. Such a class contains the classes of nonexpansive mappings, nonspreading*
mappings, and hybrid mappings in a Hilbert space. Next, we prove ﬁxed point and
nonlinear ergodic theorems for generalized hybrid mappings in a Hilbert space. Finally,
we deal with two strong convergence theorems for these nonlinear mappings in a Hilbert
space.

22

**Stochastic F´ejer-monotonicity and its applications**

Hong-Kun Xu

Department of Applied Mathematics National Sun Yat-sen University

Kaohsiung 80424, Taiwan E-mail: xuhk@math.nsysu.edu.tw

**Abstract. Let H be a real Hilbert space and let C be a nonempty closed convex subset***of H. Let (Ω,F, {F}*^{∞}_{n=1}*,P) be a ﬁltered probability space. Let {x**n**}*^{∞}* _{n=1}* be a sequence

*of random variables taking values in H which is adapted (i.e., x*

*n*is

*F*

*n*-measurable for

*each n). We say that*

*{x*

*n*

*}*

^{∞}*n=1*is stochastically quasi-F´

*ejer-monotone with respect to C*if there exists a sequence

*{ε*

*n*

*}*

^{∞}*of nonnegative random variables, with ∑*

_{n=1}

_{∞}*n=1**ε*_{n}*<* *∞*
and satisfying the property

E[

*∥x**n+1**− x∥*^{2}*|F**n*

]*≤ ∥x**n**− x∥*^{2}*+ ε*_{n}*(a.s.), n* *≥ 1, x ∈ C.*

In this talk, I will present some properties of stochastically quasi-F´ejer-monotone
sequences. I will also discuss applications to optimization, in particular, the stochastic
subgradient algorithm for solving the minimization problem min_{x}_{∈C}*f (x), where f : H* *→*
R is a convex function.

23