## Workshop on Nonlinear Analysis and Optimization

### Department of Mathematics National Taiwan Normal University

### December 17-19, 2008

### Sponsored by

### National Science Council

### National Taiwan Normal University

### Organized by

### Jein-Shan Chen, Mau-Hsiang Shih

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### Schedule of Programs

### Place : M210, Mathematics Building

Table 1: December 17, Wednesday

Chair Speaker Title

09:30 - 10:15 M-H Shih Wataru Takahashi Nonlinear operators in optimization and nonlinear analysis

Tea Break

10:25 - 11:10 M-H Shih Hang-Chin Lai Approximate solutions and error bounds for an integral functional programming problem Tea Break

11:20 - 12:05 M-H Shih Shigeo Akashi Hilbert’s 13th problem and the multidimensional numerical data compression

Lunch

13:30 - 14:15 D.S. Kim Lai-Jiu Lin Systems of set-valued vectorial version of Ekeland’s variational principle Tea Break

14:25 - 15:10 S. Schaible Ruey-Lin Sheu Global optimization for a class of fractional programming problems

Tea Break

15:20 - 16:05 M-S Shih Feng-Sheng Tsai Information storage scheme in associative memory

Table 2: December 18, Thursday

Chair Speaker Title

09:30 - 10:15 T. Tanaka Do Sang Kim Duality and stability for a class of nonconvex programs with an infinite number of constraints Tea Break

10:25 - 11:10 J-S Chen Chein-Shan Liu Novel methods to solve nonlinear algebraic equations and optimization problems Tea Break

11:20 - 12:05 J-S Chen Hong-Kun Xu Strong convergence for the proximal point algorithm in nonsmooth Banach spaces Lunch

13:30 - 14:15 H-C Lai Shue-Chin Huang Hybrid proximal point algorithms for solving constrained minimization problems in Banach spaces Tea Break

14:25 - 15:10 H-C Lai Siegfried Schaible The abstract equilibrium problem

Tea Break

15:20 - 16:05 H-C Lai Tamaki Tanaka A unified approach for scalarization on sets

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Table 3: December 19, Friday

Chair Speaker Title

09:30 - 10:10 F-S Tsai Ching-Yu Yang Recurrent neural networks for solving second-order cone programs

Tea Break

10:20 - 11:00 F-S Tsai Shen-Yi Hsu Complexity of economics

Tea Break

11:10 - 11:50 F-S Tsai Shu-Han Wu The Szemer´edi’s theorem on arithmetic progressions

Lunch

Hilbert’s 13th problem and the multidimensional numerical data compression

Shigeo Akashi

Department of Information Science Tokyo University of Science

2641 Yamazaki, Noda-shi, Chiba-ken, Japan E-mail: akashi@is.noda.tus.ac.jp

Abstract. In 1957, Kolmogorov and Arnold proved affirmatively the 13th problem which Hilbert had formulated in 1900. Actually, it is known that there still exist various kinds of unsolved problems which have been derived from Hilbert’s 13th problem. In this talk, the negative solution to the 13th problem in case of infinitely differentiable functions of several variables is given, and this result is applied to the problem asking what extent the multidimensional numerical data can be compressed to.

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Complexity of Economics

Shen-Yi Hsu

Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan

E-mail: syhsu@hp715.math.ntnu.edu.tw

Abstract. This talk will present some dynamical aspects of the Arrow-Debreu general economic equilibrium model and explain why we can regard the equilibrium as an emer- gent phenomenon.

Hybrid Proximal Point Algorithms for Solving Constrained Minimization Problems in Banach Spaces

Shue-Chin Huang

Department of Applied Mathematics National Dong Hwa University

Hualien, Taiwan

E-mail: shuang@mail.ndhu.edu.tw

Abstract. The purpose of this talk is to analyze new hybrid proximal point algorithms and solve the constrained minimization problem involving a convex functional in a uni- formly convex and uniformly smooth Banach space. The results presented in this talk improve and develop some results in the recent literature.

Co-workers: L. C. Ceng and Y. C. Liou.

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Duality and Stability for a Class of Nonconvex Programs with an Infinite Number of Constraints

Do Sang Kim

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

E-mail : dskim@pknu.ac.kr

Ta Quang Son

Nhatrang College of Education Nhatrang, Vietnam E-mail : taquangson@gmail.com

Abstract. In this talk, we introduce a class of nonconvex programming problems which has an infinite number of constraints. Duality relations for nonconvex programs cor- responding to some kinds of dual problems are established by using the property of semiconvexity. Moreover, we extend the stability and saddle point theorems for convex programming to nonconvex infinite problems.

2000 Mathematics Subject Classification. 90C26, 90C34, 49N15, 90C46.

Key words and phrases. Semiconvexity, Nonconvex infinite program, Duality, Saddle point, Stability, Augmented Lagrangian.

A new system of generalized mixed quasi-variational inclusions involving (A, η)- accretive mappings

Jong Kyu Kim

Department of Mathematics Education Kyungnam University

Masan, Kyungnam 631-730, Korea E-mail: jongkyuk@kyungnam.ac.kr

Abstract. In this talk, a new system of generalized mixed quasi-variational inclusions involving (A, η)-accretive mappings in Banach spaces is introduced and studied, which includes many kind of variational inequality (inclusion) problems as special cases. Using the resolvent operator methods associated with (A, η)-accretive mappings, an existence and uniqueness theorem of solutions for this system. Some new iterative algorithms for finding approximate solutions of this system is suggested and discussed, the convergence theorem of iterative sequence generated by new iterative algorithms is also given.

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Approximate solutions and error bounds for an integral functional programming problem

Hang-Chin Lai

Department of Applied Mathematics Chung Yuan Christian University

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

Abstract. Consider an integral functional programming as (P) max

Z T 0

f (t)^{T}x(t)dt (= hx, f i)

s.t. x ∈ L^{∞}([0, T ], IR^{q}_{+}) and for t ∈ [0, T ],
B(t)x(t) −

Z t 0

K(s, t)x(s)ds ≤_{IR}^{p}

+ g(t) in IR^{p}

where B(t) and K(s, t) are p × q matrices; f (·) ∈ C([0, T ], IR^{q}_{+}), g(·) ∈ C([0, T ], IR^{p}_{+}); and

T means transpose operation.

Purpose of this talk include

1. to find a sequence of approximate solutions to be convergent in L^{∞} and
2. to establish a formula of error bounds for the approximate solutions,

3. to evaluate the results by two examples and give the practical approximate solu- tions.

Systems of Set-Valued Vectorial Version of Ekeland’s Variational Principle

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 apply an abstract maximal element theorem to study various types of set-valued vectorial version of Ekeland’s variational principle for maps defined on metric space or product of metric spaces with quasiorder complete. Our results extend and improve many results on Ekeland’s variational principle and Caristi’s fixed point in the literature.

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Novel Methods to Solve Nonlinear Algebraic Equations and Optimization Problems

Chein-Shan Liu

Department of Mechanical and Mechatronic Engineering National Taiwan Ocean University

Keelung, Taiwan

E-mail: csliu@mail.ntou.edu.tw

Abstract. In this talk I report a novel technique to solve nonlinear algebraic equations (NAEs) basing on a new concept of fictitious time integration method (FTIM), which transforms the NAEs into a mathematical equivalent system of ordinary differential equa- tions. Then I report a novel technique to solve the nonlinear optimization problem (NOP) under multiple equality and inequality constraints. The Kuhn-Tucker optimality condi- tions are used to transform the NOP into a mixed complementarity problem (MCP).

With the aid of NCP-functions a set of nonlinear algebraic equations is obtained. Then we can apply the FTIM to solve these NAEs, and thus the NOP is solved.

The Abstract Equilibrium Problem

Siegfried Schaible

Department of Applied Mathematics Chung Yuan Christian University

Chungli 32023, Taiwan E-mail: schaible2008@gmail.com

Abstract. A unifying model of optimization problems and classical problems in nonlinear analysis is the abstract equilibrium problem. We present eight realizations of this rather general model and point out some applications through these special cases. The classical assumption of convexity and monotonicity can be relaxed to generalized convexity and generalized monotonicity to establish major results.

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Global Optimization for a Class of Fractional Programming Problems

Ruey-Lin Sheu

Department of Mathematics National Cheng Kong University

Tainan, Taiwan

E-mail: rsheu@mail.ncku.edu.tw

Abstract. This talk presents a canonical dual approach to the problem of minimizing the sum of a quadratic function and the ratio of two quadratic functions, which is a type of non-convex optimization problem subject to an elliptic constraint. We first relax the fractional structure by introducing a family of parametric subproblems. Under proper conditions on the “problem-defining” matrices associated with the three quadratic func- tions, we show that the canonical dual of each subproblem becomes a one-dimensional concave maximization problem that exhibits no duality gap. Since the infimum of the optima of the parameterized subproblems leads to a solution to the original problem, we then derive some optimality conditions and existence conditions for finding a global minimizer of the original problem. Some numerical results using quasi-Newton and line search methods are presented to illustrate our approach.

Nonlinear Operators in Optimization and Nonlinear Analysis

Wataru Takahashi

Department of Mathematical and Computing Sciences Tokyo Institute of Technology

2-12-1, Ohokayama, Meguro-ku, Tokyo 152-8552, Japan E-mail: wataru@is.titech.ac.jp

Abstract. Let E be a real Banach space and let E^{∗} be the dual space of E. Let C be a
closed convex subset of E. Let f be a bifunction from C × C to R, where R is the set of
real numbers. The equilibrium problem is to find ˆx ∈ C such that

f (ˆx, y) ≥ 0, ∀y ∈ C. (1)

The set of solutions of (1) is denoted by EP (f ). Numerous problems in physics, opti- mization, and economics reduce to find a solution of (1).

In this talk, motivated by the equilibrium problem, we first introduce some nonlinear operators in Banach spaces. Next, we prove fundamental results for the nonlinear oper- ators which are related to Optimization and Nonlinear Analysis. Finally, we apply these results to prove some important results in Banach spaces or Hilbert spaces.

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A Unified Approach for Scalarization on Sets

Tamaki Tanaka

Graduate School of Science and Technology Niigata University

Niigata 950-2181, Japan

E-mail: tamaki@math.sc.niigata-u.ac.jp

Abstract. Hamel and L¨ohne [2] proved the existence results for minimal points of subsets
of the product space X × 2^{Y} by using the result of [1], where X and Y are a separated
uniform space and a topological vector space, respectively. In this paper, we investigate
several kinds of scalarization functionals on sets and present a minimal element theorem
with set-relations, which is similar to one of those in [2] but proved by using a different
scalarization functional (see [4]).

Let Y be a real ordered topological vector space with the vector ordering ≤_{C} induced
by a convex cone C ⊂ Y ; for x, y ∈ Y , x ≤_{C} y if y − x ∈ C. We define the following
sublinear scalarization function:

h_{C}(y; k) := inf{t | y ∈ tk − C} (2)
where k ∈ int C. We get −h_{C}(−y; k) = sup{t | y ∈ tk + C}. Based on these ideas
and several scalarization functions introduced in [2], we define the following 12 types of
scalarization functions on 2^{Y} with respect to direction k ∈ C and reference set V^{0} ⊂ Y by
using ≤^{(i)}_{C} (i = 1, . . . , 6) in [3]; they are unified forms including all scalarization functions
introduced in [2].

I^{(j)}_{k,V}^{0}(V ) := inf^{n}t ∈ IR} ^{}^{}_{}V ≤^{(j)}_{C} (tk + V^{0})^{o}, (j = 1, . . . , 6) (3)

### References

[1] H. Br´ezis and F. Browder, A General Principle on Ordered Sets in Nonlinear Func- tional Analysis, Advances in Mathematics 21 (1976), 355–364.

[2] A. Hamel and A. L¨ohne, Minimal Element Theorems and Ekeland’s Principle with Set Relations, J. of Nonlinear and Convex Anal. 7 (2006), 19–37.

[3] D. Kuroiwa, T. Tanaka, and T.X.D. Ha, On Cone Convexity of Set-Valued Maps, Nonlinear Anal. 30 (1997), 1487–1496.

[4] A. Shimizu and T. Tanaka, Minimal Element Theorem with Set-Relations, to appear in J. of Nonlinear and Convex Anal. 9 (2008), 249–253.

Co-workers: Akira Shimizu, Syuuji Yamada.

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Information Storage Scheme in Associative Memory

Feng-Sheng Tsai Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan

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

Mau-Hsiang Shih Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan

E-mail: mhshih@math.ntnu.edu.tw

Abstract. A major puzzle in neural networks is understanding the information encoding principles that implement the functions of the brain systems. Population coding in neurons and plastic changes in synapses are two important subjects in attempts to explore such principles. This forms the basis of modern theory of neuroscience concerning self- organization and associative memory. Here we wish to suggest an information storage scheme based on the dynamics of evolutionary neural networks, essentially reflecting the meta-complication of the dynamical changes of neurons as well as plastic changes of synapses. The information storage scheme may lead to the development of a complete description of all the equilibrium states of Hopfield networks, a space-filling network that weaves the intricate structure of Hamming star-convexity, and a plasticity regime that encodes information based on algorithmic Hebbian synaptic plasticity.

The Szemer´edi’s Theorem on Arithmetic Progressions

Shu-Han Wu

Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan E-mail: 895400015@ntnu.edu.tw

Abstract. In 1936 Erd¨os and Turan conjectured that for every value d called density 0 < d < 1 and every integer k there is a number N (d, k) such that every subset A of {1, ..., N } of cardinality dN contains a length-k arithmetic progression, provided N >

N (d, k). In 1975 Endre Szemer´edi proved this conjecture by a combinatorial method.

In this talk, we will present a dynamical proof obtained by Harry Furstenberg of the Szemer´edi’s theorem on arithmetic progressions.

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Strong Convergence for the Proximal Point Algorithm in Nonsmooth Banach Spaces

Hong-Kun Xu

Department of Applied Mathematics National Sun Yat-sen University

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

Abstract. We provide a strongly convergent modification of Rockafellar’s proximal point algorithm in a Banach space which is uniformly convex with convexity of power two and which is not necessarily smooth.

Recurrent Neural Networks for Solving Second-Order Cone Programs

Ching-Yu Yang Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan

E-mail: yangcy@hp715.math.ntnu.edu.tw

Abstract. This paper proposes using the neural networks to efficiently solve the second- order cone programs (SOCP). To establish the neural networks, the SOCP is first reformu- lated as a second-order cone complementarity problem (SOCCP) with the Karush-Kuhn- Tucker conditions of the SOCP. The SOCCP functions, which transform the SOCCP into a set of nonlinear equations, are then utilized to design the neural networks. We pro- pose two kinds of neural networks with the different SOCCP functions. The first neural network uses the Fischer-Burmeister function to achieve an unconstrained minimization with a merit function. We show that the merit function is a Lyapunov function and this neural network is asymptotically stable. The second neural network utilizes the natural residual function with the cone projection function to achieve low computation complex- ity. It is shown to be Lyapunov stable and converges globally to an optimal solution under some condition. The SOCP simulation results demonstrate the effectiveness of the proposed neural networks.

Co-workers: Chun-Hsu Ko, Jein-Shan Chen.

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