2006 『最佳化理論及其應用』 工作坊 Workshop on Optimization, 2006

2006 『最佳化理論及其應用』 工作坊 Workshop on Optimization, 2006

國立台灣師範大學數學系 December 8-10, 2006

Sponsored by

Organized by

Schedule of Programs

Place : M210, Mathematics Building

Dec. 8 (FRI) Events Dec. 9 (SAT) Events Dec. 10 (SUN) Events 08:30 - 09:00 Registration 8:30 - 9:00 Registration 09:00 - 09:50 Plenary talk 09:00 - 09:50 Plenary talk 09:50 - 10:10 BREAK 09:50 - 10:10 BREAK 10:10 - 10:35 Wataru Takahashi 10:10 - 10:35 Defeng Sun

10:35 - 11:00 ” 10:35 - 11:00 ”

11:10 - 11:35 Hong-Kun Xu 11:10 - 11:35 Jiming Peng

11:35 - 12:00 ” 11:35 - 12:00 ”

12:00 - 13:30 LUNCH 12:00 - LUNCH

13:00 - 14:00 年會報到 13:30 - 13:55 Yasunori Kimura 14:00 - 14:40 年會開幕 13:55 - 14:20 ”

14: 40 - 15:30Plenary talk 14:20 - 14:45 Shuechin Huang 15:40 - 16:05 Han-Lin Li 14:45 - 15:05 TEA BREAK 16:05 - 16:30 ” 15:05 - 15:30 Yen-Cherng Lin 16:30 - 16:55 Xin Chen 15:30 - 15:55 Po-Feng Wu 16:55 - 17:20 ” 15:55 - 16:20 Wei-Shih Du 18:00 - 21:00 RECEPTION

2

Duality Approach to Inventory Centralization Games

Xin Chen (陳 新)

Department of Industrial and Enterprise Systems Engineering University of Illinois, Urbana-Champigain

Urbana, IL 61801 U.S.A.

[email protected]

Abstract. Linear programming (LP) duality has played a fundamental role in the analysis of cooperative games. In this lecture, we will present new applications of LP duality and stochastic LP duality in studying cooperative games arising from inventory centralization.

In particular, we show that duality theory can be used to prove the non-emptiness of cores for such inventory games and to find an element in the core.

The first example is the economic lot-sizing game, in which multiple retailers form a coalition by placing joint orders to a single supplier in order to reduce ordering cost, which is assumed to be a concave function of the order quantity. We are concerned with the issue of how to allocate the cost/benefit so that it is advantageous for every retailer to join the coalition. The standard formulation of the corresponding optimization problem is a concave minimization problem and hence LP duality does not directly apply. We suggest an integer programming formulation for this optimization problem and show that its LP relaxation admits zero integrality gap, which makes it possible to analyze the game using LP duality. We show that there exists an optimal dual solution that gives rise to an allo- cation in the core, which can be found in polynomial time. An interesting feature of our approach is that, in contrast to the duality approach for other known cooperative games, it is not necessarily true that every optimal dual solution gives rise to a core allocation.

Another example is a single-period inventory centralization game with stochastic de- mand where multiple retailers form a coalition by holding centralized inventory in order to take advantage of the effect of risk pooling. Again, we are concerned with the issue of how to allocate the cost/benefit. When the ordering cost is linear, the optimization problem corresponding to the inventory game is formulated as a stochastic program. We observe that the strong duality of stochastic LP not only directly leads to a series of recent results concerning the non-emptiness of the cores of such games, but also suggests a way to find an element in the core. We further construct a nontrivial infinite dimensional linear programming dual for the well-known newsvendor problem with concave ordering cost and prove a strong duality result for this non-convex minimization problem. This new duality result immediately implies that the corresponding game has a non-empty core. Finally,

we prove that it is NP-hard to determine whether a given allocation is in the core for the newsvendor game even in a very simple setting.

[This is a joint work with Jiawei Zhang at New York University.]

4

Iterative approximation of fixed points of asymptotically nonexpansive mappings in reflexive Banach spaces

Shuechin Huang (黃淑琴) Department of Applied Mathematics

National Dong Hwa University Hualien 974, Taiwan [email protected]

Abstract. Let X be a reflexive Banach space, C a nonempty closed convex subset of X, f : C → C a contraction, {Ti : C → C}^mi=1 a finite family of asymptotically nonexpansive mappings with sequences {ki+m(j−1)}j ⊂ [1, ∞) (1 ≤ i ≤ m), {tn} a sequence in (0, 1). We will establish the necessary and sufficient conditions for the following iterative sequence to converge to a common fixed point of T1, T2,· · · , Tm:

zn+1 =

(

1− tn

k_n

)

f (z_n) + tn

k_nT_¯n^jnz_n, n∈ N, where ¯n≡ mod n and n = ¯n + m(jn− 1).

Iterative Methods for a Sequence of Mappings in a Banach Space

Yasunori Kimura

Department of Mathematical and Computing Sciences Tokyo Institute of Technology

Tokyo 152-8552, Japan [email protected]

Abstract. Let us consider the problem of finding a solution z ∈ E of an operator inclusion 0∈ Az, where A is an accretive operator defined on a real Banach space E. One of the most popular iterative schemes of approximating this solution is the following method called the proximal point algorithm, which was first introduced by Martinet and has been studied by Rockafellar: Let x₁ ∈ E and generate a sequence {xn} by xn+1 = (I + ρ_nA)⁻¹x_n for n ∈ N, where {ρn} is a sequence of positive real numbers satisfying that infn∈Nρn > 0.

Convergence of this scheme has been studied by many researchers with various types of additional conditions. In this talk, we consider this type of iterative schemes and introduce some recent development related to this scheme.

6

On Maximal Element Theorems, Variants of Ekeland’s Variational Principle and Their Applications

Lai-Jiu Lin (林來居) Department of Mathematics

National Chang-Hua University of Education Chang-Hua 50058, Taiwan

[email protected]

Wei-Shih Du (杜威仕) Department of Mathematics

National Chang-Hua University of Education Chang-Hua 50058, Taiwan

fi[email protected]

Abstract. In this paper, we establish several different versions of generalized Ekeland’s variational and maximal element theorems for τ -functions in¹ complete metric spaces. The equivalence relations between maximal element theorem, generalized Ekeland’s variational principle, generalized Caristi’s (common) fixed point theorems and nonconvex maximal element theorems for maps are also proved. Moreover, we obtain some applications to a nonconvex minimax theorem, nonconvex vectorial equilibrium theorems and convergence theorems in complete metric spaces.

Presenter: Wei-Shih Du (杜威仕).

Making Decision on Crystal Balls

Han-Lin Li (黎漢林)

Institue of Information Management National Chiao-Tung University

Hsin-Chu 300, Taiwan [email protected]

Abstract. This study proposes a visualization method of displaying decision alternatives on spheres . Given a set of alternatives with some attributes, we intend to rank and to group these alternatives based on a decision maker’s preferences on the attributes. Following various types of specifying preferences, four models are proposed: Moral Algebra Model, Even Swap Model, Pairwise Comparison Model, and Classification Model . By examining the moving trajectory of each alternative on spheres, a decision maker can adjust his preferences thus to reach a decision more confidently. Some practical examples, such as choosing jobs, renting offices, mutual funds investment, are demonstrated.

8

Solving the F -implicit generalized variational inequalities with applications

Yen-Cherng Lin (林炎成) General Education Center

China Medical University Taichung 404, Taiwan [email protected]

Abstract. In this talk, we study the F-implicit generalized variational inequalities in a real normed space setting. Weak solutions and strong solutions are introduced. Several existence results are derived. As an application, we study the F-implicit generalized com- plementarity problems and some existence results are obtained.

0-1 Semidefinite Programming in Data Clustering: Modelling and Approximation

Jiming Peng (彭積明)

Department of Industrial and Enterprise Systems Engineering University of Illinois, Urbana-Champigain

Urbana, IL 61801 U.S.A.

[email protected]

Abstract. In this talk, we will introduce a novel optimization model called 0-1 semidefinite programming (0-1 SDP), which arises frequently in data cluster analysis. We show that various clustering models including the classical K-means clustering, the newly developed normalized cut problem and cluster ensemble models can be embedded into the 0-1 SDP model.

Next we discuss the solution techniques for the underlying 0-1 SDP problem. In par- ticular, we focus on the development of approximation algorithms based on the relaxation of the 0-1 SDP model. Numerical experiments based on approximation algorithms will be reported.

10

A Dual Optimization Approach for Inverse Quadratic Eigenvalue Problems with Partial Eigenstructure

Defeng Sun (孫德鋒) Department of Mathematics National University of Singapore

Singapore [email protected]

Abstract. The inverse quadratic eigenvalue problem (IQEP) arises in the field of structural dynamics. It aims to find three symmetric matrices, known as the mass, the damping and the stiffness matrices, respectively such that they are closest to the given analytical matrices and satisfy the measured data. The difficulty of this problem lies in the fact that in applications the mass matrix should be positive definite and the stiffness matrix positive semidefinite. Based on an equivalent dual optimization version of the IQEP, we present a quadratically convergent Newton-type method. Our numerical experiments confirm the high efficiency of the proposed method.

Proximal Point Algorithms in Optimization and Four Nonlinear Projections

Wataru Takahashi

Department of Mathematical and Computing Sciences Tokyo Institute of Technology

Tokyo 152-8552, Japan [email protected]

Abstract. Let C be a nonempty closed convex subset of a real Hilbert space H and let f : H → (−∞, ∞] be a proper convex lower semicontinuous function. Consider a convex minimization problem:

min{f(x) : x ∈ C} = α.

The number α is called an optimal value, C is called an admissible set and M ={y ∈ C : f (y) = α} is called an optimal set. Next, define a function g : H → (−∞, ∞] as follows:

g(x) =

{ f (x), x∈ C,

∞, x /∈ C.

Then, g is a proper lower semicontinuous convex function of H into (−∞, ∞]. So, we consider the convex minimization problem:

min{g(x) : x ∈ H}. (∗)

For such a g, we can define a multivalued operator ∂g on H by

∂g(x) ={x^∗ ∈ H : g(y) ≥ g(x) + (x^∗, y− x), y ∈ H}

for all x∈ H. Such a ∂g is said to be the subdifferential of g. An operator A ⊂ H × H is accretive, if for (x₁, y₁), (x₂, y₂)∈ A,

(x₁− x2, y₁− y2)≥ 0.

If A is accretive, we can define, for each positive λ, the resolvent J_λ : R(I + λA)→ D(A) by J_λ = (I + λA)⁻¹. We know that J_λ is a nonexpansive mapping. An accretive operator A ⊂ H × H is called m-accretive, if R(I + λA) = H for all λ > 0. If g : H → (−∞, ∞] is a proper lower semicontinuous convex function, then ∂g is an m-accretive operator.

We know that one method for solving (∗) is the proximal point algorithm first introduced by Martinet. The proximal point algorithm is based on the notion of resolvent J_λ, i.e.,

J_λx = arg min

{

g(z) + 1

2λkz − xk² : z ∈ H^}. 12

The proximal point algorithm is an iterative procedure, which starts at a point x₁ ∈ H, and generates recursively a sequence{xn} of points xn+1 = J_λnx_n, where{λn} is a sequence of positive numbers.

In this talk, we first prove weak and strong convergence theorems for resolvents of accretive operators and maximal monotone operators in Banach spaces. That is, we discuss weak and strong convergence of proximal point algorithms in Banach spaces with four nonlinear projections. One of four nonlinear projections is new.

Optimal State-Space Solution with Minimal Realization for Standard H2

Control Problem

Po-Feng Wu (吳柏鋒), Chee-Fai Yung (容志輝), Pei-Ju Wang (王珮如) Department of Electrical Engineering

National Taiwan Ocean University Keelung, Taiwan

[email protected], [email protected], [email protected]

Abstract. In this talk, we will show that the controllable and unobservable subspaces of both the continuous and discrete-time H2 optimal controllers can be characterized by the image and kernel spaces of two matrices Z₂ and W₂, where Z₂ and W₂ are positive semidefinite solutions of two pertinent Lyapunov equations. The coefficients of the two Lyapunov equations involve the stabilizing solutions of the two celebrated Algebraic Riccati equations used in solving theH2 optimal control problem. By suitably choosing the bases adapted to ImZ2 and KerW2, the structure of the H2 optimal controller is geometrically clear. A minimal order state-space realization ofH2 optimal controller is then given via an elegant geometric approach. In terms of the use of geometric language, all the results and proofs given are simple and clear.

Presenter: Po-Feng Wu (吳柏鋒).

14

Recent Progresses on Strong Convergence of the Proximal Point Algorithm

Hong-Kun Xu (徐洪坤)

Department of Applied Mathematics National Sun Yat-Sen University

Kaohsiung 80424, Taiwan [email protected]

Abstract. Rockafellar’s proximal point algorithm (PPA) produces a sequence {xn} by x_n+1 = (I + c_nT )⁻¹x_n + e_n, n = 0, 1,· · ·, where T is a maximal monotone operator in a Hilbert space H, {cn} is a sequence of parameters which is bounded below away from 0, and {en} is the sequence of errors such that ^∑nkenk < ∞. It is well-known that {xn} is always weakly convergent to a zero of T (if any), but not always strongly convergent if dim H =∞. In this talk, we will review some recent progresses on the strong convergence aspect of the PPA.