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## Über dieses Buch

This book presents the latest research findings and state-of-the-art solutions on optimization techniques and provides new research direction and developments. Both the theoretical and practical aspects of the book will be much beneficial to experts and students in optimization and operation research community. It selects high quality papers from The International Conference on Optimization: Techniques and Applications (ICOTA2013). The conference is an official conference series of POP (The Pacific Optimization Research Activity Group; there are over 500 active members). These state-of-the-art works in this book authored by recognized experts will make contributions to the development of optimization with its applications.

## Inhaltsverzeichnis

### Chapter 1. Analysing Human Walking Using Dynamic Optimisation

A mathematical model to simulate human walking motions and study the dynamics behind walking is developed which is adjustable to accommodate different cases such as the single and double support phases of walking. We first propose a technique for estimating joint moments and position coordinates of body segments using the method of inverse dynamics. The estimates are then used as initial joint torques for solving the model as an optimal control problem with the setup of appropriate objective functions and constraints. Numerical experiments on the developed model and solution technique have been performed and the numerical results show that the model is able to replicate human walking motions and the optimal joint torques can be calculated to produce the resulting motions.
Meiyi Tan, Leslie S. Jennings, Song Wang

### Chapter 2. Rearrangement Optimization Problems Related to a Class of Elliptic Boundary Value Problems

In this paper, we investigate two optimization problems related to a class of elliptic boundary value problems on smooth bounded domains of $$\mathbb{R}^{N}$$. These optimization problems are formulated as minimum and maximum problems related to the rearrangements of given functions. Under some suitable assumptions, we show that both problems are solvable. Moreover, we obtain a representation result of the optimal solution for the minimization problem and show that this solution is unique and symmetric if the domain is a ball centered at the origin.
Chong Qiu, Yisheng Huang, Yuying Zhou

### Chapter 3. An Extension of the MOON2/MOON2R Approach to Many-Objective Optimization Problems

A multi-objective optimization (MUOP) method that supports agile and flexible decision making to be able to handle complex and diverse decision environments has been in high demand. This study proposes a general idea for solving many-objective optimization (MAOP) problems by using the MOON2 or MOON2R method. These MUOP methods rely on prior articulation in trade-off analysis among conflicting objectives. Despite requiring only simple and relative responses, the decision maker’s trade-off analysis becomes rather difficult in the case of MAOP problems, in which the number of objective functions to be considered is larger than in MUOP. To overcome this difficulty, we present a stepwise procedure that is extensively used in the analytic hierarchy process. After that, the effectiveness of the proposed method is verified by applying it to an actual problem. Finally, a general discussion is presented to outline the direction of future work in this area.
Yoshiaki Shimizu

### Chapter 4. Existence of Solutions for Variational-Like Hemivariational Inequalities Involving Lower Semicontinuous Maps

The main aim of this chapter is to investigate the existence of solutions in connection with a class of variational-like hemivariational inequalities in reflexive Banach spaces. Some existence theorems of solutions for the variational-like hemivariational inequalities involving lower semicontinuous set-valued maps are proved under different conditions. Moreover, a necessary and sufficient condition to guarantee the existence of solutions for the variational-like hemivariational inequalities is also given.
Guo-ji Tang, Zhong-bao Wang, Nan-jing Huang

### Chapter 5. An Iterative Algorithm for Split Common Fixed-Point Problem for Demicontractive Mappings

Inspired by the inertial proximal algorithms for finding a zero of a maximal monotone operator, we propose an inertial iteration algorithm for solving the split common fixed point problem for demicontractive mappings. We prove the asymptotical convergence of the algorithm under certain mild conditions. The results extend the result of Dang and Gao (Inverse Probl, 27:015007, 2011) and Moudafi (Inverse Probl 26:055007, 6pp, 2010. doi:10.1088/0266-5611/26/5/ 055007).
Yazheng Dang, Fanwen Meng, Jie Sun

### Chapter 6. On Constraint Qualifications for Multiobjective Optimization Problems with Vanishing Constraints

In this chapter, we consider a class of multiobjective optimization problems with inequality, equality and vanishing constraints. For the scalar case, this class of problems reduces to the class of mathematical programs with vanishing constraints recently appeared in literature. We show that under fairly mild assumptions some constraint qualifications like Cottle constraint qualification, Slater constraint qualification, Mangasarian-Fromovitz constraint qualification, linear independence constraint qualification, linear objective constraint qualification and linear constraint qualification do not hold at an efficient solution, whereas the standard generalized Guignard constraint qualification is sometimes satisfied. We introduce suitable modifications of above mentioned constraint qualifications, establish relationships among them and derive the Karush-Kuhn-Tucker type necessary optimality conditions for efficiency.
S. K. Mishra, Vinay Singh, Vivek Laha, R. N. Mohapatra

### Chapter 7. A New Hybrid Optimization Algorithm for the Estimation of Archie Parameters

Archie formula, which contains three fundamental parameters (a, m, n), is the basic equation to compute the water saturation in a clean or shaly formation. These parameters are known as Archie parameters. To identify accurately the water saturation for a given reservoir condition, it depends critically on the accurate estimates of the values of Archie parameters (a, m, n). These parameters are interdependent and hence it is difficult to identify them accurately. So we present a new hybrid global optimization technique, where a gradient-based method with BFGS update is combined with an intelligent algorithm called Artificial Bee Colony. This new hybrid global optimization technique has both the fast convergence of gradient descent algorithm and the global convergence of swarm algorithm. It is used to identify Archie parameters in carbonate reservoirs. The results obtained are highly satisfactory. To further test the effectiveness of the new hybrid global optimization method, it is applied to ten non-convex benchmark problems. The outcomes are encouraging.
Jianjun Liu, Honglei Xu, Guoning Wu, Kok Lay Teo

### Chapter 8. Optimization of Multivariate Inverse Mixing Problems with Application to Neural Metabolite Analysis

A mathematical methodology is presented that optimally solves an inverse mixing problem when both the composition of the source components and the amount of each source component are unknown. The model is useful for situations when the determination of the source compositions is unreliable or infeasible. We apply the model to longitudinal proton magnetic resonance spectroscopy (1H MRS) data gathered from the brains of newborn infants. 1H MRS was used to study changes in five metabolite concentrations in two brain regions of nine healthy term neonates. Measurements were performed three times in each infant over a period of 3 months, starting from birth, for a total of 27 scans. The methodology was then used to translate the metabolite concentration data into measures of relative density for two major brain cell type populations by fitting a matrix of metabolite concentration per unit density to the data. One cell type, reflecting neuronal density, increased over time in both regions studied, but especially in the frontal regions of the brain. The second type, characterized primarily by myoinositol, reflecting glial cell content, was found to decrease in both regions over time. Our new method can provide more specific and accurate assessments of the brain cell types during early brain development in neonates. The methodology is applicable to a wide range of physical systems that involve mixing of unknown source components.
A. Tamura-Sato, M. Chyba, L. Chang, T. Ernst

### Chapter 9. Exact Regularization, and Its Connections to Normal Cone Identity and Weak Sharp Minima in Nonlinear Programming

The regularization of a nonlinear program is exact if all solutions of the regularized problem are also solutions of the original problem for all values of the regularization parameter below some positive threshold. In Deng (Pac J Optim 8(1):27–32, 2012), we show that, for a given nonlinear program, the regularization is exact if and only if the Lagrangian function of a certain selection problem has a saddle point, and the regularization parameter threshold is inversely related to the Lagrange multiplier associated with the saddle point. The results in Deng (Pac J Optim 8(1):27–32, 2012) not only provide a fresh perspective on exact regularization but also extend the main results in Friedlander and Tseng (SIAM J Optim 18:1326–1350, 2007) on a characterization of exact regularization of a convex program to that of a nonlinear (not necessarily convex) program. In this paper, we will examine inner-connections among exact regularization, normal cone identity, and the existence of a weak sharp minimum for certain associated nonlinear programs. Along the way, we illustrate by examples, how to obtain both new results and reproduce many existing results from a fresh perspective.
S. Deng

### Chapter 10. The Worst-Case DFT Filter Bank Design with Subchannel Variations

In this paper, we consider an optimal design of a DFT filter bank subject to subchannel variation constraints. The design problem is formulated as a minimax optimization problem. By exploiting the properties of this minimax optimization problem, we show that it is equivalent to a semi-infinite optimization problem in which the continuous inequality constraints are only with respect to frequency. Then, a computational scheme is developed to solve such a semi-infinite optimization problem. Simulation results show that, for a fixed distortion level, the aliasing level between different subbands is significantly reduced, in some cases up to 28 dB, when compared with that obtained by the bi-iterative optimization method without consideration of the subchannel variations.
Lin Jiang, Changzhi Wu, Xiangyu Wang, Kok Lay Teo
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