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Numerical Algebra, Control & Optimization

2012 , Volume 2 , Issue 1

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A smoothing Newton method for generalized Nash equilibrium problems with second-order cone constraints
Yanhong Yuan, Hongwei Zhang and Liwei Zhang
2012, 2(1): 1-18 doi: 10.3934/naco.2012.2.1 +[Abstract](76) +[PDF](273.3KB)
Abstract:
We consider a type of generalized Nash equilibrium problems with second-order cone constraints. The Karush-Kuhn-Tucker system can be formulated as a system of semismooth equations involving metric projectors. Furthermore, the smoothing Newton method is given to get a Karush-Kuhn-Tucker point of the problem. The nonsingularity of Clarke's generalized Jacobian of the Karush-Kuhn-Tucker system, which is needed in the convergence analysis of smoothing Newton method, is demonstrated under the so-called constraint nondegeneracy condition in generalized Nash equilibrium problems and pseudo-strong second order optimality condition. At last, we take some experiments, in which the smoothing Newton method is applied. Furthermore, we get the normalized equilibria in the constraint-shared case. The numerical results show that the smoothing Newton method has a good performance in solving this type of generalized Nash equilibrium problems.
Global convergence of an SQP algorithm for nonlinear optimization with overdetermined constraints
Chunlin Hao and Xinwei Liu
2012, 2(1): 19-29 doi: 10.3934/naco.2012.2.19 +[Abstract](102) +[PDF](207.5KB)
Abstract:
A sequential quadratic programming (SQP) algorithm is presented for solving nonlinear optimization with overdetermined constraints. In each iteration, the quadratic programming (QP) subproblem is always feasible even if the gradients of constraints are always linearly dependent and the overdetermined constraints may be inconsistent. The $\ell_2$ exact penalty function is selected as the merit function. Under suitable assumptions on gradients of constraints, we prove that the algorithm will terminate at an approximate KKT point of the problem, or there is a limit point which is either a point satisfying the overdetermined system of equations or a stationary point for minimizing the $\ell_2$ norm of the residual of the overdetermined system of equations.
A sixth order numerical method for a class of nonlinear two-point boundary value problems
Xiao-Yu Zhang and Qing Fang
2012, 2(1): 31-43 doi: 10.3934/naco.2012.2.31 +[Abstract](77) +[PDF](197.3KB)
Abstract:
In this paper, we are concerned with the numerical solution of a class of nonlinear two-point boundary value problems with general boundary conditions. We propose a new numerical method of sixth order accuracy by integrating compact finite difference methods with the Green's function approach. It is the first sixth order accurate numerical scheme on non-uniform grids for the problem. We also give numerical results of some practical problems including reaction-diffusion equations. It is remarked that our numerical method is also efficient for layer equations.
An AIS-based optimal control framework for longevity and task achievement of multi-robot systems
Raymond Ching Man Chan and Henry Ying Kei Lau
2012, 2(1): 45-56 doi: 10.3934/naco.2012.2.45 +[Abstract](73) +[PDF](318.5KB)
Abstract:
Extending the longevity of autonomous agent system in real life application is a difficult task, especially in applications which require continuous high system performance. This paper presents a novel decentralized balancing controlling architecture for longevity and achievement in multi-agent robot systems based on several artificial immune systems (AIS) designs and principles. Simulation experiments have verified the proposed architecture has good capability to efficiently minimize the trade-off in system achievement while maintaining system sustainability, even in very demanding situations.
Global optimization via differential evolution with automatic termination
Bun Theang Ong and Masao Fukushima
2012, 2(1): 57-67 doi: 10.3934/naco.2012.2.57 +[Abstract](62) +[PDF](214.1KB)
Abstract:
Evolutionary Algorithms (EAs) provide a very powerful tool for solving optimization problems. In the last decades, numerous studies have been focusing on improving the performance of EAs. However, there is a lack of studies that tackle the question of the termination criteria. Indeed, EAs still need termination criteria prespecified by the user. In this paper, we propose to combine the Differential Evolution (DE) method with novel elements, i.e., the ``Gene Matrix'' (GM), the ``Space Decomposition'' (SD) and ``Space Rotation'' (SR) mechanisms, in order to equip DE with an automatic termination criterion without resort to predefined conditions. We name this algorithm ``Differential Evolution with Automatic Termination'' (DEAT). Numerical experiments using a test bed of widely used benchmark functions in 10, 50 and 100 dimensions show the effectiveness of the proposed method.
Univariate geometric Lipschitz global optimization algorithms
Dmitri E. Kvasov and Yaroslav D. Sergeyev
2012, 2(1): 69-90 doi: 10.3934/naco.2012.2.69 +[Abstract](217) +[PDF](602.3KB)
Abstract:
In this survey, univariate global optimization problems are considered where the objective function or its first derivative can be multiextremal black-box costly functions satisfying the Lipschitz condition over an interval. Such problems are frequently encountered in practice. A number of geometric methods based on constructing auxiliary functions with the usage of different estimates of the Lipschitz constants are described in the paper.
Solvability of a class of thermal dynamical contact problems with subdifferential conditions
Samir Adly, Oanh Chau and Mohamed Rochdi
2012, 2(1): 91-104 doi: 10.3934/naco.2012.2.91 +[Abstract](80) +[PDF](189.4KB)
Abstract:
We study a class of dynamic thermal sub-differential contact problems with friction, for long memory visco-elastic materials, which can be put into a general model of system defined by a second order evolution inequality, coupled with a first order evolution equation. We present and establish an existence and uniqueness result, by using general results on first order evolution inequality, with monotone operators and fixed point methods.
A bilevel optimization approach to obtain optimal cost functions for human arm movements
Sebastian Albrecht, Marion Leibold and Michael Ulbrich
2012, 2(1): 105-127 doi: 10.3934/naco.2012.2.105 +[Abstract](127) +[PDF](716.7KB)
Abstract:
Using a bilevel optimization approach, we investigate the question how humans plan and execute their arm motions. It is known that human motions are (approximately) optimal for suitable and unknown cost functions subject to the dynamics. We investigate the following inverse problem: Which cost function out of a parameterized family (e.g., convex combinations of functions suggested in the literature) reproduces recorded human arm movements best? The lower level problem is an optimal control problem governed by a nonlinear model of the human arm dynamics. The approach is analyzed for a dynamical 3D model of the human arm. Furthermore, results for a two-dimensional experiment with human probands are presented.
An efficient algorithm for convex quadratic semi-definite optimization
Lipu Zhang, Yinghong Xu and Zhengjing Jin
2012, 2(1): 129-144 doi: 10.3934/naco.2012.2.129 +[Abstract](73) +[PDF](225.9KB)
Abstract:
We present a full-step interior-point algorithm for convex quadratic semi-definite optimization based on a simple univariate function. The algorithm uses the simple function to determine the search direction and define the neighborhood of central path. The full-step used in the algorithm has local quadratic convergence property according to the proximity function which is also constructed by the simple function. We derive the iteration complexity for the algorithm and obtain the best-known iteration bounds for convex quadratic semi-definite optimization.
A class of smoothing SAA methods for a stochastic linear complementarity problem
Jie Zhang, Yue Wu and Liwei Zhang
2012, 2(1): 145-156 doi: 10.3934/naco.2012.2.145 +[Abstract](99) +[PDF](227.6KB)
Abstract:
A class of smoothing sample average approximation (SAA) methods is proposed for solving a stochastic linear complementarity problem, where the underlying function is the expected value of stochastic function. Existence and convergence results to the proposed methods are provided and some numerical results are reported to show the efficiency of the methods proposed.
Identification of water quality model parameters using artificial bee colony algorithm
Guangzhou Chen, Guijian Liu, Jiaquan Wang and Ruzhong Li
2012, 2(1): 157-165 doi: 10.3934/naco.2012.2.157 +[Abstract](74) +[PDF](195.2KB)
Abstract:
As prime data of water quality model, water quality parameters are of importance to forecast the situation of water quality correctly. Therefore, it is a key to identify them correctly. Aimed at the parameter identification problem, it can be transformed into an optimization problem by constructing objective function that minimizes simulation errors. In this study, a novel swarm intelligence optimization algorithm-artificial bee colony algorithm was used. In the experiment, many tests were done under the various ranges of parameters, and each variable was optimized according to its own reasonable scope. In addition, the optimization effect was compared based on the two methods producing new solutions in the neighborhood. As a key parameter of the algorithm, the impact of limit value on the algorithm performance was analyzed in detail under various values. Finally, two examples were analyzed and their computation results were compared with that of artificial fish swarm algorithm, simulated annealing and genetic algorithm. The results show that artificial bee colony algorithm has good adaptability to various ranges of parameters and better optimization precision. Moreover, it needs few control parameters of algorithm. So it is an effective parameter identification method.
A DC programming approach for a class of bilevel programming problems and its application in Portfolio Selection
Le Thi Hoai An, Tran Duc Quynh and Pham Dinh Tao
2012, 2(1): 167-185 doi: 10.3934/naco.2012.2.167 +[Abstract](303) +[PDF](282.5KB)
Abstract:
In this paper, we consider a class of bilevel programming problems where the upper objective function is convex quadratic while the lower objective function and the constraints are linear. The problem is first rewritten as minimizing a convex quadratic function subject to linear constraints and one concave constraint. Then, we use an exact penalty technique to reformulate the problem as a DC program. Afterward, DCA, an efficient algorithm in nonconvex programming, is developed to solve the resulting problem.
    For globally solving the problem, we combine DCA with a Branch and Bound algorithm (BB-DCA). DCA is applied to compute upper bounds, while lower bounds are calculated via a DC relaxation of the DC constraint. The proposed algorithms, DCA and BB-DCA, are compared with the Branch-and-Bound algorithm without DCA (BB) on random data. The numerical results show that the proposed algorithms are efficient.
    Finally, we consider an application of portfolio selection problems with the historical data related to the prices in the market of France, Luxembourg, United States and England during the period 2000-2007. The experimental results confirm the efficiency and the rapidity of DCA.
On some inverse singular value problems with Toeplitz-related structure
Zheng-Jian Bai, Xiao-Qing Jin and Seak-Weng Vong
2012, 2(1): 187-192 doi: 10.3934/naco.2012.2.187 +[Abstract](77) +[PDF](120.4KB)
Abstract:
In this paper, we consider some inverse singular value problems for Toeplitz-related matrices. We construct a Toeplitz-plus-Hankel matrix from prescribed singular values including a zero singular value. Then we find a solution to the inverse singular value problem for Toeplitz matrices which have double singular values including a double zero singular value.
A filter successive linear programming method for nonlinear semidefinite programming problems
Yi Xu and Wenyu Sun
2012, 2(1): 193-206 doi: 10.3934/naco.2012.2.193 +[Abstract](140) +[PDF](228.8KB)
Abstract:
In this paper we present a successive linear programming method with filter technique for nonlinear semidefinite programming. Such a method is characterized by use of the dominance concept of multiobjective optimization,~instead of a penalty parameter. The Successive Linear Programming with Filter (SLP-Filter) was used to solve the nonlinear programming (see [8]). In this paper, we extend it to deal with nonlinear semidefinite programming, and prove the convergence of the SLP-Filter for nonlinear semidefinite programming. We report numerical experiments to show the validity of the SLP-Filter method for nonlinear semidefinite programming.
Filtering solution of nonlinear stochastic optimal control problem in discrete-time with model-reality differences
Sie Long Kek, Kok Lay Teo and Mohd Ismail Abd Aziz
2012, 2(1): 207-222 doi: 10.3934/naco.2012.2.207 +[Abstract](106) +[PDF](186.2KB)
Abstract:
In this paper, we propose an efficient algorithm for solving a nonlinear stochastic optimal control problem in discrete-time, where the true filtered solution of the original optimal control problem is obtained through solving a linear model-based optimal control problem with adjustable parameters iteratively. The adjustments of these parameters are based on the differences between the real plant and the linear model that are measured. The main feature of the algorithm proposed is the integration of system optimization and parameter estimation in an interactive way so that the correct filtered solution of the original optimal control problem is obtained when the convergence is achieved. For illustration, a nonlinear continuous stirred reactor tank problem is studied. The simulation results obtained demonstrate the efficiency of the algorithm proposed.

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