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Numerical Algebra, Control & Optimization

2012 , Volume 2 , Issue 4

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Thorsten Koch and  Xiaoling Sun
2012, 2(4): i-ii doi: 10.3934/naco.2012.2.4i +[Abstract](26) +[PDF](82.1KB)
This special issue is a collection of selected papers presented at the $1^{st}$ Sino-German Workshop on Optimization, Modeling, Methods and Applications in Industry and Management (SGWO2011), which was held on August 15-19, 2011 at the Zuse Institute Berlin, Germany.

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Gradient estimates for Gaussian distribution functions: application to probabilistically constrained optimization problems
René Henrion
2012, 2(4): 655-668 doi: 10.3934/naco.2012.2.655 +[Abstract](30) +[PDF](210.5KB)
We provide lower estimates for the norm of gradients of Gaussian distribution functions and apply the results obtained to a special class of probabilistically constrained optimization problems. In particular, it is shown how the precision of computing gradients in such problems can be controlled by the precision of function values for Gaussian distribution functions. Moreover, a sensitivity result for optimal values with respect to perturbations of the underlying random vector is derived. It is shown that the so-called maximal increasing slope of the optimal value with respect to the Kolmogorov distance between original and perturbed distribution can be estimated explicitly from the input data of the problem.
The integrated size and price optimization problem
Miriam Kiessling , Sascha Kurz and  Jörg Rambau
2012, 2(4): 669-693 doi: 10.3934/naco.2012.2.669 +[Abstract](41) +[PDF](507.9KB)
We present the Integrated Size and Price Optimization Problem (ISPO) for a fashion discounter with many branches. Based on a two-stage stochastic programming model with recourse, we develop an exact algorithm and a production-compliant heuristic that produces small optimality gaps. In a field study we show that a distribution of supply over branches and sizes based on ISPO solutions is significantly better than a one-stage optimization of the distribution ignoring the possibility of optimal pricing.
Towards globally optimal operation of water supply networks
Ambros M. Gleixner , Harald Held , Wei Huang and  Stefan Vigerske
2012, 2(4): 695-711 doi: 10.3934/naco.2012.2.695 +[Abstract](96) +[PDF](810.5KB)
This paper is concerned with optimal operation of pressurized water supply networks at a fixed point in time. We use a mixed-integer nonlinear programming (MINLP) model incorporating both the nonlinear physical laws and the discrete decisions such as switching pumps on and off. We demonstrate that for instances from our industry partner, these stationary models can be solved to $\epsilon$-global optimality within small running times using problem-specific presolving and state-of-the-art MINLP algorithms.
    In our modeling, we emphasize the importance of distinguishing between what we call real and imaginary flow, i.e., taking into account that the law of Darcy-Weisbach correlates pressure difference and flow along a pipe if and only if water is available at the high pressure end of a pipe. Our modeling solution extends to the dynamic operative planning problem.
Two-stage stochastic programs: Integer variables, dominance relations and PDE constraints
Rüdiger Schultz
2012, 2(4): 713-738 doi: 10.3934/naco.2012.2.713 +[Abstract](41) +[PDF](3193.2KB)
From a unified point-of-view, we present some recent developments in two-stage stochastic programming. Our discussion includes stochastic programs with integer variables, stochastic programs with dominance constraints, and PDE constrained stochastic programs.
Analyzing the computational impact of MIQCP solver components
Timo Berthold , Ambros M. Gleixner , Stefan Heinz and  Stefan Vigerske
2012, 2(4): 739-748 doi: 10.3934/naco.2012.2.739 +[Abstract](45) +[PDF](176.8KB)
We provide a computational study of the performance of a state-of-the-art solver for nonconvex mixed-integer quadratically constrained programs (MIQCPs). Since successful general-purpose solvers for large problem classes necessarily comprise a variety of algorithmic techniques, we focus especially on the impact of the individual solver components. The solver SCIP used for the experiments implements a branch-and-cut algorithm based on a linear relaxation to solve MIQCPs to global optimality. Our analysis is based on a set of 86~publicly available test instances.
Simulation of Lévy-Driven models and its application in finance
Rachel Chen , Jianqiang Hu and  Yijie Peng
2012, 2(4): 749-765 doi: 10.3934/naco.2012.2.749 +[Abstract](40) +[PDF](436.5KB)
Lévy processes have been widely used to model financial assets such as stock prices, exchange rates, interest rates, and commodities. However, when applied to derivative pricing, very few analytical results are available except for European options. Therefore, one usually has to resort to numerical methods such as Monte Carlo simulation method. The simulation method is attractive in that it is very general and can also handle high dimensional problems very well. In this survey paper, we provide an overview on various simulation methods for Lévy processes. In addition, we introduce two simulation based sensitivity estimation methods: perturbation analysis and the likelihood ratio method. Sensitivity estimation is useful in various applications, such as derivative pricing and parameter estimation. Finally, we provide a simple illustrative example of applying simulation and sensitivity estimation to parameter estimation of Lévy-driven stochastic volatility model.
A survey on probabilistically constrained optimization problems
Xiaodi Bai , Xiaojin Zheng and  Xiaoling Sun
2012, 2(4): 767-778 doi: 10.3934/naco.2012.2.767 +[Abstract](31) +[PDF](179.5KB)
Probabilistically constrained optimization problems are an important class of stochastic programming problems with wide applications in finance, management and engineering planning. In this paper, we summarize some important solution methods including convex approximation, DC approach, scenario approach and integer programming approach. We also discuss some future research perspectives on the probabilistically constrained optimization problems.
A note on semicontinuity to a parametric generalized Ky Fan inequality
Chunrong Chen and  Zhimiao Fang
2012, 2(4): 779-784 doi: 10.3934/naco.2012.2.779 +[Abstract](34) +[PDF](140.7KB)
In this note, the continuity results of weak vector solutions and global vector solutions to a parametric generalized Ky Fan inequality are established by using a new scalarization method. Our results improve the corresponding ones of Li and Fang (J. Optim. Theory Appl. 147: 507-515, 2010).
A multigrid method for the maximal correlation problem
Xin-Guo Liu and  Kun Wang
2012, 2(4): 785-796 doi: 10.3934/naco.2012.2.785 +[Abstract](52) +[PDF](190.2KB)
In this note, the continuity results of weak vector solutions and global vector solutions to a parametric generalized Ky Fan inequality are established by using a new scalarization method. Our results improve the corresponding ones of Li and Fang (J. Optim. Theory Appl. 147: 507-515, 2010).
On product-type generalized block AOR method for augmented linear systems
Fang Chen , Ning Gao and  Yao- Lin Jiang
2012, 2(4): 797-809 doi: 10.3934/naco.2012.2.797 +[Abstract](42) +[PDF](187.3KB)
The generalized inexact accelerated overrelaxation ( GIAOR) method was presented by Bai, Parlett and Wang (Numer. Math. 102(2005)1-38) for solving the augmented system of linear equations. In this paper, a product-type generalized block AOR ( PGBAOR ) method is proposed, which is a two-step generalization of the GIAOR method. Both convergence and semi-convergence of the PGBAOR method are proved for the nonsingular and the singular augmented linear systems.
A generalization of the positive-definite and skew-Hermitian splitting iteration
Yang Cao , Wei- Wei Tan and  Mei-Qun Jiang
2012, 2(4): 811-821 doi: 10.3934/naco.2012.2.811 +[Abstract](41) +[PDF](304.0KB)
In this paper, a generalization of the positive-definite and skew-Hermitian splitting (GPSS) iteration is considered for solving non-Hermitian and positive definite systems of linear equations. Theoretical analysis shows that the GPSS method converges unconditionally to the exact solution of the linear system, with the upper bound of its convergence factor dependent only on the spectrum of the positive-definite splitting matrices. In some situations, this new scheme can outperform the Hermitian and skew-Hermitian splitting (HSS) method, the positive-definite and skew-Hermitian splitting (PSS) method, and the generalized HSS method (GHSS) and can be used as an efficient preconditioner. Numerical experiments using discretization of convection-diffusion-reaction equations demonstrate the effectiveness of the new method.
A new Bramble-Pasciak-like preconditioner for saddle point problems
Xiao-Fei Peng and  Wen Li
2012, 2(4): 823-838 doi: 10.3934/naco.2012.2.823 +[Abstract](38) +[PDF](232.9KB)
A new Bramble-Pasciak-like preconditioner with parameter is proposed for solving a linear system in the saddle point form. The system can be reformulated as a symmetric positive definite system with respect to some inner product and thus can be solved by the Bramble-Pasciak conjugate gradient (BPCG) method. Based on the spectral condition number of the associated system, the quasi-optimal parameters can be obtained to improve the convergence rate of the BPCG method. Numerical experiments on the Stokes problem are given to illustrate our theoretical results.
Newton-MHSS methods for solving systems of nonlinear equations with complex symmetric Jacobian matrices
Ai-Li Yang and  Yu-Jiang Wu
2012, 2(4): 839-853 doi: 10.3934/naco.2012.2.839 +[Abstract](53) +[PDF](220.5KB)
Modified Hermitian and skew-Hermitian splitting (MHSS) method is an unconditionally convergent iterative method for solving large sparse complex symmetric systems of linear equations. By making use of the MHSS iteration as the inner solver for the inexact Newton method, we establish a class of inexact Newton-MHSS methods for solving large sparse systems of nonlinear equations with complex symmetric Jacobian matrices at the solution points. The local and semi-local convergence properties are analyzed under some proper assumptions. Moreover, by introducing a backtracking linear search technique, a kind of global convergence inexact Newton-MHSS methods are also presented and analyzed. Numerical results are given to examine the feasibility and effectiveness of the inexact Newton-MHSS methods.
On convergence of the inner-outer iteration method for computing PageRank
Zhong-Zhi Bai
2012, 2(4): 855-862 doi: 10.3934/naco.2012.2.855 +[Abstract](55) +[PDF](296.5KB)
Without imposing any restriction on the damping factors and the stopping tolerances, we prove the overall convergence of the inner-outer iteration method for computing the PageRank vector, which was proposed by Gleich, Gray, Greif and Lau (SIAM J. Sci. Comput. 32(2010)349-371). Based on the formula of the contraction factor of the method, we discuss possible choices of the iteration parameters, which could be practically useful for accelerating the convergence rate of the inner-outer iteration method.
On the convergence of generalized parallel multisplitting iterative methods for semidefinite linear systems
Yanxing Cui , Chuanlong Wang and  Ruiping Wen
2012, 2(4): 863-873 doi: 10.3934/naco.2012.2.863 +[Abstract](41) +[PDF](159.1KB)
In this paper, we present the generalized stationary and nonstationary multisplitting iterative methods for positive semidefinite linear systems. We study the convergence theories of new methods and show that the quotient convergence and convergence of stationary parallel multisplitting method are equivalent under a concise condition. Finally, we prove that the generalized nonstationary parallel multisplitting method is quotient convergence with general weighting matrices.




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