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Numerical Algebra, Control & Optimization

2015 , Volume 5 , Issue 1

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Preface
Yan Gao, Zhiqiang Xu, Lei Wang and Honglei Xu
2015, 5(1): i-ii doi: 10.3934/naco.2015.5.1i +[Abstract](180) +[PDF](75.3KB)
Abstract:
This Special Issue of Numerical Algebra, Control and Optimization (NACO) is dedicated to Professor Enmin Feng for his important contributions in Applied Optimization, Optimal Control, System Identification and Large Scale Computing and their Engineering Applications and on the occasion of his 75th Birthday.

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Determining the viability for hybrid control systems on a region with piecewise smooth boundary
Yanli Han and Yan Gao
2015, 5(1): 1-9 doi: 10.3934/naco.2015.5.1 +[Abstract](223) +[PDF](317.8KB)
Abstract:
This paper is devoted to determining the viability of hybrid control systems on a region which is expressed by inequalities of piecewise smooth functions. Firstly, the viability condition for the differential inclusion is discussed based on nonsmooth analysis. Secondly, the result is generalized to hybrid differential inclusion. Finally, the viability condition of differential inclusion on a region with the max-type function is given.
Delay-range dependent $H_\infty$ control for uncertain 2D-delayed systems
Li-Min Wang, Jing-Xian Yu, Jia Shi and Fu-Rong Gao
2015, 5(1): 11-23 doi: 10.3934/naco.2015.5.11 +[Abstract](309) +[PDF](436.6KB)
Abstract:
This paper proposes a delay-range dependent method to solve a two-dimensional (2D) stabilization and $H_\infty$ control problem for a class of uncertain delayed systems described by the Roessor model with a range delay. By using a new 2D Lyapunov-Krasovskii function and introducing a differential inequality to the difference Lyapunov functional for 2D systems, sufficient delay-range dependent conditions for the existence of the proposed feedback controller scheme are established in terms of linear matrix inequalities (LMIs), which depend on both the difference between the upper and lower delay bounds and the upper delay bound of the interval time-varying delay. By solving these LMIs, the 2D law is explicitly formulated, together with an adjustable robust $H_\infty$ performance level. The analysis of the application in the thermal process demonstrates the effectiveness of the proposed controller.
On the global convergence of a parameter-adjusting Levenberg-Marquardt method
Liyan Qi, Xiantao Xiao and Liwei Zhang
2015, 5(1): 25-36 doi: 10.3934/naco.2015.5.25 +[Abstract](290) +[PDF](330.1KB)
Abstract:
The Levenberg-Marquardt (LM) method is a classical but popular method for solving nonlinear equations. Based on the trust region technique, we propose a parameter-adjusting LM (PALM) method, in which the LM parameter $\mu_k$ is self-adjusted at each iteration based on the ratio between actual reduction and predicted reduction. Under the level-bounded condition, we prove the global convergence of PALM. We also propose a modified parameter-adjusting LM (MPALM) method. Numerical results show that the two methods are very efficient.
Analysis of complexity of primal-dual interior-point algorithms based on a new kernel function for linear optimization
Siqi Li and Weiyi Qian
2015, 5(1): 37-46 doi: 10.3934/naco.2015.5.37 +[Abstract](260) +[PDF](389.3KB)
Abstract:
Kernel functions play an important role in defining new search directions for primal-dual interior-point algorithm. In this paper, a new kernel function which its barrier term is integral type is proposed. We study the properties of the new kernel function, and give a primal-dual interior-point algorithm for solving linear optimization based on the new kernel function. Polynomial complexity of algorithm is analyzed. The iteration bounds both for large-update and for small-update methods are obtained, respectively. The iteration bound for small-update method is the best known complexity bound.
Optimality of piecewise thermal conductivity in a snow-ice thermodynamic system
Wei Lv and Ruirui Sui
2015, 5(1): 47-57 doi: 10.3934/naco.2015.5.47 +[Abstract](346) +[PDF](347.0KB)
Abstract:
This article is intended to provide the optimality of piecewise thermal conductivity in a snow-ice thermodynamic system. Based on the temperature distribution characteristics of snow and sea ice, we construct a piecewise smooth thermodynamic system coupled by snow and sea ice. Taking the piecewise thermal conductivities of snow and sea ice as control variables and the temperature deviations obtained from the system and the observations as the performance criterion, an identification model with state constraints is given. The dependency relationship between state and control variables is proven, and the existence of the optimal control is discussed. The work can provide a theoretical foundation for simulating temperature distributions of snow and sea ice.
Optimal dilution strategy for a microbial continuous culture based on the biological robustness
Jingang Zhai, Guangmao Jiang and Jianxiong Ye
2015, 5(1): 59-69 doi: 10.3934/naco.2015.5.59 +[Abstract](297) +[PDF](549.5KB)
Abstract:
A robust optimal parameter selection model is proposed to formulate the microbial continuous culture process of glycerol bio-dissimilation to 1,3-propanediol (1,3-PD), in which the dilution rate and the glycerol concentration in feed medium are taken as the optimization variables. In consideration of the uncertain factors that some system parameters may be changed with the change of the optimization variables, we establish a novelty mathematical model which is represented by an eight-dimensional nonlinear dynamical system with the unknown parameters. On the basis of biological robustness, we give a quantitative definition of robustness index. The concentration of 1,3-PD at the approximately steady-state time together with the robustness index are taken as the cost functional in consideration of uncertain metabolic mechanisms. A parallel particle swarm optimization -- optimal parameter selection algorithm (PPSO-OPSA) is constructed to find the optimal dilution rate and the feeding glycerol concentration. Numerical results show that, by employing the obtained optimal input strategy, not only the concentration of 1,3-PD at the approximately steady-state time can be increased considerably compared with the previous experimental results, but also the obtained optimal parameters are robust for the dynamical system.
Single machine batch scheduling problem to minimize makespan with controllable setup and jobs processing times
Chengxin Luo
2015, 5(1): 71-77 doi: 10.3934/naco.2015.5.71 +[Abstract](293) +[PDF](295.6KB)
Abstract:
This paper concerns with a single-machine scheduling problem under batch availability in which both the setup of each batch and the processing times of jobs are controllable by allocating a resource. The completion time of a job in a batch is that of the last job in the batch. Two batch scheduling problems are investigated. The objective is to determine the job sequence, the partition of the job sequence into batches and the resource allocation scheme to minimize makespan, subject to the total amount of resource is bounded by a given value $U$ in the first problem; while in the second problem is to minimize a total cost of makespan and resource without resource limitation, respectively. We show that the problems underlying can be solved in polynomial time and present optimal algorithms.
Proximal iterative Gaussian smoothing algorithm for a class of nonsmooth convex minimization problems
Sanming Liu, Zhijie Wang and Chongyang Liu
2015, 5(1): 79-89 doi: 10.3934/naco.2015.5.79 +[Abstract](313) +[PDF](334.0KB)
Abstract:
In this paper, we consider the problem of minimizing a convex objective which is the sum of three parts: a smooth part, a simple non-smooth Lipschitz part, and a simple non-smooth non-Lipschitz part. A novel optimization algorithm is proposed for solving this problem. By making use of the Gaussian smoothing function of the functions occurring in the objective, we smooth the second part to a convex and differentiable function with Lipschitz continuous gradient by using both variable and constant smoothing parameters. The resulting problem is solved via an accelerated proximal-gradient method and this allows us to recover approximately the optimal solutions to the initial optimization problem with a rate of convergence of order $O(\frac{\ln k}{k})$ for variable smoothing and of order $O(\frac{1}{k})$ for constant smoothing.

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