A globally convergent semi-smooth Newton method for control-state constrained DAE optimal control problems

We investigate a semi-smooth Newton method for the numerical solution of optimal control problems subject to differential-algebraic equations (DAEs) and mixed control-state constraints. The necessary conditions are stated in terms of a local minimum principle. By use of the Fischer-Burmeister function the local minimum principle is transformed into an equivalent nonlinear and semi-smooth equation in appropriate Banach spaces. This nonlinear and semi-smooth equation is solved by a semi-smooth Newton method. We extend known local and global convergence results for ODE optimal control problems to the DAE optimal control problems under consideration. Special emphasis is laid on the calculation of Newton steps which are given by a linear DAE boundary value problem. Regularity conditions which ensure the existence of solutions are provided. A regularization strategy for inconsistent boundary value problems is suggested. Numerical illustrations for the optimal control of a pendulum and for the optimal control of discretized Navier-Stokes equations conclude the article.     

hp-TDG/IPDG for Parabolic Obstacle Problems

We present a mixed hp-FE time DG method combined with an interior penalty DG method for parabolic obstacle problems. The discrete Lagragian multiplier set is spanned by basis functions which are biorthogonal to the basis functions of the primal variable. This allows us to write the discrete mixed problem as a problem of finding the root of a strongly semi-smooth function. In turn, this problem is solved by a locally Q-quadratic converging semi-smooth Newton method. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A minimum effort optimal control problem for the wave equation

A minimum effort optimal control problem for the undamped wave equation is considered which involves L ^∞-control costs. Since the problem is non-differentiable a regularized problem is introduced. Uniqueness of the solution of the regularized problem is proven and the convergence of the regularized solutions is analyzed. Further, a semi-smooth Newton method is formulated to solve the regularized problems and its superlinear convergence is shown. Thereby special attention has to be paid to the well-posedness of the Newton iteration. Numerical examples confirm the theoretical results.     

A semi-smooth Newton method for an inverse problem in option pricing

We present an optimal control approach using a Lagrangian framework to identify local volatility functions from given option prices. We employ a globalized sequential quadratic programming (SQP) algorithm and implement a line search strategy. The linear-quadratic optimal control problems in each iteration are solved by a primal-dual active set strategy which leads to a semi-smooth Newton method. We present first- and second-order analysis as well as numerical results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Virtual control regularization of state constrained linear quadratic optimal control problems

A numerical method for linear quadratic optimal control problems with pure state constraints is analyzed. Using the virtual control concept introduced by Cherednichenko et al. (Inverse Probl. 24:1–21, 2008) and Krumbiegel and R?sch (Control Cybern. 37(2):369–392, 2008), the state constrained optimal control problem is embedded into a family of optimal control problems with mixed control-state constraints using a regularization parameter α>0. It is shown that the solutions of the problems with mixed control-state constraints converge to the solution of the state constrained problem in the L ² norm as α tends to zero. The regularized problems can be solved by a semi-smooth Newton method for every α>0 and thus the solution of the original state constrained problem can be approximated arbitrarily close as α approaches zero. Two numerical examples with benchmark problems are provided.     

A primal-dual active-set algorithm for bilaterally constrained total variation deblurring and piecewise constant Mumford-Shah segmentation problems

In this paper, we propose a fast primal-dual algorithm for solving bilaterally constrained total variation minimization problems which subsume the bilaterally constrained total variation image deblurring model and the two-phase piecewise constant Mumford-Shah image segmentation model. The presence of the bilateral constraints makes the optimality conditions of the primal-dual problem semi-smooth which can be solved by a semi-smooth Newton’s method superlinearly. But the linear system to solve at each iteration is very large and difficult to precondition. Using a primal-dual active-set strategy, we reduce the linear system to a much smaller and better structured one so that it can be solved efficiently by conjugate gradient with an approximate inverse preconditioner. Locally superlinear convergence results are derived for the proposed algorithm. Numerical experiments are also provided for both deblurring and segmentation problems. In particular, for the deblurring problem, we show that the addition of the bilateral constraints to the total variation model improves the quality of the solutions.     

Regularized state-constrained boundary optimal control of the Navier-Stokes equations

The numerical solution of the Dirichlet boundary optimal control problem of the Navier-Stokes equations in presence of pointwise state constraints is investigated. Two different regularization techniques are considered. First, a Moreau-Yosida regularization of the problem is studied. Optimality conditions are derived and the convergence of the regularized solutions towards the original one is proved. A source representation of the control combined with a Lavrentiev type regularization strategy is also presented. The analysis concerning optimality conditions and convergence of the regularized solutions is carried out. In the last part of the paper numerical experiments are presented. For the numerical solution of each regularized problem a semi-smooth Newton method is applied.     

强部分逆网络流问题

Abstract. In this paper,a new model for inverse network flow problems,robust partial inverseproblem is presented. For a given partial solution,the robust partial inverse problem is to modify the coefficients optimally such that all full solutions containing the partial solution becomeoptimal under new coefficients. It has been shown that the robust partial inverse spanning treeproblem can be formulated as a combinatorial linear program,while the robust partial inverseminimum cut problem and the robust partial inverse assignment problem can be solved by combinatorial strongly polynomial algorithms.     

一类具有年龄结构的非线性种群扩散系统的最优收获控制

讨论了一类与年龄相关的非线性种群扩散系统的最优收获控制问题,证明了最优收获控制的存在性,并且给出了控制为最优的必要条件及其由偏微分方程组和变分不等式组成的最优性组.这些结果可为种群扩散系统最优控制问题的实际研究提供理论基础.     

Domino portrait generation: a fast and scalable approach

A domino portrait is an approximation of an image using a given number of sets of dominoes. This problem was first formulated in 1981 by Ken Knowlton in a patent application, which was finally granted in 1983. Domino portraits have been generated most often using integer linear programming techniques that provide optimal solutions, but these can be slow and do not scale well to larger portraits. In this paper we propose a new approach that overcomes these limitations and provides high quality portraits. Our approach combines techniques from operations research, artificial intelligence, and computer vision. Starting from a randomly generated template of blank domino shapes, a subsequent optimal placement of dominoes can be achieved in constant time when the problem is viewed as a minimum cost flow. The domino portraits one obtains are good, but not as visually attractive as optimal ones. Combining techniques from computer vision and large neighborhood search we can quickly improve the portraits. Empirically, we show that we obtain many orders of magnitude reduction in search time.     

条件平均场随机微分方程的最优控制问题

作者研究了一个条件平均场随机微分方程的最优控制问题.这种方程和某些部分信息下的随机最优控制问题有关,并且可以看做是平均场随机微分方程的推广.作者以庞特里雅金最大值原理的形式给出最优控制满足的必要和充分条件.此外,文中给出一个线性二次最优控制问题来说明理论结果的应用.     

A beam search implementation for the irregular shape packing problem

This paper investigates the irregular shape packing problem. We represent the problem as an ordered list of pieces to be packed where the order is decoded by a placement heuristic. A placement heuristic from the literature is presented and modified with a more powerful nofit polygon generator and new evaluation criteria. We implement a beam search algorithm to search over the packing order. Using this approach many parallel partial solutions can be generated and compared. Computational results for benchmark problems show that the algorithm generates highly competitive solutions in significantly less time than the best results currently in the literature.     

Solving optimal control problems for the unsteady Burgers equation in COMSOL Multiphysics

The optimal control of unsteady Burgers equation without constraints and with control constraints are solved using the high-level modelling and simulation package COMSOL Multiphysics. Using the first-order optimality conditions, projection and semi-smooth Newton methods are applied for solving the optimality system. The optimality system is solved numerically using the classical iterative approach by integrating the state equation forward in time and the adjoint equation backward in time using the gradient method and considering the optimality system in the space-time cylinder as an elliptic equation and solving it adaptively. The equivalence of the optimality system to the elliptic partial differential equation (PDE) is shown by transforming the Burgers equation by the Cole-Hopf transformation to a linear diffusion type equation. Numerical results obtained with adaptive and nonadaptive elliptic solvers of COMSOL Multiphysics are presented both for the unconstrained and the control constrained case.     

Total Completion Time in a Two-machine Flowshop with Deteriorating Tasks

This paper deals with two-machine flowshop problems with deteriorating tasks, i.e. tasks whose processing times are a nondecreasing function that depend on the length of the waiting periods. We consider the so-called Restricted Problem. This problem can be defined as follows: for a given permutation of tasks, find an optimal placement on two machines so that the total completion time is minimized. We will show that the Restricted Problem is nontrivial. We give some properties for the optimal placement and we propose an optimal placement algorithm.      

Path-following for optimal control of stationary variational inequalities

Moreau-Yosida based approximation techniques for optimal control of variational inequalities are investigated. Properties of the path generated by solutions to the regularized equations are analyzed. Combined with a semi-smooth Newton method for the regularized problems these lead to an efficient numerical technique.     

单部件可修系统的一个几何过程维修模型

本文研究了单部件、一个修理工组成的可修系统的最优更换问题,假定系统不能修复如新,以系统年龄T为策略,利用几何过程求出了最优的策略T^*,使得系统经长期运行单位时间内期望效益达到最大,并求出了系统经长期运行单位时间内期望效益的显式表达式。在一定条件下证明了T^*的唯一存在性。最后还证明了策略T^*比文献[6]中的策略T^*优。     

Model Predictive Control for Nonlinear Parabolic Differential Equations Based on a Linear Quadratic Gaussian Design

We consider optimal control problems for semilinear parabolic partial differential equations where process and measurement noise can occur. If we apply a Model Predictive Control (MPC) scheme we obtain optimal control problems on small time intervals. The resulting “smaller problems” can be linearized around a reference and solved by using a Linear Quadratic Gaussian (LQG) design. We present some theoretical background of the strategy above as well as results of a numerical implementation for a 3D problem. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Actuator placement based on reachable set optimization for expected disturbance

This paper examines the role of the control objective and the control time in determining fuel-optimal actuator placement for structural control. A general theory is developed that can be easily extended to include alternative performance metrics such as energy and timeoptimal control. The performance metric defines a convex admissible control set which leads to a max-min optimization problem expressing optimal location as a function of initial conditions and control time. A solution procedure based on a nested genetic algorithm is presented and applied to an example problem. Results indicate that the optimal placement varies widely as a function of both control time and disturbance location. An approximate fitness function is presented to alleviate the computational burden associated with finding exact solutions. This function is shown to accurately predict the optimal actuator locations for a 6th-order system, and is further demonstrated on a 12th-order system.This work was supported by the US Department of Energy at Sandia National Laboratories under Contract DE-AC04-76DP00789.     

云环境下面向复杂资源需求的虚拟机能效部署研究

针对云环境下在线虚拟机部署这一矢量装箱问题进行了研究,提出了多维空间划分模型和在线虚拟机能效部署算法OEEVMP。多维空间划分模型可以引导虚拟机部署,避免多维资源的不均衡利用;基于此模型,提出的OEEVMP算法在物理机运行数量局部最优和全局最优之间取得均衡,从而提高虚拟机部署能效。通过仿真实验,将OEEVMP算法与MFFD算法进行了对比,实验结果验证了所提算法的可行性和有效性。最后,对控制模型的两个参数进行了分析,给出了最佳的参数组合。     

Mathematical analysis of the optimal location of wastewater outfalls

In this work we deal with the design of wastewater treatmentsystems, mainly the optimal placement of underwater outfalls.This problem can be formulated as a state constrained optimalcontrol problem where the control is the position of the outfalls,the cost function is the sum of the distances to the wastewaterfarms and the state equations are those modelling dissolvedoxygen and biochemical oxygen demand concentrations. We discretizethe problem by means of a characteristic Galerkin method andwe propose an interior point algorithm for the numerical resolutionof the discretized optimization problem.