Optimal Thickness of a Cylindrical Shell Under a Time-Dependent Force

We discuss thickness optimization problems for cylindrical tubes that are loaded by time-dependent applied force. This is a problem of shape optimization that leads to optimal control in linear elasticity theory. We determine the optimal thickness of a cylindrical tube by minimizing the deformation of the tube under the influence of an external force. The main difficulty is that the state equation is a hyperbolic partial differential equation of the fourth order. The first order necessary conditions for the optimal solution are derived. Based on them, a numerical method is set up and numerical examples are presented.     

$$L^1$$ penalization of volumetric dose objectives in optimal control of PDEs

This work is concerned with a class of PDE-constrained optimization problems that are motivated by an application in radiotherapy treatment planning. Here the primary design objective is to minimize the volume where a functional of the state violates a prescribed level, but prescribing these levels in the form of pointwise state constraints leads to infeasible problems. We therefore propose an alternative approach based on \(L^1\) penalization of the violation that is also applicable when state constraints are infeasible. We establish well-posedness of the corresponding optimal control problem, derive first-order optimality conditions, discuss convergence of minimizers as the penalty parameter tends to infinity, and present a semismooth Newton method for their efficient numerical solution. The performance of this method for a model problem is illustrated and contrasted with an alternative approach based on (regularized) state constraints.     

A multigrid method for constrained optimal control problems

We consider the fast and efficient numerical solution of linear-quadratic optimal control problems with additional constraints on the control. Discretization of the first-order conditions leads to an indefinite linear system of saddle point type with additional complementarity conditions due to the control constraints. The complementarity conditions are treated by a primal-dual active set strategy that serves as outer iteration. At each iteration step, a KKT system has to be solved. Here, we develop a multigrid method for its fast solution. To this end, we use a smoother which is based on an inexact constraint preconditioner.We present numerical results which show that the proposed multigrid method possesses convergence rates of the same order as for the underlying (elliptic) PDE problem. Furthermore, when combined with a nested iteration, the solver is of optimal complexity and achieves the solution of the optimization problem at only a small multiple of the cost for the PDE solution.     

Optimal discrete-valued control computation

In this paper, we consider an optimal control problem in which the control takes values from a discrete set and the state and control are subject to continuous inequality constraints. By introducing auxiliary controls and applying a time-scaling transformation, we transform this optimal control problem into an equivalent problem subject to additional linear and quadratic constraints. The feasible region defined by these additional constraints is disconnected, and thus standard optimization methods struggle to handle these constraints. We introduce a novel exact penalty function to penalize constraint violations, and then append this penalty function to the objective. This leads to an approximate optimal control problem that can be solved using standard software packages such as MISER. Convergence results show that when the penalty parameter is sufficiently large, any local solution of the approximate problem is also a local solution of the original problem. We conclude the paper with some numerical results for two difficult train control problems.     

An augmented Lagrangian method for optimization problems with structured geometric constraints

This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints are nonconvex and possibly complicated, but allow for a fast computation of projections onto this nonconvex set. Typical problem classes which satisfy this requirement are optimization problems with disjunctive constraints (like complementarity or cardinality constraints) as well as optimization problems over sets of matrices which have to satisfy additional rank constraints. The key idea behind our method is to keep these complicated constraints explicitly in the constraints and to penalize only the remaining constraints by an augmented Lagrangian function. The resulting subproblems are then solved with the aid of a problem-tailored nonmonotone projected gradient method. The corresponding convergence theory allows for an inexact solution of these subproblems. Nevertheless, the overall algorithm computes so-called Mordukhovich-stationary points of the original problem under a mild asymptotic regularity condition, which is generally weaker than most of the respective available problem-tailored constraint qualifications. Extensive numerical experiments addressing complementarity- and cardinality-constrained optimization problems as well as a semidefinite reformulation of MAXCUT problems visualize the power of our approach.

     

Optimal Design of Plane Frames under Uncertainty: A Comparison of Two Approaches

Using the first collapse theorem, the necessary and sufficient survival conditions of an elasto-plastic structure consist of the yield condition and the equilibrium condition. In practical applications several random model parameters have to be taken into account. This leads to a stochastic optimization problem which cannot be solved using the traditional methods. Instead of that, appropriate (deterministic) substitute problems must be formulated. Here, the design of plane frames is considered, where the applied load is supposed to be stochastic. In the first approach the recourse problem will be formulated in the standard form of stochastic linear programming (SLP). In order to apply efficient numerical solution procedures (LP-solvers), approximate recourse problems based on discretization (RPD) and the expected value problem (EVP) are introduced. In the second approach – based on the yield condition – a quadratic cost function will be introduced. After the formulation of the stochastic optimization problem, the expected cost based optimization problem (ECBOP) and the minimum expected cost problem (MECP) are formulated as representatives of appropriate substitute problems. Subsequently, comparative numerical results using these methods are presented. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Tracking a sharp crested wave front in hyperbolic heat transfer

In this article we investigate the numerical oscillations encountered when approximating the solution to the hyperbolic heat conduction equation. We consider a benchmark problem and show that it is not well-posed, unless a jump condition is specified. The alternative is to “smooth” the jump which leads to a sharp crested wave front, but with no discontinuity. To track the wave front we split the problem into auxiliary problems and solve these using different methods. The resulting solution is oscillation-free.     

MIP-based instantaneous control of mixed-integer PDE-constrained gas transport problems

We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations on a graph. We propose an instantaneous control approach in which suitable Euler discretizations yield systems of ordinary differential equations on a graph. This networked system of ordinary differential equations is shown to be well-posed and affine-linear solutions of these systems are derived analytically. As a consequence, finite-dimensional mixed-integer linear optimization problems are obtained for every time step that can be solved to global optimality using general-purpose solvers. We illustrate our approach in practice by presenting numerical results on a realistic gas transport network.     

A method for the numerical solution of the one-dimensional inverse Stefan problem

Summary In this paper we suggest the use of complete families of solutions of the heat equation for the numerical solution of the inverse Stefan problem. Our approach leads to linear optimization problems which can be established and solved easily. Convergence results are proved. In a final section the method is applied to some examples.     

On an Inequality of C. Sundberg: A Computational Investigation via Nonlinear Programming

The main goal of this article is to discuss a numerical method for finding the best constant in a Sobolev type inequality considered by C. Sundberg, and originating from Operator Theory. To simplify the investigation, we reduce the original problem to a parameterized family of simpler problems, which are constrained optimization problems from Calculus of Variations. To decouple the various differential operators and nonlinearities occurring in these constrained optimization problems, we introduce an appropriate augmented Lagrangian functional, whose saddle-points provide the solutions we are looking for. To compute these saddle-points, we use an Uzawa–Douglas–Rachford algorithm, which, combined with a finite difference approximation, leads to numerical results suggesting that the best constant is about five times smaller than the constant provided by an analytical investigation.     

Efficient Trefftz collocation algorithms for elliptic problems in circular domains

We consider the numerical solution of certain elliptic boundary value problems in disks and annuli using the Trefftz collocation method. In particular we examine boundary value problems for the Laplace, Helmholtz, modified Helmholtz and biharmonic equations in such domains. It is shown that this approach leads to systems in which the matrices possess specific structures. By exploiting these structures we propose efficient algorithms for the solution of the systems. The proposed algorithms are applied to standard test problems.     

A variational approach to define robustness for parametric multiobjective optimization problems

In contrast to classical optimization problems, in multiobjective optimization several objective functions are considered at the same time. For these problems, the solution is not a single optimum but a set of optimal compromises, the so-called Pareto set. In this work, we consider multiobjective optimization problems that additionally depend on an external parameter ${\lambda \in \mathbb{R}}$ , so-called parametric multiobjective optimization problems. The solution of such a problem is given by the λ-dependent Pareto set. In this work we give a new definition that allows to characterize λ-robust Pareto points, meaning points which hardly vary under the variation of the parameter λ. To describe this task mathematically, we make use of the classical calculus of variations. A system of differential algebraic equations will turn out to describe λ-robust solutions. For the numerical solution of these equations concepts of the discrete calculus of variations are used. The new robustness concept is illustrated by numerical examples.     

A New Conical Regularization for Some Optimization and Optimal Control Problems: Convergence Analysis and Finite Element Discretization

In this article, we study an abstract constrained optimization problem that appears commonly in the optimal control of linear partial differential equations. The main emphasis of the present study is on the case when the ordering cone for the optimization problem has an empty interior. To circumvent this major difficulty, we propose a new conical regularization approach in which the main idea is to replace the ordering cone by a family of dilating cones. We devise a general regularization approach and use it to give a detailed convergence analysis for the conical regularization as well as a related regularization approach. We showed that the conical regularization approach leads to a family of optimization problems that admit regular multipliers. The approach remains valid in the setting of general Hilbert spaces and it does not require any sort of compactness or positivity condition on the operators involved. One of the main advantages of the approach is that it is amenable for numerical computations. We consider four different examples, two of them elliptic control problems with state constraints, and present numerical results that completely support our theoretical results and confirm the numerical feasibility of our approach. The motivation for the conical regularization is to overcome the difficulties associated with the lack of Slater's type constraint qualification, which is a common hurdle in numerous branches of applied mathematics including optimal control, inverse problems, vector optimization, set-valued optimization, sensitivity analysis, variational inequalities, among others.     

Tropical optimization problems in time-constrained project scheduling

We consider a project that consists of activities to be performed in parallel under various temporal constraints, which include start-start, start-finish and finish-start precedence relationships, release times, deadlines and due dates. Scheduling problems are formulated to find optimal schedules for the project with respect to different objective functions to be minimized, such as the project makespan, the maximum deviation from the due dates, the maximum flow-time and the maximum deviation of finish times. We represent these problems as optimization problems in terms of tropical mathematics, and then solve them by applying direct solution methods of tropical optimization. As a result, new direct solutions of the scheduling problems are obtained in a compact vector form, which is ready for further analysis and practical implementation. The solutions are illustrated by simple numerical examples.     

B-spline solution of a singularly perturbed boundary value problem arising in biology

We use B-spline functions to develop a numerical method for solving a singularly perturbed boundary value problem associated with biology science. We use B-spline collocation method, which leads to a tridiagonal linear system. The accuracy of the proposed method is demonstrated by test problems. The numerical result is found in good agreement with exact solution.     

张量分析和多项式优化的若干进展

张量分析 (也称多重数值线性代数) 主要包括张量分解和张量特征值的理论和算法,多项式优化主要包括目标和约束均为多项式的一类优化问题的理论和算法. 主要介绍这两个研究领域中若干新的研究结果. 对张量分析部分,主要介绍非负张量H-特征值谱半径的一些性质及求解方法,还介绍非负张量最大 (小) Z-特征值的优化表示及其解法;对多项式优化部分,主要介绍带单位球约束或离散二分单位取值、目标函数为齐次多项式的优化问题及其推广形式的多项式优化问题和半定松弛解法. 最后对所介绍领域的发展趋势做了预测和展望.     

Starting-point strategy for interior-point algorithms for lower-bound shakedown analysis

Shakedown analysis by using the lower-bound theorem leads to computationally intensive nonlinear convex optimization problems with a large number of unknowns and constraints. Interior-point algorithms such as recently developed by the authors have proven to be efficient for the solution of these problems. For convergence and efficiency of the iterative process of these algorithms the choice of the starting-point is crucial. It should be inside of the feasible region and well-centered for fast convergence. No general method exists for the construction of optimum starting-points and only few investigations have been published on this issue. In this paper the physical meaning of the involved variables in shakedown problems is used to optimize starting-points. The efficiency of the new method is illustrated by a numerical examples and comparison with an alternative approach. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solving a free boundary problem with nonconstant coefficients

The present article is concerned with the numerical solution of a free boundary problem for an elliptic state equation with nonconstant coefficients. We maximize the Dirichlet energy functional over all domains of fixed volume. The domain under consideration is represented by a level set function, which is driven by the objective's shape gradient. The state is computed by the finite element method where the underlying triangulation is constructed by means of a marching cubes algorithm. We show that the combination of these tools lead to an efficient solver for general shape optimization problems.     

On phenomenon of scattering on resonances associated with discretisation of systems with fast rotating phase

Numerical integration of ODEs by standard numerical methods reduces continuous time problems to discrete time problems. Discrete time problems have intrinsic properties that are absent in continuous time problems. As a result, numerical solution of an ODE may demonstrate dynamical phenomena that are absent in the original ODE. We show that numerical integration of systems with one fast rotating phase leads to a situation of such kind: numerical solution demonstrates phenomenon of scattering on resonances that is absent in the original system.