Preconditioning by Projectors in the Solution of Contact Problems: A Parallel Implementation

A non-overlapping domain decomposition algorithm of the Neumann–Neumann type for solving contact problems of elasticity is presented. Using the duality theory of convex programming, the discretized problem turns into a quadratic one with equality and bound constraints. The dual problem is modified by orthogonal projectors to the natural coarse space. The resulting problem is solved by an augmented Lagrangian algorithm. The projectors ensure an optimal convergence rate for the solution of the auxiliary linear problems by the preconditioned conjugate gradient method. Relevant aspects on the numerical linear algebra of these problems are presented, together with an efficient parallel implementation of the method.     

On minimax eigenvalue problems via constrained optimization

In this paper, we consider the problem of minimizing the maximum eigenvalues of a matrix. The aim is to show that this optimization problem can be transformed into a standard nonlinearly constrained optimization problem, and hence is solvable by existing software packages. For illustration, two examples are solved by using the proposed method.     

Maximin Approach to the Ship Collision Avoidance Problem via Multiple-Subarc Sequential Gradient-Restoration Algorithm

The ideal strategy for ship collision avoidance under emergency conditions is to maximize wrt the controls the timewise minimum distance between a host ship and an intruder ship. This is a maximin problem or Chebyshev problem of optimal control in which the performance index being maximinimized is the distance between the two ships. Based on the multiple-subarc sequential gradient-restoration algorithm, a new method for solving the maximin problem is developed.Key to the new method is the observation that, at the maximin point, the time derivative of the performance index must vanish. With the zero derivative condition being treated as an inner boundary condition, the maximin problem can be converted into a Bolza problem in which the performance index, evaluated at the inner boundary, is being maximized wrt the controls. In turn, the Bolza problem with an added inner boundary condition can be solved via the multiple-subarc sequential gradient-restoration algorithm (SGRA).The new method is applied to two cases of the collision avoidance problem: collision avoidance between two ships moving along the same rectilinear course and collision avoidance between two ships moving along orthogonal courses. For both cases, we are basically in the presence of a two-subarc problem, the first subarc corresponding to the avoidance phase of the maneuver and the second subarc corresponding to the recovery phase. For stiff systems, the robustness of the multiple-subarc SGRA can be enhanced via increase in the number of subarcs. For the ship collision avoidance problem, a modest increase in the number of subarcs from two to three (one subarc in the avoidance phase, two subarcs in the recovery phase) helps containing error propagation and achieving better convergence results.     

Numerical solution of singular perturbation problems by a terminal boundary-value technique

In this paper, we present a new approach for numerically solving linear singularly perturbed two-point boundary-value problems in ordinary differential equations with a boundary layer on the left end of the interval. The original problem is divided into outer and inner region problems. A terminal boundary condition in implicit form is introduced. Then, the outer region problem is solved as a two-point boundary-value problem (TPBVP), and an explicit terminal boundary condition is obtained. In turn, the inner region problem is modified and solved as a TPBVP using the explicit terminal boundary condition. The proposed method is iterative on the terminal point of the inner region. Three numerical examples have been solved to demonstrate the applicability of the method.     

A generalized bottleneck assignment problem

A solution procedure is presented for a generalization of the standard bottleneck assignment problem in which a secondary criterion is automatically provided. A partitioning problem is modeled by this bottleneck problem to provide an example of its application.     

Optimal Trajectories and Guidance Schemes for Ship Collision Avoidance

The best strategy for collision avoidance under emergency conditions is to maximize wrt the controls the timewise minimum distance between the host ship and the intruder ship. In a restricted waterway area, two main constraints must be satisfied: the lateral deviation of the host ship from the original course is to be contained within certain limits; the longitudinal distance covered by the host ship is to be subject to a prescribed bound. At the maximin point of the encounter, the time derivative of the relative distance vanishes; this yields an inner boundary condition (orthogonality between the relative position vector and the relative velocity vector) separating the main phases of the maneuver: the avoidance and recovery phases. In this way, the optimal trajectory problem (a Chebyshev problem) can be converted into a Bolza problem with an inner boundary condition. Numerical solutions are obtained via the multiple-subarc sequential gradient-restoration algorithm (SGRA). Because the optimal trajectory is not suitable for real-time implementation, a guidance scheme approximating the optimal trajectory in real time is to be developed. For ship collision avoidance, the optimal trajectory results show that the rudder angle time history has a bang-bang form characterized by the alternation of saturated control subarcs of opposite signs joined by rapid transitions. Just as the optimal trajectory can be partitioned into three phases (avoidance phase, recovery phase, steady phase), a guidance trajectory can be constructed in the same way. For the avoidance and recovery phases, use of decomposition techniques leads to an algorithm computing the time lengths of these phases in real time. For the steady phase, a feedback control scheme is used to maneuver the ship steadily. Numerical results are presented. Portions of this paper were presented by the senior author at the 13th International Workshop on Dynamics and Control, Wiesensteig, Germany, 22-26 May 2005, in honor of George Leitmann. This research was supported by NSF Grant CMS-02-18878.     

A time-decomposition algorithm for a stiff linear two-point boundary-value problem and its application to a nonlinear optimal control problem

An algorithm is proposed to solve a stiff linear two-point boundary-value problem (TPBVP). In a stiff problem, since some particular solutions of the system equation increase and others decrease rapidly as the independent variable changes, the integration of the system equation suffers from numerical errors. In the proposed algorithm, first, the overall interval of integration is divided into several subintervals; then, in each subinterval a sub-TPBVP with arbitrarily chosen boundary values is solved. Second, the exact boundary values which guarantee the continuity of the solution are determined algebraically. Owing to the division of the integration interval, the numerical error is effectively reduced in spite of the stiffness of the system equation. It is also shown that the algorithm is successfully imbedded into an interaction-coordination algorithm for solving a nonlinear optimal control problem.The authors would like to thank Mr. T. Sera and Mr. H. Miyake for their help with the calculations.     

A class of singularly perturbed,nonlinear, fixed-endpoint control problems

Singular perturbation techniques are applied to a class of nonlinear, fixed-endpoint control problems to decompose the full-order problem into three lower-order problems, namely, the reduced problem and the left and right boundary-layer problems. The boundary-layer problems are linear-quadratic and, contrary to previous singular perturbation works, the reduced problem has a simple formulation. The solutions of these lower-order problems are combined to yield an approximate solution to the full nonlinear problem. Based on the properties of the lower-order problems, the full problem is shown to possess an asymptotic series solution.This work was supported in part by the National Science Foundation under Grant No. ENG-47-20091 and in part by the US Air Force under Grant No. AFOSR-73-2570.The author acknowledges the helpful suggestions of Professor P. V. Kokotovic, University of Illinois, Urbana, Illinois.     

Dual gradient method for linearly constrained,strongly convex,separable mathematical programming problems

A new dual gradient method is given to solve linearly constrained, strongly convex, separable mathematical programming problems. The dual problem can be decomposed into one-dimensional problems whose solutions can be computed extremely easily. The dual objective function is shown to have a Lipschitz continuous gradient, and therefore a gradient-type algorithm can be used for solving the dual problem. The primal optimal solution can be obtained from the dual optimal solution in a straightforward way. Convergence proofs and computational results are given.     

Application of the method of multipliers to state space constrained optimal control problems with discrete time

The paper deals with convergence conditions of multiplier algorithms for solving optimal control problems with discrete time suggested by J. Bjbvonek in some earlier papers. In this approach the original state space constrained problem is converted into a control-constrained problem by introducing an additional control variable and an equality constraint which is taken into consideration by a multiplier method. Convergence conditions for the multiplier Iteration of global and local nature are given for exact and inexact solution of the subproblems.     

Mathematical modeling of flows through inelastic porous solids

We propose a framework, based on classical mixture theory, to describe the isothermal flow of an incompressible fluid through a deformable inelastic porous solid. The modeling of the behavior of the inelastic solid takes into account changes in the elastic response due to evolution in the microstructure of the material. We apply the model to a compression layer problem. The mathematical problem generated by the model is a free boundary problem.     

Conjugate maps and duality in multiobjective optimization

This paper considers duality in convex vector optimization. A vector optimization problem requires one to find all the efficient points of the attainable value set for given multiple objective functions. Embedding the primal problem into a family of perturbed problems enables one to define a dual problem in terms of the conjugate map of the perturbed objective function. Every solution of the stable primal problem is associated with a certain solution of the dual problem, which is characterized as a subgradient of the perturbed efficient value map. This pair of solutions also provides a saddle point of the Lagrangian map.     

Optimal Control of Continuous Casting by Nondifferentiable Multiobjective Optimization

A new version of an interactive NIMBUS method for nondifferentiable multiobjective optimization is described. It is based on the reference point idea and the classification of the objective functions. The original problem is transformed into a single objective form according to the classification information. NIMBUS has been designed especially to be able to handle complicated real-life problems in a user-friendly way.The NIMBUS method is used for solving an optimal control problem related to the continuous casting of steel. The main goal is to minimize the defects in the final product. Conflicting objective functions are constructed according to certain metallurgical criteria and some technological constraints. Due to the phase changes during the cooling process there exist discontinuities in the derivative of the temperature distribution. Thus, the problem is nondifferentiable.Like many real-life problems, the casting model is large and complicated and numerically demanding. NIMBUS provides an efficient way of handling the difficulties and, at the same time, aids the user in finding a satisficing solution. In the end, some numerical experiments are reported and compared with earlier results.     

Abort landing in the presence of windshear as a minimax optimal control problem,part 1: Necessary conditions

The landing of a passenger aircraft in the presence of windshear is a threat to aviation safety. The present paper is concerned with the abort landing of an aircraft in such a serious situation. Mathematically, the flight maneuver can be described by a minimax optimal control problem. By transforming this minimax problem into an optimal control problem of standard form, a state constraint has to be taken into account which is of order three. Moreover, two additional constraints, a first-order state constraint and a control variable constraint, are imposed upon the model. Since the only control variable appears linearly, the Hamiltonian is not regular. Thus, well-known existence theorems about the occurrence of boundary arcs and boundary points cannot be applied. Numerically, this optimal control problem is solved by means of the multiple shooting method in connection with an appropriate homotopy strategy. The solution obtained here satisfies all the sharp necessary conditions including those depending on the sign of certain multipliers. The trajectory consists of bang-bang and singular subarcs, as well as boundary subarcs induced by the two state constraints. The occurrence of boundary arcs is known to be impossible for regular Hamiltonians and odd-ordered state constraints if the order exceeds two. Additionally, a boundary point also occurs where the third-order state constraint is active. Such a situation is known to be the only possibility for odd-ordered state constraints to be active if the order exceeds two and if the Hamiltonian is regular. Because of the complexity of the optimal control, this single problem combines many of the features that make this kind of optimal control problems extremely hard to solve. Moreover, the problem contains nonsmooth data arising from the approximations of the aerodynamic forces and the distribution of the wind velocity components. Therefore, the paper can serve as some sort of user's guide to solve inequality constrained real-life optimal control problems by multiple shooting.An extended abstract of this paper was presented at the 8th IFAC Workshop on Control Applications of Nonlinear Programming and Optimization, Paris, France, 1989 (see Ref. 1).This paper is dedicated to Professor Hans J. Stetter on the occasion of his 60th birthday.     

Abort landing in the presence of windshear as a minimax optimal control problem,part 2: Multiple shooting and homotopy

In Part 1 of the paper (Ref. 2), we have shown that the necessary conditions for the optimal control problem of the abort landing of a passenger aircraft in the presence of windshear result in a multipoint boundary-value problem. This boundary-value problem is especially well suited for numerical treatment by the multiple shooting method. Since this method is basically a Newton iteration, initial guesses of all variables are needed and assumptions about the switching structure have to be made. These are big obstacles, but both can be overcome by a so-called homotopy strategy where the problem is imbedded into a one-parameter family of subproblems in such a way that (at least) the first problem is simple to solve. The solution data to the first problem may serve as an initial guess for the next problem, thus resulting in a whole chain of problems. This process is to be continued until the objective problem is reached.Techniques are presented here on how to handle the various changes of the switching structure during the homotopy run. The windshear problem, of great interest for safety in aviation, also serves as an excellent benchmark problem: Nearly all features that can arise in optimal control appear when solving this problem. For example, the candidate for an optimal trajectory of the minimax optimal control problem shows subarcs with both bang-bang and singular control functions, boundary arcs and touch points of two state constraints, one being of first order and the other being of third order, etc. Therefore, the results of this paper may also serve as some sort of user's guide for the solution of complicated real-life optimal control problems by multiple shooting.The candidate found for an optimal trajectory is discussed and compared with an approximate solution already known (Refs. 3–4). Besides the known necessary conditions, additional sharp necessary conditions based on sign conditions of certain multipliers are also checked. This is not possible when using direct methods.An extended abstract of this paper was presented at the 8th IFAC Workshop on Control Applications of Nonlinear Programming and Optimization, Paris, France, 1989 (see Ref. 1).This paper is dedicated to Professor Hans J. Stetter on the occasion of his 60th birthday.     

Numerical solution of minimax optimal control problems by multiple shooting technique

This paper presents the application of the multiple shooting technique to minimax optimal control problems (optimal control problems with Chebyshev performance index). A standard transformation is used to convert the minimax problem into an equivalent optimal control problem with state variable inequality constraints. Using this technique, the highly developed theory on the necessary conditions for state-restricted optimal control problems can be applied advantageously. It is shown that, in general, these necessary conditions lead to a boundary-value problem with switching conditions, which can be treated numerically by a special version of the multiple shooting algorithm. The method is tested on the problem of the optimal heating and cooling of a house. This application shows some typical difficulties arising with minimax optimal control problems, i.e., the estimation of the switching structure which is dependent on the parameters of the problem. This difficulty can be overcome by a careful application of a continuity method. Numerical solutions for the example are presented which demonstrate the efficiency of the method proposed.     

A critical case of stability in a free boundary problem

We study the stability of the planar travelling wave solution to a free boundary problem for the heat equation in the whole . We turn the problem into a fully nonlinear parabolic system and establish a stability result which is the proper generalization of the one-dimensional case. The curvature terms contribute a gradient squared corresponding to critical growth. The latter is eliminated by means of the Hopf-Cole transformation. Received August 18, 2000, accepted September 27, 2000.     

Comment to “interactive fuzzy goal programming approach for bilevel programming problem” by S.R. Arora and R. Gupta

The fuzzy bilevel programming problem is solved in the paper [1] using an approach of fuzzy goal programming. Using a simple example we show that this will not lead to a satisfactory solution. Especially the fuzzy constraint that the upper level variable is near some desired one has no influence on the computed solution in the example. This constraint is used to model the hierarchy in the approach in [1]. At the end of the paper we suggest one straightforward possibility for modeling the fuzzy bilevel programming problem and converting it into a crisp substitute.     

Generalized Lagrange multiplier technique for nonlinear programming

Our aim here is to present numerical methods for solving a general nonlinear programming problem. These methods are based on transformation of a given constrained minimization problem into an unconstrained maximin problem. This transformation is done by using a generalized Lagrange multiplier technique. Such an approach permits us to use Newton's and gradient methods for nonlinear programming. Convergence proofs are provided, and some numerical results are given.     

A Haar wavelets based approximation for nonlinear inverse problems influenced by unknown heat source

In this discussion, a new numerical algorithm focused on the Haar wavelet is used to solve linear and nonlinear inverse problems with unknown heat source. The heat source is dependent on time and space variables. These types of inverse problems are ill-posed and are challenging to solve accurately. The linearization technique converted the nonlinear problem into simple nonhomogeneous partial differential equation. In this Haar wavelet collocation method (HWCM), the time part is discretized by using finite difference approximation, and space variables are handled by Haar series approximation. The main contribution of the proposed method is transforming this ill-posed problem into well-conditioned algebraic equation with the help of Haar functions, and hence, there is no need to implement any sort of regularization technique. The results of numerical method are efficient and stable for this ill-posed problems containing different noisy levels. We have utilized the proposed method on several numerical examples and have valuable efficiency and accuracy.