Computational experiments in the optimal slewing of flexible structures

Numerical experiments on the problem of moving a flexible beam are discussed. An optimal control problem is formulated and transcribed into a form which can be solved using semi-infinite optimization techniques. All experiments were carried out on a SUN 3 microcomputer

Computational methods in optimization: a unified approach

Computational Methods in Optimizatio

c○1998 Kluwer Academic Publishers B.V. On an Approach to Optimization Problems with a Probabilistic Cost and or Constraints

We present a new approach to a class of probability constrained optimization problems that arise in the context of optimal engineering design. These problems are characterized by the fact that the probability of failure of one or several components either must be minimized or must not exceed a preassigned threshold. Our approach is interactive: it consists of replacing the original optimal design problem in which either the cost function or a constraint are expressed in terms of a probability of failure, by a constrained minimax problem. Once the minimax problem is solved, the actual probability of failure is computed. Depending on the outcome of this computation, we provide heuristic rules for modifying the minimax problem and repeating this process a couple of times. An important feature of our new approach is that it decouples optimization and probability of failure calculations. This decoupling allows independent selection of methods for the solution of the optimization and the reliability subproblems. We present an example to demonstrate the effectiveness of our approach

Optimal design with probabilistic objective and constraints

Title: Optimal design with probabilistic objective and constraints Journal Issue: Journal of Engineering Mechanics-ASCE, 132(1) Publication Date: 01-01-2006.Significant challenges are associated with solving optimal structural design problems involving the failure probability in the objective and constraint functions. In this paper, we develop gradient-based optimization algorithms for estimating the solution of three classes of such problems in the case of continuous design variables. Our approach is based on a sequence of approximating design problems, which is constructed and then solved by a semi-infinite optimization algorithm. The construction consists of two steps: First, the failure probability terms in the objective function are replaced by auxiliary variables resulting in a simplified objective function. The auxiliary variables are determined automatically by the optimization algorithm. Second, the failure probability constraints are replaced by a parameterized first-order approximation. The parameter values are determined in an adaptive manner based on separate estimations of the failure probability. Any computational reliability method, including FORM, SORM and Monte Carlo simulation, can be used for this purpose. After repeatedly solving the approximating problem, an approximate solution of the original design problem is found, which satisfies the failure probability constraints at a precision level corresponding to the selected reliability method. The approach is illustrated by a series of examples involving optimal design and maintenance planning of a reinforced concrete bridge girder

On the use of augmented Lagrangians in the solution of generalized semi-infinite min-max problems

We present an approach for the solution of a class of generalized semi-infinite optimization problems. Our approach uses augmented Lagrangians to transform generalized semi-infinite min-max problems into ordinary semi-infinite min-max problems, with the same set of local and global solutions as well as the same stationary points. Once the transformation is effected, the generalized semi-infinite min-max problems can be solved using any available semi-infinite optimization algorithm. We illustrate our approach with two numerical examples, one of which deals with structural design subject to reliability constraints

Building design optimization using a convergent pattern search algorithm with adaptive precision simulations

We propose a simulation-precision control algorithm that can be used with a family of derivative free optimization algorithms to solve optimization problems in which the cost function is defined through the solutions of a coupled system of differential algebraic equations (DAEs). Our optimization algorithms use coarse precision approximations to the solutions of the DAE system in the early iterations and progressively increase the precision as the optimization approaches a solution. Such schemes often yield a significant reduction in computation time. We assume that the cost function is smooth but that it can only be approximated numerically by approximating cost functions that are discontinuous in the design parameters. We show that this situation is typical for many building energy optimization problems. We present a new building energy and daylighting simulation program, which constructs approximations to the cost function that converge uniformly on bounded sets to a smooth function as precision is increased. We prove that for our simulation program, our optimization algorithms construct sequences of iterates with stationary accumulation points. We present numerical experiments in which we minimize the annual energy consumption of an office building for lighting, cooling and heating. In these examples, our precision control algorithm reduces the computation time up to a factor of four. (c) 2004 Elsevier B.V. All rights reserved

Generalized pattern search algorithms with adaptive precision function evaluations

In the literature on generalized pattern search algorithms, convergence to a stationary point of a once continuously differentiable cost function is established under the assumption that the cost function can be evaluated exactly. However, there is a large class of engineering problems where the numerical evaluation of the cost function involves the solution of systems of differential algebraic equations. Since the termination criteria of the numerical solvers often depend on the design parameters, computer code for solving these systems usually defines a numerical approximation to the cost function that is discontinuous with respect to the design parameters. Standard generalized pattern search algorithms have been applied heuristically to such problems, but no convergence properties have been stated. In this paper we extend a class of generalized pattern search algorithms to a form that uses adaptive precision approximations to the cost function. These numerical approximations need not define a continuous function. Our algorithms can be used for solving linearly constrained problems with cost functions that are at least locally Lipschitz continuous. Assuming that the cost function is smooth, we prove that our algorithms converge to a stationary point. Under the weaker assumption that the cost function is only locally Lipschitz continuous, we show that our algorithms converge to points at which the Clarke generalized directional derivatives are nonnegative in predefined directions. An important feature of our adaptive precision scheme is the use of coarse approximations in the early iterations, with the approximation precision controlled by a test. Such an approach leads to substantial time savings in minimizing computationally expensive functions