Warren Hare

A survey of non-gradient optimization methods in structural engineering

In this paper, we present a review on non-gradient optimization methods with applications to structural engineering. Due to their versatility, there is a large use of heuristic methods of optimization in structural engineering. However, heuristic methods do not guarantee convergence to (locally) optimal solutions. As such, recently, there has been an increasing use of derivative-free optimization techniques that guarantee optimality. For each method, we provide a pseudo code and list of references with structural engineering applications. Strengths and limitations of each technique are discussed. We conclude with some remarks on the value of using methods customized for a desired application.

A proximal bundle method for nonsmooth nonconvex functions with inexact information

For a class of nonconvex nonsmooth functions, we consider the problem of computing an approximate critical point, in the case when only inexact information about the function and subgradient values is available. We assume that the errors in function and subgradient evaluations are merely bounded, and in principle need not vanish in the limit. We examine the redistributed proximal bundle approach in this setting, and show that reasonable convergence properties are obtained. We further consider a battery of difficult nonsmooth nonconvex problems, made even more difficult by introducing inexactness in the available information. We verify that very satisfactory outcomes are obtained in our computational implementation of the inexact algorithm.

Optimizing horizontal alignment of roads in a specified corridor

Finding an optimal alignment connecting two end-points in a specified corridor is a complex problem that requires solving three interrelated sub-problems, namely the horizontal alignment, vertical alignment and earthwork optimization problems. In this research, we developed a novel bi-level optimization model combining those three problems. In the outer level of the model, we optimize the horizontal alignment and in the inner level of the model a vertical alignment optimization problem considering earthwork allocation is solved for a fixed horizontal alignment. Derivative-free optimization algorithms are used to solve the outer problem. The result of our model gives an optimal horizontal alignment in the form of a linear-circular curve and an optimal vertical alignment in the form of a quadratic spline. Our model is tested on real-life data. The numerical results show that our approach improves the road alignment designed by civil engineers by 27% on average, resulting in potentially millions of dollars of savings. HighlightsSolution 27% cheaper than the one manually built by civil engineers.Bi-level optimization: Derivative-free algorithm for horizontal alignment.Mixed integer linear programming for vertical alignment and earth-work.Piecewise linear-circular horizontal; piecewise quadratic vertical.

A mixed-integer linear programming model to optimize the vertical alignment considering blocks and side-slopes in road construction

In the vertical alignment phase of road design, one minimizes the cost of moving material between different sections of the road while maintaining safety and building code constraints. Existing vertical alignment models consider neither the side-slopes of the road nor the natural blocks like rivers, mountains, etc., in the construction area. The calculated cost without the side-slopes can have significant errors (more than 20 percent), and the earthwork schedule without considering the blocks is unrealistic. In this study, we present a novel mixed integer linear programming model for the vertical alignment problem that considers both of these issues. The numerical results show that the approximation of the side-slopes can generate solutions within an acceptable error margin specified by the user without increasing the time complexity significantly.

Derivative-free optimization methods for finite minimax problems

Derivative-free optimization focuses on designing methods to solve optimization problems without the analytical knowledge of the function. In this paper, we consider the problem of designing derivative-free methods for finite minimax problems: min x max i=1, 2, …, N f i (x). In order to solve the problem efficiently, we seek to exploit the smooth substructure within the problem. Using ideas developed by Burke et al. [J.V. Burke, A.S. Lewis, and M.L. Overton, Approximating subdifferentials by random sampling of gradients, Math. Oper. Res. 27(3) (2002), pp. 567–584; J.V. Burke, A.S. Lewis, and M.L. Overton, A robust gradient sampling algorithm for nonsmooth, nonconvex optimization, SIAM J. Optim. 15(3) (2005), pp. 751–779 (electronic)], we create the idea of a robust simplex gradient descent direction and use it to accelerate convergence. Convergence is proven by showing that the resulting algorithm fits into the directional direct-search framework. Numerical tests demonstrate the algorithms effectiveness on finite minimax problems.

Configuration optimization of dampers for adjacent buildings under seismic excitations

Passive coupling of adjacent structures is known to be an effective method to reduce undesirable vibrations and structural pounding effects. Past results have shown that reducing the number of dampers can considerably decrease the cost of implementation and does not significantly decrease the efficiency of the system. The main objective of this study was to find the optimal arrangement of a limited number of dampers to minimize interstorey drift. Five approaches to solving the resulting bi-level optimization problem are introduced and examined (exhaustive search, inserting dampers, inserting floors, locations of maximum relative velocity and a genetic algorithm) and the numerical efficiency of each method is examined. The results reveal that the inserting damper method is the most efficient and reliable method, particularly for tall structures. It was also found that increasing the number of dampers does not necessarily increase the efficiency of the system. In fact, increasing the number of dampers can exacerbate the dynamic response of the system.

Models and strategies for efficiently determining an optimal vertical alignment of roads

Selecting an optimal vertical alignment while satisfying safety and design constraints is an important task during road construction. The amount of earthwork operations depends on the design of the vertical alignment, so a good vertical alignment can have a profound impact on final construction costs. In this research, we improve the performance of a previous mixed-integer linear programming model, and we propose a new quasi-network flow model. Both models use a piecewise quadratic curve to compute the minimum cost vertical alignment and take earthwork operations into account. The models consider several features such as side-slopes, and physical blocks in the terrain. In addition to improving the precision, we propose several techniques that speed up the search for a solution, so that it is possible to make interactive design tools. We report numerical tests that validate the accuracy of the models, and reduce the calculation time.

Dichotomization: 2 × 2 (×2 × 2 × 2...) categories: infinite possibilities

BackgroundConsumers of epidemiology may prefer to have one measure of risk arising from analysis of a 2-by-2 table. However, reporting a single measure of association, such as one odds ratio (OR) and 95% confidence interval, from a continuous exposure variable that was dichotomized withholds much potentially useful information. Results of this type of analysis are often reported for one such dichotomization, as if no other cutoffs were investigated or even possible.MethodsThis analysis demonstrates the effect of using different theory and data driven cutoffs on the relationship between body mass index and high cholesterol using National Health and Nutrition Examination Survey data. The recommended analytic approach, presentation of a graph of ORs for a range of cutoffs, is the focus of most of the results and discussion.ResultsThese cutoff variations resulted in ORs between 1.1 and 1.9. This allows investigators to select a result that either strongly supports or provides negligible support for an association; a choice that is invisible to readers. The OR curve presents readers with more information about the exposure disease relationship than a single OR and 95% confidence interval.ConclusionAs well as offering results for additional cutoffs that may be of interest to readers, the OR curve provides an indication of whether the study focuses on a reasonable representation of the data or outlier results. It offers more information about trends in the association as the cutoff changes and the implications of random fluctuations than a single OR and 95% confidence interval.

On the Range of the Douglas–Rachford Operator

The problem of finding a minimizer of the sum of two convex functions - or, more generally, that of finding a zero of the sum of two maximally monotone operators - is of central importance in variational analysis. Perhaps the most popular method of solving this problem is the Douglas-Rachford splitting method. Surprisingly, little is known about the range of the Douglas-Rachford operator. In this paper, we set out to study this range systematically. We prove that for 3* monotone operators a very pleasing formula can be found that reveals the range to be nearly equal to a simple set involving the domains and ranges of the underlying operators. A similar formula holds for the range of the corresponding displacement mapping. We discuss applications to subdifferential operators, to the infimal displacement vector, and to firmly nonexpansive mappings. Various examples and counter-examples are presented, including some concerning the celebrated Brezis-Haraux theorem.

Generalized Solutions for the Sum of Two Maximally Monotone Operators

A common theme in mathematics is to define generalized solutions to deal with problems that potentially do not have solutions. A classical example is the introduction of least squares solutions via the normal equations associated with a possibly infeasible system of linear equations. In this paper, we introduce a “normal problem” associated with finding a zero of the sum of two maximally monotone operators. If the original problem admits solutions, then the normal problem returns this same set of solutions. The normal problem may yield solutions when the original problem does not admit any; furthermore, it has attractive variational and duality properties. Several examples illustrate our theory.