Keywords: subgradient, associated gradient, discontinuous objective function, gradient-only optimization, shape optimization, remeshing, positive projection point.

In this paper we propose the modification of a class of subgradient methods for structural shape optimization that uses remeshing. It is well known that remeshing introduces discontinuities in the numerically approximated objective function. Although subgradient methods were developed for non-smooth continuous functions, a subtle modification is proposed to make these methods suitable for piece-wise smooth step discontinuous functions. The subgradient methods are modified such that they find a positive projection point to solve the gradient-only optimization problem instead of finding a minimizer of an objective function.

A recent proposal aimed to optimize these piece-wise smooth step discontinuous objective functions by solving a gradient-only optimization problem instead of the conventional mathematical programming minimization problem [1]. The gradient-only optimization problem relies on an associated gradient [1], to obtain a gradient field that is everywhere defined for the non-differentiable piece-wise smooth step discontinuous functions.

By definition a subgradient is a global under-estimator of an objective function. Consequently, subgradients are everywhere defined for non-differentiable convex functions but not for piece-wise smooth step discontinuous functions. Hence, subgradient methods are not directly applicable to solve piece-wise smooth step discontinuous optimization problems. However, only two simple modifications are required to extend subgradient methods to piece-wise smooth discontinuous functions [1].

Four modified subgradient methods are considered in this study on two popular unconstrained shape optimization problems with a weighted compliance volume approach namely a cantilever design and a Michell structure. In addition, we consider a previously proposed gradient-only variant of the quasi-Newton BFGS algorithm [1], for comparison. The four subgradient algorithms considered are the non-summable diminishing step size (SG-NSDS), constant step size (SG-CS), constant step length (SG-CL) and the Open Opt implementation of Shor's r-algorithm (R-ALG) [2].

We obtained competitive results with R-ALG as well as the constant step length subgradient algorithm (SG-CL) on both problems. We found that the constant step size subgradient algorithm (SG-CS) converged within a region of a positive projection point whereas poor convergence was achieved with the non-summable diminishing step size subgradient algorithm (SG-NSDS).

1

D. Wilke, "Approaches to accommodate remeshing in shape optimization", PhD thesis, University of Pretoria, Pretoria, Aug. 2010.

2

D. Kroshko, "Open Opt", 2009.

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