Target shape design optimization by evolving B-splines with cooperative coevolution

• A framework was proposed to evolve the two components of B-spline in a reasonable cooperated fashion. It could increase the freedom of knot vector greatly, which is desired for a better exploration in the search space.
• The difficulty of TSDOPs is reduced with the natural decomposition of control points and knot vector. It is also helpful for EAs exploring the search space easier in a tight time budget.
• An algorithm named CMA-ES-CC was formulated to evaluate the performance of the new coevolutionary paradigm.Experiments were conducted based on two target shapes. It was found that the performance of CMA-ES-CC was not only good, but also stable. The comparison with six other EAs for TSDOPs suggested that the results of CMA-ES-CC were significantly better.

With high reputation in handling non-linear and multi-model problems with little prior knowledge, evolutionary algorithms (EAs) have successfully been applied to design optimization problems as robust optimizers. Since real-world design optimization is often computationally expensive, target shape design optimization problems (TSDOPs) have been frequently used as efficient miniature model to check algorithmic performance for general shape design. There are at least three important issues in developing EAs for TSDOPs, i.e., design representation, fitness evaluation and evolution paradigm. Existing work has mainly focused on the first two issues, in which (1) an adaptive encoding scheme with B-spline has been proposed as a representation, and (2) a symmetric Hausdorff distance based metric has been used as a fitness function. But for the third issue, off-the-shelf EAs were used directly to evolve B-spline control points and/or knot vector. In this paper, we first demonstrate why it is unreasonable to evolve the control points and knot vector simultaneously. And then a new coevolutionary paradigm is proposed to evolve the control points and knot vector of B-spline separately in a cooperative manner. In the new paradigm, an initial population is generated for both the control points, and the knot vector. The two populations are evolved mostly separately in a round-robin fashion, with only cooperation at the fitness evaluation phase. The new paradigm has at least two significant advantages over conventional EAs. Firstly, it provides a platform to evolve both the control points and knot vector reasonably. Secondly, it reduces the difficulty of TSDOPs by decomposing the objective vector into two smaller subcomponents (i.e., control points and knot vector). To evaluate the efficacy of the proposed coevolutionary paradigm, an algorithm named CMA-ES-CC was formulated. Experimental studies were conducted based on two target shapes. The comparison with six other EAs suggests that the proposed cooperative coevolution paradigm is very effective for TSDOPs.

Since real-world design optimization is often computationally expensive, target shape design optimization problems (TSDOPs) was used as miniature model to check algorithmic performance for general shape design. With B-spline and adaptive encoding as representation, the TSDOPs can be modeled as complex numerical optimization problems which can be decomposed into two subproblems. i.e. components for control points and knot vector. A new cooperative coevolution paradigm is proposed to evolve the control points and knot vector. The final designed approaches are expected to be effective for the original real-world design optimization problems. Figure optionsDownload as PowerPoint slide