We are concerned with solving linear programming problems arising in the plastic truss layout optimization. We follow the ground structure approach with all possible connections between the nodal points. For very dense ground structures, the solutions of such problems converge to the so-called generalized Michell trusses. Clearly, solving the problems for large nodal densities can be computationally prohibitive due to the resulting huge size of the optimization problems. A technique called member adding that has correspondence to column generation is used to produce a sequence of smaller sub-problems that ultimately approximate the original problem. Although these sub-problems are significantly smaller than the full formulation, they still remain large and require computationally efficient solution techniques. In this article, we present a special purpose primal-dual interior point method tuned to such problems. It exploits the algebraic structure of the problems to reduce the normal equations originating from the algorithm to much smaller linear equation systems. Moreover, these systems are solved using iterative methods. Finally, due to high degree of similarity among the sub-problems after preforming few member adding iterations, the method uses a warm-start strategy and achieves convergence within fewer interior point iterations. The efficiency and robustness of the method are demonstrated with several numerical experiments.

中文翻译：

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一种用于塑料桁架布局优化的专门的双对偶内点法。

我们关注解决在塑料桁架布局优化中出现的线性编程问题。我们遵循节点之间所有可能连接的地面结构方法。对于非常密集的地面结构，此类问题的解决方案收敛到所谓的广义米歇尔桁架。显然，由于优化问题的结果是庞大的，因此对于大节点密度的问题的解决可能在计算上是禁止的。与列生成相对应的一种称为成员添加的技术用于产生一系列较小的子问题，这些子问题最终近似于原始问题。尽管这些子问题远小于完整公式，但它们仍然很大，并且需要计算有效的求解技术。在这篇文章中，我们提出了一种针对此类问题的特殊用途的原始对偶内点法。它利用问题的代数结构将源自算法的正态方程式简化为更小的线性方程组。此外，使用迭代方法解决了这些系统。最后，由于在执行几个成员添加迭代后，子问题之间具有高度相似性，因此该方法使用了热启动策略，并在较少的内部点迭代中实现了收敛。通过几个数值实验证明了该方法的效率和鲁棒性。这些系统使用迭代方法求解。最后，由于在执行几个成员添加迭代后，子问题之间具有高度相似性，因此该方法使用了热启动策略，并在较少的内部点迭代中实现了收敛。通过几个数值实验证明了该方法的效率和鲁棒性。这些系统使用迭代方法求解。最后，由于在执行几个成员添加迭代后，子问题之间具有高度相似性，因此该方法使用了热启动策略，并在较少的内部点迭代中实现了收敛。通过几个数值实验证明了该方法的效率和鲁棒性。