A. Weldeyesus, J. Gondzio, Linwei He, M. Gilbert, P. Shepherd, A. Tyas
Abstract
Truss layout optimization problems with global stability constraints
are nonlinear and nonconvex and hence very challenging to solve,
particularly when problems become large. In this paper, a relaxation
of the nonlinear problem is modeled as a (linear) semidefinite
programming problem for which we describe an efficient primal-dual
interior point method capable of solving problems of a scale that
would be prohibitively expensive to solve using standard methods.
The proposed method exploits the sparse structure and low-rank
property of the stiffness matrices involved, greatly reducing
the computational effort required to process the associated
linear systems. Moreover, an adaptive ‘member adding’ technique
is employed which involves solving a sequence of much smaller
problems, with the process ultimately converging on the solution
for the original problem. Finally, a warm-start strategy is used
when successive problems display sufficient similarity, leading
to fewer interior point iterations being required. We perform
several numerical experiments to show the efficiency of the method
and discuss the status of the solutions obtained.
Key words: Truss structures, Global stability, Semidefinite programming, Interior point methods.