Global weight optimization of frame structures with polynomial programming

TL;DR

This paper presents a novel approach using polynomial programming and the Lasserre hierarchy to globally optimize the weight of frame structures, overcoming non-convexity challenges.

Contribution

It introduces a method to compute global minimizers and feasible bounds for weight optimization of frame structures using semi-algebraic relaxation techniques.

Findings

Global minimizers can be computed using Lasserre hierarchy.

Feasible solutions can be projected from relaxed solutions.

Optimality gap converges to zero for convex minimizer sets.

Abstract

Weight optimization of frame structures with continuous cross-section parametrization is a challenging non-convex problem that has traditionally been solved by local optimization techniques. Here, we exploit its inherent semi-algebraic structure and adopt the Lasserre hierarchy of relaxations to compute the global minimizers. While this hierarchy generates a natural sequence of lower bounds, we show, under mild assumptions, how to project the relaxed solutions onto the feasible set of the original problem and thus construct feasible upper bounds. Based on these bounds, we develop a simple sufficient condition of global $\varepsilon$-optimality. Finally, we prove that the optimality gap converges to zero in the limit if the set of global minimizers is convex. We demonstrate these results by means of two academic illustrations.