On optimum design of frame structures

TL;DR

This paper compares four optimization methods for frame structures, demonstrating that polynomial optimization guarantees global optimality and bounds, unlike local methods which may converge to suboptimal or infeasible solutions.

Contribution

It introduces polynomial optimization as a robust approach for globally optimal frame structure design, overcoming limitations of traditional local optimization methods.

Findings

Polynomial optimization guarantees global optimality.

Local methods may converge to suboptimal or infeasible solutions.

Polynomial optimization has higher computational demands but provides bounds.

Abstract

Optimization of frame structures is formulated as a~non-convex optimization problem, which is currently solved to local optimality. In this contribution, we investigate four optimization approaches: (i) general non-linear optimization, (ii) optimality criteria method, (iii) non-linear semidefinite programming, and (iv) polynomial optimization. We show that polynomial optimization solves the frame structure optimization to global optimality by building the (moment-sums-of-squares) hierarchy of convex linear semidefinite programming problems, and it also provides guaranteed lower and upper bounds on optimal design. Finally, we solve three sample optimization problems and conclude that the local optimization approaches may indeed converge to local optima, without any solution quality measure, or even to infeasible points. These issues are readily overcome by using polynomial optimization, which exhibits a finite convergence, at the prize of higher computational demands.