Truss Topology Design by Conic Linear Optimization

This chapter can be viewed as a complement to Chapter 2 (Truss Topology Design by Linear Optimization). We will use the same mechanical model of trusses and, whenever possible, the same notation. In Chapter 2, the truss topology design problem is formulated and solved as a linear optimization (LO) problem. In this chapter, we will introduce alternative formulations using conic linear optimization (CLO). In particular, we will present linear second-order cone optimization (SOCO) and linear semidefinite Optimization (SDO) formulations of the minimum volume and minimum compliance problems. All formulations will be developed in the “primal” variables (bar cross-sectional areas) and the “dual” variables (displacements). We will start with the nonlinear (and nonconvex) formulation of the basic truss topology problem, prove the existence of a solution, and show that the Lagrangian dual to this problem is a convex quadratically constrained quadratic problem. Then we introduce the SOCO formulations of the problem, both primal and dual, and the SDO formulations, again primal and dual. In the last section, we will demonstrate why we need these conic formulations when we already have the LO formulations from Chapter 2. In particular, we will show that by adding new important constraints
to the basic problem, the conic formulations will prove to be very useful.