# The MINLP Approach to Topology, Shape and Discrete Sizing Optimization of Trusses
## 4 Solution of the MINLP Truss Synthesis Problem
### The Proposed MINLP Strategies for Truss Synthesis
#### 4.4.1 The Modified Two-Phase (MTP) MINLP Strategy  
The Modified TP strategy (see Figure 3) starts (at \(n=1\)) with topology, shape and sizing optimization where cross-section dimensions are relaxed into continuous parameters (MINLP phase T). The process proceeds until the convergence of the OA/ER algorithm is achieved, i.e., when the lower bound, obtained at the MILP master problem, exceeds the upper bound yielded by the best NLP sub-problem. The obtained topology (with the theoretical value of the objective \(z_{n}^{\text{st}}\)) is adopted as the optimal topology of the firstphase. After that, the pre-screening of discrete dimensions for selected bars is applied. The calculation process proceeds with the second phase (MINLP phase D), where the shape and discrete sizing optimization is performed at the fixed optimal topology from the first phase. The obtained solution (with the discrete value of the objective \(z_{n}^{\ddagger}\)) now represents the structure with optimal topology, shape and discrete/standard values of cross-section dimensions (\(z_{\text{OPT}}=z_{n}^{\ddagger}\)). Up to this stage, the process coincides with the classical Two-Phase MINLP approach. However, by terminating the process at this point, one would not gain any information about the possible influence of the incorporation of discrete sizing variables on the subsequent topology optimization.  
The process is therefore returned back to the first phase with the topology optimization in which the discrete sizing variables are again relaxed into continuous parameters. At this stage, the accumulated linearizations of the previous discrete sizing optimization phase are also included in the optimization. As the >>integer cut<< equations prevent all topologies obtained in previous iterations from being re-calculated, former topologies are now excluded from the superstructure, which may cut off some good solutions. For this reason, all the integer cuts with the exception of the one resulting from the optimal topology are thus deactivated after the conclusion of the initial Two-Phase procedure.  
In the continuation (\(n=n+1\)), each time a topology is obtained, which would yield a theoretical objective value \(z_{n}^{\ddagger}\) better than the current optimal solution (\(z_{n}^{\ddagger}<z_{\text{OPT}}\)), a successive discrete sizing optimization at the fixed current \(n\)-th topology is performed. The process proceeds until a defined number (\(n_{\text{fin}}\)) of subsequent new topologies, which do  
Figure 3: The Modified Two-Phase (MTP) MINLP strategy.  
not fulfil the criterion \(z_{n}^{t}<z_{\text{OPT}}\), is reached. The number \(n_{\text{fin}}\) is defined on the basis of the estimation of the non-convexity problem, when the objective is not improved strictly monotonically, and a new better solution is expected to be gained after some worse intermediate solutions. However, a small number (i.e., from two to five) of MINLP iterations, which do not yield an improvement in the objective, is generally sufficient to terminate the optimization process.