# The MINLP Approach to Topology, Shape and Discrete Sizing Optimization of Trusses
## 4 Solution of the MINLP Truss Synthesis Problem
### Multi-Level MINLP Strategies  
Besides many non-linearities and non-convexities, a very high number of discrete variables, particularly those associated with discrete dimensions, may be included in the optimization. The simultaneous topology and standard dimension optimization can generally be performed in a single MINLP phase. With the Single-Phase MINLP approach [77], the optimal result is reached directly in a single optimization process together with the optimal topology and standard dimension. All the binary variables included in the optimization are initialized in a single full set. The disadvantage of this approach is that the initialization scheme is weak, since the chosen initial standard dimensions may be infeasible or a bad choice. This strategy also exhibits slow convergence when applied to large-scale MINLP problems, and the result obtained may be a poor local solution. In order to overcome these problems, we applied multi-level MINLP [60; 77; 78; 79; 80]. From a number of multi-level strategies, two were applied to truss synthesis:  
1. The Two-Phase (TP) MINLP approach [80], and
2. The Linked Two-Phase (LTP) MINLP strategy [60].  
The TP approach performs topology, shape and standard dimension optimization separately in two phases. In the first phase, the simultaneous topology, shape and continuous sizing optimization is performed with the dimensions of the cross-sections being temporarily relaxed into continuous parameters. When the optimal topology is obtained, the discrete dimensions of cross-sections are re-established, and the process continues with the second phase, where the shape and standard dimension optimization is performed until the optimal solution is obtained. In the second phase, the optimization is carried out at the fixed optimal topology obtained in the first phase. The TP approach is thus performed through a hierarchic decomposition of discrete binary variables into two sub-sets. The first sub-set is used to describe topology alternatives and the second one to describe discrete dimension alternatives.  
The main disadvantage of the TP strategy lies in the fact that the final solution is not necessarily the optimal one since the topology and standard dimensions are optimized consecutively. A simultaneous consideration of the standard dimensions may cause a change in the optimal topology [60]. In order to avoid the disadvantages of the TP approach, the LTP strategy was developed. While the first phase of the LTP strategy is identical to the first phase of the TP approach, the second phase differs in the fact that by using the LTP strategy, the topology is not fixed. The second phase of the LTP strategy thus performs simultaneous topology, shape and standard dimension optimization.  
As the majority of binary variables is attributed to discrete/standard dimensions, the design space in the previously described second phases is still very extensive, and in some cases, even when the TP strategy is used, a very high number of MINLP iterations is needed before the first feasible solution is attained. In order to additionally reduce the discrete space, after the first phase is complete (topology optimization with continuous sizing variables), a special pre-screening of the discrete dimensions is applied. With the use of the pre-screening procedure the discrete solution is sought only in some pre-defined neighborhood of the continuous solution obtained in the first phase.