# The MINLP Approach to Topology, Shape and Discrete Sizing Optimization of Trusses
## 2 Optimization of Trusses
### Topology Optimization  
The first examples of truss optimization included only sizing optimization, i.e., the calculation of optimal cross-section areas of bars. Soon after that, it was discovered that it would be reasonable to exclude some unfavorable or needless elements from the original truss structure. As a result, not only the optimal cross-sections would be obtained but also the optimal number and configuration of bars, namely the optimal topology. One of the most important contributions to the area of truss topology optimization is the so-called ground structure approach [5], where a grid of nodes and a set of inter-connections between nodes, representing bars, are defined. In the process of optimization, some bars are eliminated, and the optimal topology is represented by the sub-set of existing bars. Since trusses are analyzed using joint equilibrium equations, the set of feasible solutions includes only statically determinate structures. Further, only stress constraints are applied, and consequently, the problem can be formulated as a linear programming (LP) problem.  
By including the deflection constraints and statically indeterminate solutions, the discussed truss problem becomes a non-linear one. Numerous topology optimization problems of trusses, solved by the non-linear programming (NLP) approach, were therefore subsequently presented [6; 7; 8; 9; 10; 11]. Using continuous optimization methods, i.e., the LP or the NLP, the truss topology optimizations were performed by either allowing zero values or by defining a small lower bound of the cross-section areas of bars. While the former approach caused problems to be potentially insoluble due to the possible appearance of singularities, the latter generally lead to non-exact solutions, which require reanalysis. Moreover, allowing zero or non-zero values led to non-convexities, which usually cause the optimization to become stuck in poor local optima.  
All the mentioned difficulties led to the idea to optimize the truss topology by different discrete optimization methods. The topology is optimized inside a discrete space by introducing binary 0-1 variables. An extra binary variable \(y\) is associated with each truss element, indicating whether this particular element is included in (\(y=1\)) or excludedfrom (\(y=0\)) the current topology. By means of discrete optimization, numerous authors optimize the truss topology applying the genetic algorithm (GA) [12; 13; 14; 15; 16], the penalty function method [10] and the simulated annealing [17; 18; 19].  
The nature of the considered truss topology problem is that the optimization models are mixed discrete/continuous, with discrete variables for topology and continuous variables for cross-section dimensions, nodal coordinates, stresses, strains, etc. As the NLP is regarded as the most effective for solving continuous non-linear optimization problems, some attempts have been made to link the NLP with different discrete optimization techniques. Such examples are the linkage of GA and NLP [20] and the tabu search [21].  
Besides the widely used ground structure approach, a second approach to topology optimization of trusses is the homogenization method [22; 23; 24; 25], where the optimal material distribution in a continuous design domain is determined and interpreted as a discrete truss.