By the term optimization we mean the matemathical (or numerical) problem of minimizaing or maximizing a function in a domain. The function depends on a number of parameters (design variables). Frequently, an optimization problem is characterized from the needing of optimizing more than one function at the same time. In this case, we define the problem as a multi-objective optimization. When the design is subjected to some limitation (constraints) on parameters we talk about a constrained optimization.

When talking about Structural optimization we mean the application of optimization concepts to structural design. In this case, the objective function is usually one or more output of a finite element calculation (FEM) obtained through a numerical structural analysis whose parameters characterize the design of the structural component.

For structural optimizations, constraints are often consisting in the respect of the structural integrity considering some kind of failure criterion. In most cases the stress state inside the structure represents the constraint set of a structural optimization.

As we mentioned, a designer frequently faces the problem of looking for a best compromise solution between more than one contraddictory objectives. If any of objectives are competing, there is no unique solution to this problem. Some optimility criteria has to be used in this case (i.e. Pareto optimality criterion) to identify the unique set of parameters giving the best compromise solution among the objectives.
In multi-objective optimization so, we want to optimize, at the same time, more than one objective function. Multiobjective optimization concerns with the optimization of a vector of objectives subject to a number of constraints.

Topological optimization solves the problem of "filling" a volume of a body with only the strictly necessary material to sustain loads the body itself is subjected to.
In topological optimization the body is discretized in a number of elements, as in FEM analyses, and design variables are the amount of material inside each element. Material is then added only where it is necessary to carry structural loads.

Combinatorial optimization aims to the determination of an optimal sequence of objects. The problem is discrete because we want to determine an optimal (ordered) sequence to maximize or minimize a function depending on the order of these objects.
A typical example of combinatorial optimization is represented by the so-called Traveling salesman problem. In this problem, a salesman has to visit 'n' cities but wants to minimize total distance.

A large number of techniques can be used to solve optimization problems: the optimization engineer has to choose the most appropriate tool depending on problem's features. Among the most widely used techniques we may cite:

gradient-based methods are based on the calculation of the gradient of the objective function; genetic algorithms are based on the concept that a population of individuals (candidate design solutions) evolves improving some objective, imitating the evolution of species in Nature; particle swarm optimization imitates the behaviour of swarms in Nature; ant colony optimization simulates the collective social behaviour of ants looking for food; and many others.

An opportune choice of the solution technique is crucial to determine an optimal solution by containing the solution cost. The solution cost is the number of function evaluations needed to obtain the optimal solution.