Though there is a subtle difference in the way that a criterion function and an objective function is defined,
in a broad sense we treat them here as identical. ...
One of the striking difference between single-objective and multi-objective optimization is that in multi-objective optimization the objective functions constitute a multi-dimensional space, in addition to the usual decision variable space. This additional space is called the objective space, Z.
For each solution x in the decision variable space, there exists a point in the objective space, denoted by f(x) = z = (z1, z2, ..., zm)T
The mapping takes place between an n-dimensional solution vector and an M-dimensional objective vector.
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Multi-objective optimization is sometimes referred to as vector optimization, because a vector of objectives, instead of a single objective, is optimized.
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Linear and Nonlinear MOOP
- multi-objective linear program (MOLP): all objective functions and constraint functions are linear
- nonlinear problems: the solution techniques often do not have convergence proofs. Since most real-world multi-objective optimization problems are nonlinear in nature, we do not assume any particular structure of the objective and constraint functions here.
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Convex and nonconvex MOOP
- for a convex function, a local minimum is always a global minimum
- ...???
The convexity of an MOOP is an important matter, which we shall see in subsequent chapters. There exist many algorithms which can handle convex MOOPs well, but face difficulty in solving nonconvex MOOPs. Since an MOOP has two spaces, the convexity in each space (objective and decision variable space) is important to a multi-objective optimization algorithm. Moreover, although the search space can be nonconvex, the Pareto-optimal front may be convex.
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