Introduction to Unconstrained Optimization - Scilab
Introduction to Unconstrained Optimization - Scilab
Introduction to Unconstrained Optimization - Scilab
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f(x)<br />
strong local<br />
optimum<br />
weak local<br />
optimum<br />
strong global<br />
optimum<br />
x<br />
Figure 5: Different types of optimum – strong local, weak local and strong global.<br />
These types of optimum are presented in figure 5.<br />
Definition 1.1. (Strong local minimum) The point x ⋆ is a strong local minimum<br />
of the constrained optimization problem 2–4 if there exists δ > 0 such that the two<br />
following conditions are satisfied :<br />
• f(x) is defined on N(x ⋆ , δ), and<br />
• f(x ⋆ ) < f(y), ∀y ∈ N(x ⋆ , δ), y ≠ x ⋆ .<br />
Definition 1.2. (Weak local minimum) The point x ⋆ is a weak local minimum of<br />
the constrained optimization problem 2–4 if there exists δ > 0 such that the three<br />
following conditions are satisfied<br />
• f(x) is defined on N(x ⋆ , δ),<br />
• f(x ⋆ ) ≤ f(y), ∀y ∈ N(x ⋆ , δ),<br />
• x ⋆ is not a strong local minimum.<br />
Definition 1.3. (Strong global minimum) The point x ⋆ is a strong global minimum<br />
of the constrained optimization problem 2–4 if there exists δ > 0 such that the two<br />
following conditions are satisfied :<br />
• f(x) is defined on the set of feasible points,<br />
• f(x ⋆ ) < f(y), for all y ∈ R n feasible and y ≠ x ⋆ .<br />
Most algorithms presented in this document are searching for a strong local<br />
minimum. The global minimum may be found in particular situations, for example<br />
when the cost function is convex. The difference between weak and strong local<br />
minimum is also of very little practical use, since it is difficult <strong>to</strong> determine what<br />
are the values of the function for all points except the computed point x ⋆ .<br />
In practical situations, the previous definitions does not allow <strong>to</strong> get some insight<br />
about a specific point x ⋆ . This is why we will derive later in this document first<br />
order and second order necessary conditions, which are computable characteristics<br />
of the optimum.<br />
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