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4 Elements of optimization theory In this short chapter we consider the NLP minimize f (x) (4.1a) x∈R s. t. g (x)= 0, i ∈ E , (4.1b) g (x) ≥ 0, i ∈ I , (4.1c) n n m where f : R → R and g : R → R are twice continuously differentiable functions and the sets E and I form a partition of {1,..., m} =: m = E ∪ I . In the case of E = m, I = {}, NLP (4.1) is called Equality Constrained Optimization Problem (ECOP). 4.1 Basic deﬁnitions We follow Nocedal and Wright [121] in the presentation of the following basic deﬁnitions. Deﬁnition 4.1. The set F = {x ∈ R | g (x)= 0, i ∈ E , g (x) ≥ 0, i ∈ I } i i is called feasible set. Deﬁnition 4.2. A point x ∈ F is called feasible point. ∗ n ∗ Deﬁnition 4.3. A point x ∈ R is called global solution if x ∈ F and f (x ) ≤ f (x) for all x ∈ F . ∗ n ∗ Deﬁnition 4.4. A point x ∈ R is called local solution
Published: Nov 30, 2013
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