Unconstrained Optimization Problems. Set-Constrained Optimization Problems. Classical Constrained Optimization Problems

Giorgio Giorgi (), Bienvenido Jiménez () and Vicente Novo ()
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Giorgio Giorgi: University of Pavia
Bienvenido Jiménez: National University of Distance Education
Vicente Novo: National University of Distance Education

Chapter Chapter 4 in Basic Mathematical Programming Theory, 2023, pp 83-122 from Springer

Abstract: Abstract In this section we shall treat problem $$(P_{1}),$$ ( P 1 ) , i.e. $$\begin{aligned} (P_{1}):{ \ \ \ \ }\min f(x),{ \ \ subject to }x\in S\subset {\mathbb {R}}^n, \end{aligned}$$ ( P 1 ) : min f ( x ) , s u b j e c t t o x ∈ S ⊂ R n , where $$f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}$$ f : R n → R and S is an open set (for example, $$S={\mathbb {R}}^n$$ S = R n ) or, more generally, where for the optimal point $$x^0$$ x 0 it holds $$x^0\in \mathrm{{int}}(S).$$ x 0 ∈ int ( S ) . In other words, we assume that the optimal points of $$(P_{1})$$ ( P 1 ) are interior to S. A first basic result is given by the following necessary optimality conditions.

Date: 2023
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DOI: 10.1007/978-3-031-30324-1_4

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