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Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives. (English) Zbl 0698.90070

The paper introduces second-order optimality conditions for finite- dimensional smooth and nonsmooth nonlinear programming problems. The classical results are generalized by replacing graphs of functions by epigraphs and graphical convergence by epigraphical convergence.
The concept of epiconvergence is simple and useful. A family of functions \(\psi_ t\) on \({\mathbb{R}}^ n\), defined for \(t>0\), is said to epiconverge to a function \(\psi\) if the epigraph sets epi \(\psi_ t\) converge as \(t\to 0\) to epi \(\psi\). The notions of epi-differentiability and twice epi-differentiability follow directly from the definition of epiconvergence applied to the well known first and second-order difference quotients.
The above mentioned optimality conditions are then applied to nonlinear programming problems represented in the general form “Minimize \(\{\) g(F(x)): F(x)\(\in D\}\)”, where g, F and D depend on the specific problem to be considered (represented by penalties or augmented Lagrangian functions).
Reviewer: J.C.Geromel

MSC:

90C30 Nonlinear programming
49J50 Fréchet and Gateaux differentiability in optimization
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