Reduction of weak exhausters and optimality conditions via reduced weak exhausters. (English) Zbl 1322.49029
Let \(h: \mathbb{R}^n\to\mathbb{R}\) be a positively homogeneous function with associated weak subdifferential \(\underline\partial^w h(0_n)\) and weak superdifferential \(\overline\partial^w h(0_n)\) according to
\[
\begin{aligned} \underline\partial^w h(0_n) &= \{(x^*,c)\in\mathbb{R}^n\times \mathbb{R}_+:-c\| x\|+\langle x^*,x\rangle\leq h(x),\;\forall x\in\mathbb{R}^n\},\\ \overline\partial^w h(0_n) &= \{(x^*,c)\in\mathbb{R}^n\times \mathbb{R}_+: c\| x\|+\langle x^*,x\rangle\geq h(x),\;\forall x\in\mathbb{R}^n\}.\end{aligned}
\]
A family \(E\) of nonempty compact and convex sets in \(\mathbb{R}^n\) is called
\(*\) a lower exhauster of \(h\) iff \(h(x)= \sup_{C\in E}\min_{u\in C}\langle u,x\rangle\), \(\forall x\in\mathbb{R}^n\),
\(*\) an upper exhauster of \(h\) iff \(h(x)= \text{inf}_{C\in E}\max_{v\in C}\langle v,x\rangle\), \(\forall x\in\mathbb{R}^n\).
It is pointed out that for lower-semicontinuous functions we have \(\underline\partial^w h(0_n)\neq\varnothing\) and that the family \[ E_w= \{\overline B(x^*,c): (x^*,c)\in\underline\partial^w h(0_n)\} \] forms a weak lower exhauster of \(h\). In the same manner, for upper-semicontinuous functions we have \(\overline\partial A^w h(0_n)\neq\varnothing\) and the family \[ E^w= \{\overline B(x^*, c): (x^*,c)\in \overline\partial^w h(0_n)\} \] forms a weak upper exhauster of \(h\).
Hence, the weak exhausters are constructed by infinitely many closed balls correspond to all weak subgradients and supergradients, respectively. The aim of the paper is the reduction of the weak exhausters, i.e. their description by smaller families of balls. So it is proved that a suitable reduction is possible if the weak subdifferential/superdifferential of \(h\) can be represented as the sum of a subset of it and the weak subdifferential \(\partial^w\mathbf{0}(0_n)\) of the zero function, and especially if this subset is the set of minimal elements of \(\underline\partial^w h(0_n)\) or \(\overline\partial^w h(0_n)\) with respect to the ordering cone \(\partial^w\mathbf{0}(0_n)\).
At the end of the paper, some optimality conditions are given using the introduced reduced weak exhausters.
\(*\) a lower exhauster of \(h\) iff \(h(x)= \sup_{C\in E}\min_{u\in C}\langle u,x\rangle\), \(\forall x\in\mathbb{R}^n\),
\(*\) an upper exhauster of \(h\) iff \(h(x)= \text{inf}_{C\in E}\max_{v\in C}\langle v,x\rangle\), \(\forall x\in\mathbb{R}^n\).
It is pointed out that for lower-semicontinuous functions we have \(\underline\partial^w h(0_n)\neq\varnothing\) and that the family \[ E_w= \{\overline B(x^*,c): (x^*,c)\in\underline\partial^w h(0_n)\} \] forms a weak lower exhauster of \(h\). In the same manner, for upper-semicontinuous functions we have \(\overline\partial A^w h(0_n)\neq\varnothing\) and the family \[ E^w= \{\overline B(x^*, c): (x^*,c)\in \overline\partial^w h(0_n)\} \] forms a weak upper exhauster of \(h\).
Hence, the weak exhausters are constructed by infinitely many closed balls correspond to all weak subgradients and supergradients, respectively. The aim of the paper is the reduction of the weak exhausters, i.e. their description by smaller families of balls. So it is proved that a suitable reduction is possible if the weak subdifferential/superdifferential of \(h\) can be represented as the sum of a subset of it and the weak subdifferential \(\partial^w\mathbf{0}(0_n)\) of the zero function, and especially if this subset is the set of minimal elements of \(\underline\partial^w h(0_n)\) or \(\overline\partial^w h(0_n)\) with respect to the ordering cone \(\partial^w\mathbf{0}(0_n)\).
At the end of the paper, some optimality conditions are given using the introduced reduced weak exhausters.
Reviewer: Jörg Thierfelder (Ilmenau)
MSC:
| 49J52 | Nonsmooth analysis |
| 49J53 | Set-valued and variational analysis |
| 49K10 | Optimality conditions for free problems in two or more independent variables |
| 90C30 | Nonlinear programming |
| 90C46 | Optimality conditions and duality in mathematical programming |
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