Abstract
This article is devoted to the Error Bound property (also named R-regularity) in mathematical programming problems. This property plays an important role in analyzing the convergence of numerical optimization algorithms, a topic covered by multiple publications, and at the same time it is a relatively generic constraint qualification that guarantees the satisfaction of the necessary Kuhn – Tucker optimality conditions in mathematical programming problems. In the article, new sufficient conditions for the error bound property are described, and it’s also shown that several known necessary conditions are insufficient. The sufficient conditions obtained can be used to prove the regularity of a large class of sets including sets that cannot be proven regular by other known constraints.
Highlights
This article is devoted to the Error Bound property in mathematical programming problems
This property plays an important role in analyzing the convergence of numerical optimization algorithms, a topic covered by multiple publications, and at the same time it is a relatively generic constraint qualification that guarantees the satisfaction of the necessary Kuhn – Tucker optimality conditions in mathematical programming problems
The sufficient conditions obtained can be used to prove the regularity of a large class of sets including sets that cannot be proven regular by other known constraints
Summary
{ } ΓC ( y=) y ∈ R m | 〈∇hi ( y), y〉 ≤ 0, i ∈ I ( y), 〈∇hi ( y), y=〉 0, i ∈ I 0 , соответственно касательный конус, касательный конус Кларка [22] и линеаризованный касательный конус к множеству C в точке y ∈C. Что в точке y 0 ∈C выполнено ослабленное условие регулярности Ман гасаряна – Фромовица (RMFCQ), если система векторов { hi ( y), i I 0 I a ( y)} имеет постоянный ранг в некоторой окрестности этой точки. Что наиболее общие условия регулярности формулируются непосредственно в терминах касательных конусов к множеству допустимых точек. Что в точке y ∈C выполнено условие регулярности Абади (ACQ), если TC ( y) = ΓC ( y). Условие регулярности Кларка [24] выполняется в y ∈C, если TC ( y) = TC ( y). В общем случае выполнение условий регулярности Абади и Кларка для множества не влечет его R-регулярность
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