Geodesic
Optimality conditions and newton-type methods for mathematical programs with vanishing constraints
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 7, pp. 1184-1196

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A new class of optimization problems is discussed in which some constraints must hold in certain regions of the corresponding space rather than everywhere. In particular, the optimal design of topologies for mechanical structures can be reduced to problems of this kind. Problems in this class are difficult to analyze and solve numerically because their constraints are usually irregular. Some known first- and second-order necessary conditions for local optimality are refined for problems with vanishing constraints, and special Newton-type methods are developed for solving such problems. 
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     author = {A. F. Izmailov and A. L. Pogosyan},
     title = {Optimality conditions and newton-type methods for mathematical programs with vanishing constraints},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1184--1196},
     publisher = {mathdoc},
     volume = {49},
     number = {7},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_7_a5/}
}
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A. F. Izmailov; A. L. Pogosyan. Optimality conditions and newton-type methods for mathematical programs with vanishing constraints. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 7, pp. 1184-1196. http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_7_a5/