TY - CHAP

T1 - Alternance Form of Optimality Conditions in the Finite-Dimensional Space

AU - Demyanov, V.F.

AU - Malozemov, V.N.

PY - 2014

Y1 - 2014

N2 - In solving optimization problems, necessary and sufficient optimality conditions play an outstanding role. They allow, first of all, to check whether a point under study satisfies the conditions, and, secondly, if it does not, to find a “better” point. This is why such conditions should be “constructive” letting to solve the above-mentioned problems. For the class of directionally differentiable functions in, a necessary condition for an unconstrained minimum requires for the directional derivative to be non-negative in all directions. This condition becomes efficient for special classes of directionally differentiable functions. For example, in the case of convex and max-type functions, the necessary condition for a minimum takes the form where is a convex compact set. The problem of verifying this condition is reduced to that of finding the point of C which is the nearest to the origin. If the origin does not belong to C, we easily find the steepest descent direction and are able to construct a numerical me

AB - In solving optimization problems, necessary and sufficient optimality conditions play an outstanding role. They allow, first of all, to check whether a point under study satisfies the conditions, and, secondly, if it does not, to find a “better” point. This is why such conditions should be “constructive” letting to solve the above-mentioned problems. For the class of directionally differentiable functions in, a necessary condition for an unconstrained minimum requires for the directional derivative to be non-negative in all directions. This condition becomes efficient for special classes of directionally differentiable functions. For example, in the case of convex and max-type functions, the necessary condition for a minimum takes the form where is a convex compact set. The problem of verifying this condition is reduced to that of finding the point of C which is the nearest to the origin. If the origin does not belong to C, we easily find the steepest descent direction and are able to construct a numerical me

KW - Necessary optimality conditions

KW - Alternance form

KW - Directionally differentiable functions

U2 - 10.1007/978-1-4614-8615-2_12

DO - 10.1007/978-1-4614-8615-2_12

M3 - Chapter

SN - 978-1-4614-8615-2

SP - 265, 185-203

BT - Springer Optimization and Its Applications. Vol.87: Constructive Nonsmooth Analysis and Related Topics /Editors: Demyanov, Vladimir F., Pardalos, Panos M., Batsyn, Mikhail V.

PB - Springer Nature

ER -