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K(x, yo) 0; -

y0v

g(

0))x0

-

0;

• Yo(g(xo)- b) - 0; • g(xo) _ 0. Therefore and subject to the assumptions made, the Karush-Kuhn-Tucker conditions provide necessary and sufficient conditions for an optimal solution of the mathematical programming problem (5). This result has had a tremendous impact on the development of algorithms to solve mathematical programming problems. Moreover, if the perturbation function ¢(d) is also differentiable at b then it is approximated by the right-hand side of (9). This happens in many applications. Thus with numerically small deviations d - b the vector y0 measures the marginal effects on the optimal value of the objective function of (5). Due to this property Y0 is often denoted as the vector of shadow prices and as such it plays a central role in the discussion and interpretation of results obtained by mathematical programs, in particular when modeled over problems in economics.

See also" I n e q u a l i t y - c o n s t r a i n e d n o n l i n e a r o p t i m i z a t i o n ; E q u a l i t y - c o n s t r a i n e d nonlinear p r o g r a m m i n g : K K T n e c e s s a r y optim a l i t y conditions; S e c o n d o r d e r o p t i m a l i t y c o n d i t i o n s for n o n l i n e a r o p t i m i z a t i o n ; Lag r a n g i a n d u a l i t y : Basics; F i r s t o r d e r cons t r a i n t qualifications; S e c o n d o r d e r cons t r a i n t qualifications; K u h n - T u c k e r optim a l i t y conditions; R o s e n ' s m e t h o d , global convergence, and Powell's conjecture. References [1] KUHN, H.W., AND TUCKER, A.W.: 'Nonlinear programming': Proc. Second Berkeley Symp. Math. Statist. Probab., Univ. Calif. Press, 1951, pp. 481-490. [2] NEUMANN, J. VON, AND MORGENSTERN, O.: Theory of games and economic behaviour, Princeton Univ. Press, 1947. [3] ROCKAFELLAR, R.T." Convex analysis, Princeton Univ. Press, 1970.

Jcrgen Tind Univ. Copenhagen Denmark E-mail address" tind~math.ku.dk MSC 2000:90C06 Key words and phrases: optimization, saddle point.

SECOND ORDER CONSTRAINT QUALIFICATIONS A second order constraint qualification (SOCQ) is a condition which is imposed on the analytical description of the constraint set of an optimization problem and which usually involves second order approximations of the data functions. second order constraint qualifications are essential in order to establish second order optimality conditions, but they play also a certain role in the perturbation analysis for optimization problems. Roughly speaking, second order constraint qualifications establish a link between the geometry of the given set and certain kinds of second order approximations of the analytical data. Second order constraint qualifications are closely related to first order constraint qualifications (cf. F i r s t o r d e r c o n s t r a i n t qualifications), in particular, see that article for notions such as LICQ, MFCQ, Robinson CQ and others. Historically, SOCQs were first introduced in studies of second order optimality conditions for 69

Second order constraint qualifications sm