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.... v) r x (B(x) ffl Y ) u i r x c i (x) v j r x d j (x) 15) Note that in (14) the gradients of the vector valued functions c(x) and d(x) are defined as r x c(x) D x c(x) and r x d(x) D x d(x) If the problem (12) is convex and satisfies Slater s condition [13], then for each optimal solution x of (12) there exists an m Theta m matrix Y 0 and vectors u 2 IR , u 0, and v 2 IR such that the quadruple (x; Y; u; v) is a saddle point of the Lagrangian (13) L. More generally, for nonconvex problems (12) let x 2 IR be a feasible point of (12) ....
....Y 0 and vectors u 2 IR , u 0, and v 2 IR such that the quadruple (x; Y; u; v) is a saddle point of the Lagrangian (13) L. More generally, for nonconvex problems (12) let x 2 IR be a feasible point of (12) and assume that the Robinson or Mangasarian Fromovitz constraint qualification [13, 18, 19] is satisfied at x, i.e. the matrix D x d(x) has full rank and there exists a vector Deltax 6= 0 such that B(x) D x B(x) Deltax] OE 0, c(x) D x c(x) Deltax 0, and D x d(x) Deltax = 0. Then, if x is a local minimizer of (12) the first order optimality condition is satisfied, i.e. there ....
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Mangasarian, O.L. (1969): Nonlinear Programming. MvGraw-Hill, New York
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....developing a stopping rule for solving (3.14) or, equivalently, an approximation criterion for projection directions to be used in the PVD scheme. For algorithmic purposes, it is important to make this criterion constructive and implementable. Assuming some constraint quali cation condition [13], we have that z solves (3.14) i.e. z = P C [x rf(x) if, and only if, the pair ( z; u) 2 satis es the KKT system r z L( z; u) z x rf(x) c u = 0 ; c( z) 0 ; u 0 and h u; c( z)i = 0 ; 3.15) L(z; u) 1 hu; c(z)i (3.16) is the standard Lagrangian ....
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