# Accurate Reanalysis of Structures by A Preconditioned by Kirsch U., Kocvara M., Zowe J.

By Kirsch U., Kocvara M., Zowe J.

By Kirsch U., Kocvara M., Zowe J.

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M+r ) ∈ IR m+r λi ≥ 0 , λi ϕi (¯ x ) = 0 for i = 1, . . , m with si (¯ x ) = ϕi (¯ x ), i = 1, . . , m, and that F(x) = s(x), 0) + (0, ϕm+1 (x), . . 31) for the above F, where s := (s1 , . . 11(iii) with tiable around x¯. Thus the condition y ∗ ∈ N (¯ y ∗ = (λ1 , . . 25) as i = 1, . . , m. 31) are equivalent to the SNC property of f = (ϕm+1 , . . 70. 11(iii) holds if and only if either Ω or f is SNC at x¯. 22) with x1∗ ∈ D ∗ F(¯ is equivalent to the conditions m λi ∇si (¯ x ) + x ∗ + x ∗, 0= (λ0 , .

49) holds as equality if X is ﬁnite-dimensional. 49). 26 (mixed subdiﬀerential conditions for local minima). 23), where the space X is Asplund, where the set Ω is locally closed around x¯, and where all the functions ϕi are locally Lipschitzian around this point. Assume also that there exists a 36 5 Constrained Optimization and Equilibria convex closed subcone M of T (¯ x ; Ω) with M ∗ ⊂ N (¯ x ; Ω) and that the funcx ) ∪ {0}, admit upper convex approximations at x¯, which are tions ϕi , i ∈ I (¯ continuous at some point of M.

The next optimization result we are going to obtain in the form of the Lagrange principle, which says that necessary optimality conditions in constrained problems can be given as necessary conditions for unconstrained local minima of some Lagrange functions (Lagrangian) built upon the original constraints with suitable multipliers. 23) we consider the standard Lagrangian L(x, λ0 , . . , λm+r ) := λ0 ϕ0 (x) + . . 35) involving the cost function and the functional (but not geometric) constraints, and also the essential Lagrangian L Ω (x; λ0 , .