Calculus of variations by Miersemann E.

By Miersemann E.

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4) that L (x, λ)w, w ≤ 0. 5) contradict the assumption of the theorem. Since the proof shows that T0 = {0} if x is no strict local minimizer, it follows that x defines a strict local minimum if T0 = {0}. ✷ Remark. The above proof is mainly based on the observation that the sequence t2k G(xk , λ) remains bounded from above. In the general case of a set V which defines the side condition we have that the sequence t2k f (x), xk −x remains bounded from above. In the infinitely dimensional case we must exploit this fact much more then in the above finitely dimensional case where it was enough to use the conclusion that f (x), w = 0.

Definition. A subset V ⊆ X is said to be convex if for any two vectors x, y ∈ V the inclusion λx + (1 − λ)y ∈ V holds for all 0 ≤ λ ≤ 1. Definition. We say that a functional f defined on a convex subset V ⊆ X is convex if f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) for all x, y ∈ V and for all 0 ≤ λ ≤ 1, and f is strictly convex if the strict inequality holds for all x, y ∈ V , x = y, and for all λ, 0 < λ < 1. 1. If f is a convex functional on a convex set V ⊆ X, then any local minimum of f in V is a global minimum of f in V .

Then there exists λj ∈ R, λj ≥ 0 if j ∈ I0 , such that f (x) + λj gj (x) = 0. j∈E∪I0 Proof. 3) with K = {z ∈ Rn : gj (x), z ≥ 0 and −gj (x), z ≥ 0 for each j ∈ E, and −gj (x), z ≥ 0 for each j ∈ I0 } (1) is satisfied, there exists nonnegative real numbers µj if j ∈ I0 , µj and (2) µj if j ∈ E such that f (x) = j∈I0 = − µj −gj (x) + (1) j∈E j∈E µj j∈E (1) µj gj (x) + j∈I0 (2) µj gj (x) + (2) µj − µj gj (x). 2. 6: let A be a real matrix with m rows and n columns and let b ∈ Rn , then b, y ≥ 0 ∀y ∈ Rn with Ay ≥ 0 if and only if there exists an x ∈ Rm , such that x ≥ 0 and AT x = b.

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