Optimality Conditions in Convex Optimization

On the solution of convex bilevel optimization problems

“…By Dhara and Dutta [39,Theorem 2.73], Klatte and Kummer [40] the convex function ϕ(·) is locally Lipschitz continuous. Using compactness of the set {x ∈ X : ∃y satisfying g(x, y) ≤ 0}, there exists a Lipschitz constant L < ∞ for ϕ(·) on X :…”

Generalized bipartite quantum state discrimination problems with sequential measurements

“…Since f ⋆ = −∞ implies that M • A is empty, for any ∈ M A , there exists j ∈ I J such that η j (Â) > 0. Therefore, the set W ′ ≡ {{β j ≤ 0} J−1 j=0 ∈ R J } has no intersecton with W. We can easily verify that W is compact and W ′ is closed; thus, by a separating hyperplane theorem (e.g., [59]), there exist q ≡ {q j } J−1 j=0 ∈ R J + and 0 < ǫ ∈ R + such that…”

On a weakly C-$\epsilon$-vector saddle point approach in weak vector problems

“…, m are also convex functions. By [5,Definition 2.109] of the ε-subdifferential, one has By using Lemma 2.1, there exists no x ∈ F such that f (x) − f (x) + 2ε ∈ −intC, which means thatx is a weakly C-2ε-solution of (VP).…”

“…Besides, according to [5,Chapter 10], there are many approaches to establish approximate optimality conditions for a scalar (and also possible for vector case by scalarization methods) optimization problem; for example, ε-subdifferential approach, max-function approach, ε-saddle point approach, exact penalization approach, and duality-based approach to ε-optimality.…”