Continuity of Optimal Solution Functions and their Conditions on Objective Functions

Inverse maximum theorems and some consequences

“…The above corollary generalizes Theorem 3.1 in [33]. (i) for every (x, y) ∈ gra(K), the condition θ(x, z) > θ(x, y), for some z ∈ K(x), implies that there exists a neighborhood V (x,y) of (x, y) such that, for any (x ′ , y ′ ) ∈ V (x,y) ∩ gra(K), there exists…”

“…Terazono and Matani, 2015). Let us assume that K : R k ⇒ R ℓ is closed-valued, convex-valued, and continuous; θ : gra(K) → R is continuous and θ(x, •) is quasi-concave, for each x ∈ R k ; and m : X → (−∞, ∞] be defined as m(x) = sup y∈K(x) θ(x, y).…”

Berge's maximum theorem to vector-valued functions with some applications

“…The well-known Berge's maximum theorem is as follows: Many results about this theorem have been achieved in the literatures and the literatures therein (see [5,7,8,18,23,24,28]). In the early years, Dutta and Mitra [5] presented a maximum theorem for convex structures with weaker continuity requirements and applied to the problem of optimal intertemporal allocation.…”

“…Soon after, Feidenberg and Kasyanov [7] obtained the local Berge's maximum theorem for noncompact feasible sets and showed that it is more general than the recently established Berge's maximum theorem in other literature. Terazono and Matani [23] gave two theorems and show us two variants of the Berge's maximum theorem and the inverse of Berge's maximum theorem, where the condition of compact-valuedness of feasible solution function was replaced with other assumptions.…”

Optimal Force Allocation for Overconstrained Cable-Driven Parallel Robots: Continuously Differentiable Solutions With Assessment of Computational Efficiency

“…The results are, however, less novel from a mathematical perspective. Proofs and results that with some effort can be shown to cover Theorem 1 and Theorem 2 can be found in[31][32][33]…”