An elementary proof of the Karush–Kuhn–Tucker theorem in normed linear spaces for problems with a finite number of inequality constraints

Abstract

We present an elementary proof of the Karush-Kuhn-Tucker theorem for the problem with a finite number of nonlinear inequality constraints in normed linear spaces under the linear independence constraint qualification. Most proofs in the literature rely on advanced concepts and results such as the convex separation theorem and Farkas, lemma. By contrast, the proofs given in this article, including a proof of the lemma, employ only basic results from linear algebra. The lemma derived in this article represents an independent theoretical result.

Infinite programming and theorems of the alternative

“…The theorem of the alternative given in Theorem allows us to characterize the solutions of the infinite program —under the infsup‐convexity of a certain family of functions associated with our infinite program as well as an adequate Slater's condition—in terms of some Karush–Kuhn–Tucker conditions. We extend the classical result of Karush–Kuhn–Tucker (see Karush and Kuhn and Tucker) and, for instance, Brezhnevaa and Tretyakov and its generalizations for weak convexity assumptions …”

An elementary proof of the Lagrange multiplier theorem in normed linear spaces

“…The proof for the finite-dimensional case is given in [6]. The article complements our recent article [5], where we gave an elementary proof of the Karush-Kuhn-Tucker theorem for optimization problems with inequality constraints in normed linear spaces.…”

“…Now, define matrix A m (x) by As follows from the proof of Lemma 1 in [5], all leading principal submatrices of A mþ1 (x à ), defined in (4), are invertible, and, therefore, matrix A m (x à ) is invertible. By the assumption that the set fg 0 i ðx Ã Þ j i ¼ 1, .…”

“…Let () be defined by (5) and be defined in (8). Choose 2 40 so that by Taylor's theorem, there exist a constant C 2 40 and function !ðt, Þ such that…”

Effect of Pressure and Temperature on Densification in Electric Field-Assisted Sintering of Inconel 718 Superalloy

“…In the early stage of sintering, the growth of nucleus and shear significantly affect the change of porosity Δ ϕ : where is the increase of porosity when crystal nuclei grow, Δ ϕ s = k w ϕn D ε is the increase of porosity accompanying the shear growth, ε N is the nucleus strain, S N is the standard deviation, k w is the material shear parameter, n D is the deflection tensor coaxial with the stress tensor, w is the stress tensor and ε is the plastic strain rate that depends on the porous plastic model. The relationship between ε and Σ is where , E is the modulus of elasticity, and λ is a non-negative multiplier, satisfying the Kuhn-Tuker condition [ 25 ]: …”