Our idea is that we first show the metric D is identical with the first Wasserstein metric on P(K) thanks to the

duality theorem of Kantorovich-Rubinstein [KR58] (see also Villani's book [V09, Particular Case 5.16]), and then use several definitions for weak contractions which are equivalent to Browder's definition.

Drawing inspiration from Brouwer's work and Poincare's results, in 1922 Alexander formulated his own

duality theorem, attaining to the contextual properties of mathematical objects through the analysis of their internal structure (Lautman, 2011).

Furthermore, if the weak

duality Theorem 20 holds for all feasible solutions of the problems (MP) and (MDI), then ([bar.x], [bar.[lambda]], [bar.[mu]]) is a weakly efficient solution of (MDI).

Terai Alexander

duality theorem and Stanley-Reisner rings Surikaisekikenkyusho Kokyu- ruko(1999) no.

Because [Q.sub.1] [??] 0, the convex set S has an interior point and objective function [[xi].sup.T] [Q.sub.1][xi] + [([q.sub.1] + x).sup.T] [xi] + [r.sub.1] in the left-hand side of (10) is bounded below on S, and from the conic

duality theorem, we know that (10) can be transformed into the following problem:

Both model (5) and its dual problem are feasible, so by the

duality theorem of linear programming they have bounded optimal solutions.

By the weak

duality theorem, ((u,v), [bar.[tau]], [bar.[lambda]], [bar.v]) is an efficient solution of (FD2).

Calderon's remarkable

duality theorem [3, Section 12.1 p.

Invoking the

Duality Theorem of Linear Programming [22], we arrive at

Chapters discuss duality, linear mappings, matrices, determinant and trace, spectral theory, Euclidean structure, calculus of vector- and matrix-valued functions, matrix inequalities, kinematics and dynamics, convexity, the

duality theorem, normed liner spaces, linear mappings between normed linear spaces, positive matrices, and solutions of systems of linear equations.

Keywords: Evolution equation, Gross Laplacian, potential function, white noise analysis, generalized functions, convolution operator, Laplace transform,

duality theorem.

Teo: A converse

duality theorem on higher-order dual models in nondifferentiable mathematical programming, Optim.