Sentence examples for minimum norm problem from inspiring English sources

Fixed point theory plays a major role in many applications including variational and linear inequalities, optimization and applications in the field of approximation theory and minimum norm problem.

Then the sequence { x n } converges strongly to a point x ˜ in F ( T ) ∩ ( A + B ) − 1 0, which is the unique solution of the minimum norm problem (3.26).

Then the net { x t } defined by the implicit method (3.3) converges strongly, as t → 0, to x ˜, which solves the following minimum norm problem: find x ˜ ∈ F ( T ) ∩ ( A + B ) − 1 0 such that ∥ x ˜ ∥ = min x ∈ F ( T ) ∩ ( A + B ) − 1 0 ∥ x ∥. (3.26).

Also, as applications, some corollaries for solving the minimum-norm problems are also included.

It is easy to see that F is Lipschitz continuous with constant L = 1 and strongly monotone with constant β = 1 on ℋ. Problem (BVI) becomes the minimum-norm problems of the solution set of the variational inequalities.

However, they cover some classes of mathematical programs with equilibrium constraints (see [6]), bilevel minimization problems (see [7]), variational inequalities (see [8 13]), minimum-norm problems of the solution set of variational inequalities (see [14, 15]), bilevel convex programming models (see [16]) and bilevel linear programming in [17].

In a special case F ( x ) = x for all x ∈ C, problem (BVI) becomes the minimum-norm problems of the solution set of variational inequalities as follows: Find  x ∗ ∈ C  such that  x ∗ = Pr Sol ( G, C ) ( 0 ), where Pr Sol ( G, C ) ( 0 ) is the projection of 0 onto Sol ( G, C ).

Remark 3.3 A special form of the optimization problem is to take h ( x ) = ∥ x ∥, which is known as the minimum norm point problem.

The topic of this paper is the study of four real, linear, possibly constrained minimum norm approximation problems, which arise in connection with the design of linear-phase nonrecursive digital filters and are distinguished by the type of used trigonometric approximation functions.

This suggests an important question: can the two-layer iteration method (1.5) be modified to prove its strong convergence to the minimum norm solution of problem (1.12).

This paper analyzes the conditions under which the minimum norm solution to this problem is non-unique, provides a physical interpretation of the issue, proposes a simple method to normalize the minimum norm solution and make it unique, and extends these principles to unilaterally constrained multibody systems.