w1EOOnorm

The w1EOOnorm is a hypothetical norm used in optimization discussions to illustrate how a weighted L1 penalty can be combined with a Euclidean penalty to balance sparsity and stability. While not a standard in formal texts, it serves as a concrete example of how composite regularizers may be constructed for convex optimization problems.

Definition: Let x be a vector in R^n, let w = (w1, ..., wn) with each w_i ≥ 0, and let λ ≥ 0. The w1EOOnorm of x is defined as

w1EOOnorm(x) = ∑_{i=1}^n w_i |x_i| + λ ||x||_2,

where ||x||_2 is the Euclidean norm. The first term is a weighted L1 norm and the second term is the L2 norm scaled by λ. Since both terms are norms and the sum of norms is again a norm, w1EOOnorm is a norm on R^n when w_i ≥ 0 and λ ≥ 0.

Properties: As a norm, w1EOOnorm is convex, positively homogeneous, and satisfies the triangle inequality. The weighted L1 component encourages sparsity while the L2 component provides stability and discourages large overall magnitudes. The choice of weights w_i allows anisotropic penalization across coordinates.

Computation and optimization: In regularized optimization, w1EOOnorm can be used as a regularizer in objective functions. Its proximal operator does not generally have a closed form and is typically handled via splitting methods such as ADMM or proximal gradient algorithms. Efficient implementation relies on standard convex optimization toolchains.

Variants and applications: Variants include adjusting the weights w_i or replacing the L2 term with other norms, yielding a family of regularizers that trade sparsity against small-norm behavior. Potential applications include sparse regression, feature selection, and compressed sensing where balanced sparsity and stability are desirable. See also: L1 norm, L2 norm, elastic net, proximal methods.