Minimalizova

Minimalizova is a theoretical framework in optimization describing a family of iterative methods for minimizing an objective under constraints by alternating descent steps with localized minimization subproblems. It aims to construct trial points via a small auxiliary problem that yields better progress than a plain gradient step while maintaining feasibility.

Origin and terminology: The name blends minimize with the Slavic suffix -ova, and it appears in optimization literature as a label for constrained descent schemes. The concept emerged in the early 2000s and has since evolved into several variants.

Definition: Let f: R^n → R be differentiable with Lipschitz continuous gradient, and C ⊆ R^n a closed convex feasible set. A minimalizova step at xk ∈ C seeks d with xk + d ∈ C that minimizes a local model m_k(d) = ∇f(xk)^T d + (1/2) d^T B_k d, where B_k ≽ 0. The next iterate is xk+1 = xk + α_k d with α_k ∈ (0,1] chosen by line search to ensure decrease of f.

Variants and properties: Proximal minimalizova adds a proximal term; accelerated variants use momentum. Convergence typically requires convex f, convex C, and Lipschitz gradient, with standard step-size conditions.

Applications: Used in constrained machine learning, engineering design, resource allocation, and economics, especially when the feasible set encodes simple constraints such as non-negativity or linear equalities.

See also: gradient projection method; projected gradient; proximal gradient method; augmented Lagrangian methods.