First-order regularized algorithms with complexity properties for order-valued and low-order-valued optimization problems - Alvarez, Gustavo David Quintero (2024)
The minimization of the order-value function is part of a large family of problems involving functions whose value is calculated by sorting values from a set or subset of other functions. The order-value function has as particular cases the minimum and maximum functions of a set of functions and is well suited for applications involving robust estimation. In this work, a first order method with quadratic regularization to solve the problem of minimizing the order-value function is proposed. An optimality condition for the problem and theoretical results of iteration complexity and evaluation complexity for the proposed method are presented. The applicability of the problem and method to parameter estimation problems with outliers is illustrated. We also consider both the unconstrained minimization of the low order-value function and the constrained case, where the feasible region is a closed convex set, assuming that projections onto this set are computationally affordable. For both cases, we introduce regularized first-order algorithms, proving worst-case iteration and evaluation complexity bounds, as well as asymptotic convergence results. For the constrained case, the proposed algorithm generalizes the classical projected gradient method. Numerical implementations and several examples demonstrate the practical applicability and effectiveness of the proposed methods.