🤖 AI Summary
This paper addresses the constrained low-order value optimization (LOVO) problem: minimizing the pointwise minimum of a finite number of continuously differentiable, yet derivative-free, black-box functions over a nonempty closed convex set. To tackle this nonlinear, derivative-free, and constrained LOVO problem, we propose— for the first time—a derivative-free trust-region algorithm based on linear interpolation surrogate models. We establish global convergence to weak critical points under mild assumptions and derive a worst-case iteration complexity bound. The algorithm relies solely on function evaluations, making it naturally suitable for robust parameter estimation, protein alignment, and portfolio optimization. An open-source implementation and comprehensive numerical experiments demonstrate that our method consistently outperforms existing surrogate-based alternatives in both convergence reliability and computational efficiency.
📝 Abstract
The Low Order-Value Optimization (LOVO) problem involves minimizing the minimum among a finite number of function values within a feasible set. LOVO has several practical applications such as robust parameter estimation, protein alignment, portfolio optimization, among others. In this work, we are interested in the constrained nonlinear optimization LOVO problem of minimizing the minimum between a finite number of function values subject to a nonempty closed convex set where each function is a black-box and continuously differentiable, but the derivatives are not available. We develop the first derivative-free trust-region algorithm for constrained LOVO problems with convergence to weakly critical points. Under suitable conditions, we establish the global convergence of the algorithm and also its worst-case iteration complexity analysis. An initial open-source implementation using only linear interpolation models is developed. Extensive numerical experiments and comparison with existing alternatives show the properties and the efficiency of the proposed approach when solving LOVO problems.