An unconstrained-like control-based dynamic method for optimization problems with simple bounds

TL;DR

This paper introduces an unconstrained-like dynamic control method for optimization problems with simple bounds, leveraging Lyapunov control principles and anti-windup integrators to improve efficiency and simplicity over existing methods.

Contribution

The paper proposes a novel dynamic method that avoids programming sub-problems and active constraint estimation, ensuring global convergence and higher efficiency for bounded optimization problems.

Findings

Proposed method guarantees global convergence without strict complementarity.

Achieves higher efficiency than existing dynamic and iterative methods.

Simplifies implementation by avoiding active constraint estimation.

Abstract

The optimization problems with simple bounds are an important class of problems. To facilitate the computation of such problems, an unconstrained-like dynamic method, motivated by the Lyapunov control principle, is proposed. This method employs the anti-windup limited integrator to address the bounds of parameters upon the dynamics for unconstrained problem, and then solves the transformed Initial-value Problems (IVPs) with mature Ordinary Differential Equation (ODE) integration methods. It is proved that when the gain matrix is diagonal, the result is equivalent to that of the general dynamic method which involves an intermediate Quadratic Programming (QP) sub-problem. Thus, the global convergence to the optimal solution is guaranteed without the requirement of the strict complementarity condition. Since the estimation of the right active constraints is avoided and no programming sub-problem is involved in the computation process, it shows higher efficiency than the general dynamic method and other common iterative methods through the numerical examples. In particular, the implementation is simple, and the proposed method is easy-to-use.