Modeling Approaches for Addressing Simple Unrelaxable Constraints with Unconstrained Optimization Methods

TL;DR

This paper introduces a novel domain warping approach to transform nonlinear optimization problems with unrelaxable bound constraints into unconstrained problems, enabling the use of existing unconstrained optimization methods.

Contribution

It proposes a new reformulation technique using domain warping, analyzes its properties, and develops an algorithm that guarantees convergence to a stationary point.

Findings

Theoretical analysis of multioutput sigmoidal warping.

Development of an algorithm with convergence guarantees.

Practical insights into applying unconstrained methods to bound-constrained problems.

Abstract

We explore novel approaches for solving nonlinear optimization problems with unrelaxable bound constraints, which must be satisfied before the objective function can be evaluated. Our method reformulates the unrelaxable bound-constrained problem as an unconstrained optimization problem that is amenable to existing unconstrained optimization methods. The reformulation relies on a domain warping to form a merit function; the choice of the warping determines the level of exactness with which the unconstrained problem can be used to find solutions to the bound-constrained problem, as well as key properties of the unconstrained formulation such as smoothness. We develop theory when the domain warping is a multioutput sigmoidal warping, and we explore the practical elements of applying unconstrained optimization methods to the formulation. We develop an algorithm that exploits the structure of the sigmoidal warping to guarantee that unconstrained optimization algorithms applied to the merit function will find a stationary point to the desired tolerance.