Global Optimization on an Interval

Thomas A Weber

doi:10.1007/s10957-016-1006-y

Abstract

This paper provides expressions for solutions of a one-dimensional global optimization problem using an adjoint variable which represents the available one-sided improvements up to the interval “horizon.” Interpreting the problem in terms of optimal stopping or optimal starting, the solution characterization yields two-point boundary problems as in dynamic optimization. Results also include a procedure for computing the adjoint variable, as well as necessary and sufficient global optimality conditions.

Highlights

The generic nonconcavity of maximization problems generally leads to multiple local optima
Standard optimality conditions tend to be local, and techniques for global optimization are usually algorithmic in nature, restricting the search for the best solution to subsets of the domain
For the simple case where the domain is an interval, a global maximizer of a continuously differentiable function can be found by using techniques from dynamic systems, notably by introducing global information in the form of an adjoint variable

Summary

Introduction

The generic nonconcavity of maximization problems generally leads to multiple local optima. For the simple case where the domain is an interval, a global maximizer of a continuously differentiable function can be found by using techniques from dynamic systems, notably by introducing global information in the form of an adjoint variable. In this manner, we construct expressions for solutions to a global. In addition to providing a full characterization of solutions to a global optimization problem over an interval, the adjoint variable can be used locally to formulate necessary and sufficient optimality conditions for one-sided subproblems of the original global optimization problem

Objectives

Results

Conclusion