The Natural Gas Cash-Out Problem: A Bilevel Optimal Control Approach

Vyacheslav V. Kalashnikov,Patrick Mehlitz,Francisco Benita

doi:10.1155/2015/286083

Abstract

The aim of this paper is threefold: first, it formulates the natural gas cash-out problem as a bilevel optimal control problem (BOCP); second, it provides interesting theoretical results about Pontryagin-type optimality conditions for a general BOCP where the upper level boasts a Mayer-type cost function and pure state constraints, while the lower level is a finite-dimensional mixed-integer programming problem with exactly one binary variable; and third, it applies these theoretical results in order to find possible local minimizers of the natural gas cash-out problem.

Highlights

Bilevel programs are hierarchical optimization problems in the sense that their constraints are defined in part by a second parametric optimization problem
The aim of this paper is threefold: first, it formulates the natural gas cash-out problem as a bilevel optimal control problem (BOCP); second, it provides interesting theoretical results about Pontryagin-type optimality conditions for a general BOCP where the upper level boasts a Mayer-type cost function and pure state constraints, while the lower level is a finite-dimensional mixed-integer programming problem with exactly one binary variable; and third, it applies these theoretical results in order to find possible local minimizers of the natural gas cash-out problem
We formulate the gas cash-out problem as a bilevel optimal control problem (BOCP) where the upper level is equipped with a Mayer-type cost function and pure state constraints, whereas the lower level is formulated as a finite-dimensional mixed-integer programming problem (MIP) with only one binary variable

Summary

Introduction

Bilevel programs are hierarchical optimization problems in the sense that their constraints are defined in part by a second parametric optimization problem. Hierarchical structures can be found in diverse scientific disciplines including environmental studies, classification theory, databases, network design, transportation, game theory, economics, and new applications, such as the gas cash-out problem. In its turn, this stimulates the development of both new theoretical results and efficient algorithms to solve bilevel programming problems. A particular case of the bilevel programming problem is presented by the following optimal control problem arising in the natural gas industry This problem has been formulated as a mixed-integer model [1–3] arising from the minimization of cash-out penalty costs of a natural gas shipping company.

The Natural Gas Cash-Out Problem

General Setting and Optimality Conditions

Optimality Conditions of the Natural Gas Cash-Out Problem

Concluding Remarks