New Algorithm to Solve Mixed Integer Quadratically Constrained Quadratic Programming Problems Using Piecewise Linear Approximation

Loay Alkhalifa,Hans Mittelmann

doi:10.3390/math10020198

Loay Alkhalifa, Hans Mittelmann

Open Access

https://doi.org/10.3390/math10020198

Copy DOI

Journal: Mathematics	Publication Date: Jan 9, 2022
Citations: 4	License type: CC BY 4.0

Affiliation: Qassim University, Arizona State University

Abstract

Techniques and methods of linear optimization underwent a significant improvement in the 20th century which led to the development of reliable mixed integer linear programming (MILP) solvers. It would be useful if these solvers could handle mixed integer nonlinear programming (MINLP) problems. Piecewise linear approximation (PLA) is one of most popular methods used to transform nonlinear problems into linear ones. This paper will introduce PLA with brief a background and literature review, followed by describing our contribution before presenting the results of computational experiments and our findings. The goals of this paper are (a) improving PLA models by using nonuniform domain partitioning, and (b) proposing an idea of applying PLA partially on MINLP problems, making them easier to handle. The computational experiments were done using quadratically constrained quadratic programming (QCQP) and MIQCQP and they showed that problems under PLA with nonuniform partition resulted in more accurate solutions and required less time compared to PLA with uniform partition.

Highlights

The rapid advances of Linear Programming (LP), Non-Linear Programming (NLP), and Mixed Integer Linear Programming (MILP) techniques and algorithms in the 20th century led to the development of robust MILP solvers that can handle problems with millions of variables
Even with the current improvements in the Mixed Integer NonLinear Programming (MINLP) solvers, it would be a great step forward if MINLP problems could be solved globally by MILP solvers. This can be done by approximating the MINLP problem by an MILP one, but solving this approximation by an MILP solver will probably be harder than solving the original problem by an MINLP solver
It can be concluded that Piecewise linear approximation (PLA) models can be improved by using nonuniform partitioning depending on local solutions instead of uniform partitioning

Summary

Introduction

The rapid advances of Linear Programming (LP), Non-Linear Programming (NLP), and Mixed Integer Linear Programming (MILP) techniques and algorithms in the 20th century led to the development of robust MILP solvers that can handle problems with millions of variables. Methods that deal with Mixed Integer NonLinear Programming (MINLP), which is the most difficult class of optimization, started to improve recently. Even with the current improvements in the MINLP solvers, it would be a great step forward if MINLP problems could be solved globally by MILP solvers. In principle, this can be done by approximating the MINLP problem by an MILP one, but solving this approximation by an MILP solver will probably be harder than solving the original problem by an MINLP solver. For some real life optimization applications, we refer to [8,9,10,11]

Objectives

Findings

Discussion

Conclusion