Homotopy Continuation Enhanced Branch and Bound Algorithms for Strongly Nonconvex Mixed-Integer Nonlinear Programming Problems

TL;DR

This paper introduces homotopy continuation enhanced branch and bound algorithms for solving large-scale, strongly nonlinear, nonconvex mixed-integer nonlinear programming problems, demonstrating improved solution quality and efficiency over existing methods.

Contribution

The paper develops two novel HCBB algorithms integrating homotopy continuation with branch and bound for better solving complex MINLPs, showing superior performance in practice.

Findings

HCBB algorithms find optimal solutions from various initial points.

HCBB-RB outperforms HCBB-FP in computational efficiency.

Proposed methods outperform existing MINLP algorithms on benchmark problems.

Abstract

Large-scale strongly nonlinear and nonconvex mixed-integer nonlinear programming (MINLP) models frequently appear in optimisation-based process synthesis, integration, intensification, and process control. However, they are usually difficult to solve by existing algorithms within an acceptable time. In this work, we propose two robust homotopy continuation enhanced branch and bound (HCBB) algorithms (denoted as HCBB-FP and HCBB-RB) where the homotopy continuation method is employed to gradually approach the optimum of the NLP subproblem at a node from the solution at its parent node. A variable step length is adapted to effectively balance feasibility and computational efficiency. The computational results from solving four existing process synthesis problems demonstrate that the proposed HCBB algorithms can find the same optimal solution from different initial points, while the existing MINLP algorithms fail or find much worse solutions. In addition, HCBB-RB is superior to HCBB-FP due to the much lower computational effort required for the same locally optimal solution.