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An Adaptive Refinement Algorithm for Discretizations of Nonconvex QCQP

Authors Akshay Gupte , Arie M. C. A. Koster , Sascha Kuhnke

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Author Details

Akshay Gupte
  • School of Mathematics, The University of Edinburgh, UK
Arie M. C. A. Koster
  • Lehrstuhl II für Mathematik, RWTH Aachen University, Germany
Sascha Kuhnke
  • Lehrstuhl II für Mathematik, RWTH Aachen University, Germany

Funding

The second and third authors' visit was supported by a grant from DAAD, the German Academic Exchange Service.
  • Gupte, Akshay: Initial phase of this research supported by NSF grant DMS-1913294.

Acknowledgements

This research was initiated during the second and third authors' visit to Clemson University, USA, in 2019, where the first author was a faculty member.

Cite AsGet BibTex

Akshay Gupte, Arie M. C. A. Koster, and Sascha Kuhnke. An Adaptive Refinement Algorithm for Discretizations of Nonconvex QCQP. In 20th International Symposium on Experimental Algorithms (SEA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 233, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SEA.2022.24

Abstract

We present an iterative algorithm to compute feasible solutions in reasonable running time to quadratically constrained quadratic programs (QCQPs), which form a challenging class of nonconvex continuous optimization. This algorithm is based on a mixed-integer linear program (MILP) which is a restriction of the original QCQP obtained by discretizing all quadratic terms. In each iteration, this MILP restriction is solved to get a feasible QCQP solution. Since the quality of this solution heavily depends on the chosen discretization of the MILP, we iteratively adapt the discretization values based on the MILP solution of the previous iteration. To maintain a reasonable problem size in each iteration of the algorithm, the discretization sizes are fixed at predefined values. Although our algorithm did not always yield good feasible solutions on arbitrary QCQP instances, an extensive computational study on almost 1300 test instances of two different problem classes - box-constrained quadratic programs with complementarity constraints and disjoint bilinear programs, demonstrates the effectiveness of our approach. We compare the quality of our solutions against those from heuristics and local optimization algorithms in two state-of-the-art commercial solvers and observe that on one instance class we clearly outperform the other methods whereas on the other class we obtain competitive results.

Subject Classification

ACM Subject Classification
  • Theory of computation → Mixed discrete-continuous optimization
  • Mathematics of computing → Nonconvex optimization
Keywords
  • Quadratically Constrained Quadratic Programs
  • Mixed Integer Linear Programming
  • Heuristics
  • BoxQP
  • Disjoint Bilinear

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