Skip to main content
SpringerLink
Log in

Finding local optima in quadratic optimization problems in RP

  • Application of soft computing
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

This paper describes a new randomized algorithm for calculating local optima in standard quadratic optimization problems (StQPs) over the standard simplex. The new algorithm transforms StQPs into a new form by using a fuzzification technique. This paper proves that: (1) the computational complexity of the new algorithm for computing local optima in StQPs is in the class of randomized polynomial time; (2) the solution of the new algorithm satisfies \(\epsilon -\delta \) condition. Examples are given that demonstrate that the new algorithm outperforms IBM ILOG CPLEX (CPLEX). Numerical experiments indicate that for calculating the first local optima of StQPs, the average computational time of the new algorithm is one hundred time faster than that of CPLEX when the number of variables is 1000.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Data availability

Enquiries about data availability should be directed to the authors.

References

  • Ahmadi AA, Zhang J (2020) On the complexity of finding a local minimizer of a quadratic function over a polytope. Optimization Online 2020 at arXiv:2008.05558v2

  • Ammar E (2009) On fuzzy random multiobjective quadratic programming. Eur J Oper Res 193:329–341

    Article  MathSciNet  Google Scholar 

  • Ammar E (2008) On solutions of fuzzy random multiobjective quadratic programming with applications in portfolio problem. Inf Sys 178(2008):468–484

    MathSciNet  Google Scholar 

  • Ammar E (2005) On solution analysis of quadratic programming with fuzzy random coefficients and variables. Il Nuovo Cimento, vol 120 B. N.1, Italy, pp 5–17

  • Ammar E, Khalifa AM (2003) Fuzzy portfolio optimization a quadratic programming approach. J Chaos Solitons Fractals 18(5):1045–1054

    Article  MathSciNet  Google Scholar 

  • Arora S, Barak B (2009) Computational complexity: a modern approach. Cambridge University Press, New York, p 2009

    Book  Google Scholar 

  • Bhanja S, Karunaratne DK, Panchumarthy R, Rajaram S, Sarkar S (2015) Non-Boolean computing with nanomagnets for computer vision applications. Nat Nanotechnol 11:177–183

    Article  Google Scholar 

  • Buló SR, Pelillo M, Bomze IM (2011) Graph-based quadratic optimization: a fast evolutionary approach. Comput Vis Image Underst 115:984–995

    Article  Google Scholar 

  • Bomze IM, Schachinger W, Ullrich R (2017) The complexity of simple models—a study of worst and typical hard cases for the standard quadratic optimization problem. Math Oper Res 43(2):651–674

    Article  MathSciNet  Google Scholar 

  • Bomze IM (2012) Copositive optimization–recent developments and applications. Eur J Oper Res 216(2012):509–520

    Article  MathSciNet  Google Scholar 

  • Bomze IM, De Klerk E (2002) Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. J Glob Optim 24(2):163–185

    Article  MathSciNet  Google Scholar 

  • Bomze IM (1998) On standard quadratic optimization problems. J Glob Optim 13:369–387

    Article  MathSciNet  Google Scholar 

  • Bomze IM (2002) Branch-and-bound approaches to standard quadratic optimization problems. J Glob Optim 22:17–37

    Article  MathSciNet  Google Scholar 

  • Bonami P, Lodi A, Schweiger J, Tramontani A (2016) Solving standard quadratic programming by cutting planes. Technical Report, DS4DM-2016-001, Polytechnique Montréal

  • Burer S, Vandenbussche D (2008) A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations. Math Progr 113(2008):259–282

    Article  MathSciNet  Google Scholar 

  • Butler A, Kwon RH (2021) Efficient differentiable quadratic programming layers: an ADMM approach. https://doi.org/10.48550/arXiv.2112.07464

  • Chen J, Burer S (2012) Globally solving nonconvex quadratic programming problems via completely positive programming. Math Progr Comput 4(1):33–52

    Article  MathSciNet  Google Scholar 

  • Cruz C, Silva RC, Verdegay JL, Yamakami A (2009) A parametric approach to solve quadratic programming problems with fuzzy environment in the set of constraints. In: The proceeding of IFSA-EUSFLAT 2009:1158–1163

  • De Klerk E (2008) The complexity of optimizing over a simplex, hypercube or sphere: a short survey. CEJOR 16:111–125

    Article  MathSciNet  Google Scholar 

  • De Klerk E, den Hertog D, Elabwabi G (2008) On the complexity of optimization over the standard simplex. Eur J Oper Res 191:773–785

    Article  MathSciNet  Google Scholar 

  • Chen X, Pittel B (2021) On sparsity of the solution to a random quadratic optimization problem. Math Progr 186:309–336

    Article  MathSciNet  Google Scholar 

  • Gao J, Lu M, Liu L (2004) Chance-constrained programming for fuzzy quadratic minimum spanning tree problem. In: The proceeding of 2004 IEEE international conference on fuzzy systems, pp 983–987

  • Gao J, Lu M (2005) Fuzzy quadratic minimum spanning tree problem. Appl Math Comput 164(2005):773–788

    MathSciNet  Google Scholar 

  • Gao L (2020) An approximation algorithm for solving standard quadratic optimization problems. J Intell Fuzzy Syst 39(3):4383–4392

    Article  Google Scholar 

  • Gao L (2020) An algorithm for finding approximate Nash equilibria in bimatrix games. Soft Comput. https://doi.org/10.1007/s00500-020-05213-y

    Article  Google Scholar 

  • Gao L (1999) The fuzzy arithmetic mean. Fuzzy Sets Syst 107:335–348

    Article  MathSciNet  Google Scholar 

  • Gondzio J, Yıldırım EA (2021) Global solutions of nonconvex standard quadratic programs via mixed integer linear programming reformulations. J Glob Optim 81:293–321

    Article  MathSciNet  Google Scholar 

  • Huang Y, Palomer DP (2014) Randomized algorithms for optimal solutions of double-sided QCQP with applications in signal processing. IEEE Trans Signal Process. https://doi.org/10.1109/TSP.2013.2297683

    Article  MathSciNet  Google Scholar 

  • IBM (2020) IBM ILOG. CPLEX Optimization Studio. https://www.ibm.com/products/ilog-cplex-optimization-studio. Accessed 21 Oct 2020

  • IBM (2017) IBM ILOG CPLEX Optimization Studio CPLEX User’s Manual Version 12 Release 8. IBM Corp., 2017

  • Ichnowski J, Jain P, Stellato B, Banjac G, Luo M, Borrelli F, Gonzalez JE, Stoica I, Goldberg K (2021) Accelerating quadratic optimization with reinforcement learning. In: The Proceeding of 35th conference on neural information processing systems (NeurIPS 2021)

  • Liu ST (2007) Solving quadratic programming with fuzzy parameters based on extension principle. In: Proceeding of IEEE international conference fuzzy systems, FUZZ-IEEE, pp 1–5

  • Liu B, Liu Y-K (2002) Expected value of fuzzy variable and fuzzy expected value models. IEEE Trans Fuzzy Syst 10(4):445–450

    Article  Google Scholar 

  • Liu YK, Gao J (2007) The independence of fuzzy variables with applications to fuzzy random optimization. Int J Uncertain Fuzziness Knowl Based Syst. https://doi.org/10.1142/S021848850700456X

    Article  MathSciNet  Google Scholar 

  • Liuzzi G, Locatelli M, Piccialli V (2019) A new branch-and-bound algorithm for standard quadratic programming problems. Optim Methods Softw 34(1):79–97. https://doi.org/10.1080/10556788.2017.1341504

    Article  MathSciNet  Google Scholar 

  • Mirmohseni SM, Nasseri SH (2017) A quadratic programming with triangular fuzzy numbers. J Appl Math Phys 2017(5):2218–2227

    Article  Google Scholar 

  • Momot A, Momot M (2009) Fuzzy weighted averaging using criterion function minimization. ICMMI 2009:273-280

    Google Scholar 

  • Motwani R, Raghavan P (1995) Randomized algorithms. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Motzkin TS, Straus EG (1965) Maxima for graphs and a new proof of a theorem of T’uran. Can J Math 17:533–540

  • Nesterrov Y (2003) Random walk in a simplex and quadratic optimization over convex polytopes. Tech. Rep. No. 2003/71, CORE-UCL

  • Nowak I (1999) A new semidefinite programming bound for indefinite quadratic forms over a simplex. J Glob Optim 14:357–364

    Article  MathSciNet  Google Scholar 

  • Palagi L, Piccialli V, Rendl F, Rianldi G, Wiegele A (2012) Chapter 28 computational approaches to max-cut. In: Anjos MF, Lasserre JB (eds) Handbook on semidefinite, conic and polynomial optimization. Springer, Berlin

  • Williams R (2007) Matrix-vector multiplication in sub-quadratic time: (some preprocessing required). In: Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms(2007), pp 995–1001

  • Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1:3–28

    Article  MathSciNet  Google Scholar 

  • Zhou X, Cao B, Nasseri SH (2014) Optimality conditions for fuzzy number quadratic programming with fuzzy coefficients. J Appl Math. https://doi.org/10.1155/2014/489893

    Article  MathSciNet  Google Scholar 

  • Zimmermann H-J (2001) Fuzzy set theory and its application, 4th edn. Kluwer Academic Publishers, Amsterdam

    Book  Google Scholar 

Download references

Acknowledgements

The author thanks Dr. Andrea Raiconi and the anonymous reviewers for their careful reading of the paper and for their constructive comments, which are greatly appreciated.

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. Department of Physics and Computer Science, Wilfrid Laurier University, 75 University Ave. W., Waterloo, ON, N2L3C5, Canada

    Lunshan Gao

Authors
  1. Lunshan Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lunshan Gao.

Ethics declarations

Conflict of interest

The authors have not disclosed any competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gao, L. Finding local optima in quadratic optimization problems in RP. Soft Comput 28, 495–508 (2024). https://doi.org/10.1007/s00500-023-08262-1

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-023-08262-1

Keywords

Search

Navigation