Symmetry Detection for Quadratic Optimization Using Binary Layered Graphs
Abstract
:1. Introduction
2. Formulation Symmetry for QuadraticallyConstrained Quadratic Optimization Problems
2.1. Symmetry Detection with Expression Graphs
2.2. Symmetry Detection with Tensors
2.2.1. Sparse Tensor Representation
2.2.2. Converting Matrices to EdgeLabeled, VertexColored Graphs
 If ${I}_{i}={J}_{i}$, i.e., a quadratic term, then $E=E\cup \{\left\{{({v}_{{I}_{i}},{v}_{{K}_{i}})}^{r}\right\}\cap \left\{{\left({v}_{{I}_{i}}\right)}^{r}\right\}\}$.
 else for bilinear term, ${I}_{i}\ne {J}_{i}$, then $E=E\cup \{\left\{{({v}_{{I}_{i}},{v}_{{K}_{i}})}^{r}\right\}\cap \left\{{({v}_{{J}_{i}},{v}_{{K}_{i}})}^{r}\right\}\cap \left\{{({v}_{{I}_{i}},{v}_{{J}_{i}})}^{r}\right\}\}.$
3. Formulation Symmetry Detection via Binary Layered Graphs
Algorithm 1 Algorithm constructing the vertex set 

Algorithm 2 Algorithm constructing the edge set 

4. Numerical Discussion and Comparison to the StateoftheArt
5. Exploiting Symmetry in the Point Packing Problem
Exhaustive Search and 2D Symmetry Removal
Algorithm 3 Algorithm 1—Exhaustive Search 
Input: number of points n, number of grids k, number of occupied grids m. Output: The optimal solution ${d}^{*}$

Algorithm 4 Algorithm 2—2D Symmetry Removal 
Input: number of points n, number of grids k, number of occupied grids m. Output: The optimal solution ${d}^{*}$

 One point on ${V}_{1}$, 2 points on ${V}_{4}$ and 2 points on ${V}_{5}$. This is from ${O}_{3}$.
 Two points on ${H}_{1}$, 1 point on ${H}_{2}$ and 2 points on ${H}_{5}$. This is from ${O}_{4}$.
6. Results and Comparisons
7. Conclusions
Author Contributions
Funding
Conflicts of Interest
Abbreviations
MINLP  Mixedinteger nonlinear optimization 
QCQP  Quadraticallyconstrained quadratic optimization 
References
 Margot, F. Symmetry in Integer Linear Programming. In 50 Years of Integer Programming 1958–2008: From the Early Years to the StateoftheArt; Springer: Berlin/Heidelberg, Germany, 2010; pp. 647–686. [Google Scholar]
 Liberti, L. Reformulations in mathematical programming: Automatic symmetry detection and exploitation. Math. Program. 2012, 131, 273–304. [Google Scholar] [CrossRef]
 Costa, A.; Hansen, P.; Liberti, L. On the impact of symmetrybreaking constraints on spatial BranchandBound for circle packing in a square. Discret. Appl. Math. 2013, 161, 96–106. [Google Scholar] [CrossRef]
 Puget, J.F. Automatic Detection of Variable and Value Symmetries. In Proceedings of the 11th International Conference on Principles and Practice of Constraint Programming—CP 2005, Sitges, Spain, 1–5 October 2005; van Beek, P., Ed.; Springer: Berlin/Heidelberg, Germany, 2005; pp. 475–489. [Google Scholar]
 Salvagnin, D. A Dominance Procedure for Integer Programming. Master’s Thesis, University of Padua, Padua, Italy, 2005. [Google Scholar]
 Berthold, T.; Pfetsch, M. Detecting Orbitopal Symmetries. In Proceedings of the Annual International Conference of the German Operations Research Society (GOR), Augsburg, Germany, 3–5 September 2008; Springer: Berlin/Heidelberg, Germany, 2009; pp. 433–438. [Google Scholar]
 Bremner, D.; Dutour Sikirić, M.; Pasechnik, D.V.; Rehn, T.; Schürmann, A. Computing symmetry groups of polyhedra. LMS J. Comput. Math. 2014, 17, 565–581. [Google Scholar] [CrossRef] [Green Version]
 Knueven, B.; Ostrowski, J.; Pokutta, S. Detecting almost symmetries of graphs. Math. Program. Comput. 2018, 10, 143–185. [Google Scholar] [CrossRef]
 Sherali, H.D.; Smith, J.C. Improving Discrete Model Representations via Symmetry Considerations. Manag. Sci. 2001, 47, 1396–1407. [Google Scholar] [CrossRef]
 Liberti, L. Automatic Generation of SymmetryBreaking Constraints. In Combinatorial Optimization and Applications; Yang, B., Du, D.Z., Wang, C.A., Eds.; Springer: Berlin/Heidelberg, Germany, 2008; pp. 328–338. [Google Scholar] [Green Version]
 Liberti, L.; Ostrowski, J. Stabilizerbased symmetry breaking constraints for mathematical programs. J. Glob. Optim. 2014, 60, 183–194. [Google Scholar] [CrossRef]
 Ghoniem, A.; Sherali, H.D. Defeating symmetry in combinatorial optimization via objective perturbations and hierarchical constraints. IIE Trans. 2011, 43, 575–588. [Google Scholar] [CrossRef]
 Ostrowski, J.; Linderoth, J.; Rossi, F.; Smriglio, S. Constraint Orbital Branching. In Proceedings of the 13th International Conference on Integer Programming and Combinatorial Optimization IPCO, Bertinoro, Italy, 26–28 May 2008; pp. 225–239. [Google Scholar]
 Ostrowski, J.; Linderoth, J.; Rossi, F.; Smiriglio, S. Orbital branching. Math. Program. 2011, 126, 147–178. [Google Scholar] [CrossRef]
 Margot, F. Pruning by isomorphism in branchandcut. Math. Program. 2002, 94, 71–90. [Google Scholar] [CrossRef]
 Kaibel, V.; Peinhardt, M.; Pfetsch, M.E. Orbitopal fixing. Discret. Optim. 2011, 8, 595–610. [Google Scholar] [CrossRef] [Green Version]
 Faenza, Y.; Kaibel, V. Extended Formulations for Packing and Partitioning Orbitopes. Math. Oper. Res. 2009, 34, 686–697. [Google Scholar] [CrossRef] [Green Version]
 Pfetsch, M.E.; Rehn, T. A computational comparison of symmetry handling methods for mixed integer programs. Math. Program. Comput. 2019, 11, 37–93. [Google Scholar] [CrossRef]
 Margot, F. Small covering designs by branchandcut. Math. Program. 2003, 94, 207–220. [Google Scholar] [CrossRef]
 Costa, A.; Liberti, L.; Hansen, P. Formulation symmetries in circle packing. Electron. Notes Discret. Math. 2010, 36, 1303–1310. [Google Scholar] [CrossRef] [Green Version]
 Ostrowski, J.; Vannelli, A.; Anjos, M.F. Symmetry in Scheduling Problems; Cahier du GERAD G201069; GERAD: Montreal, QC, Canada, 2010. [Google Scholar]
 Ostrowski, J.; Wang, J.; Liu, C. Exploiting Symmetry in Transmission Lines for Transmission Switching. IEEE Trans. Power Syst. 2012, 27, 1708–1709. [Google Scholar] [CrossRef]
 Ostrowski, J.; Wang, J. Network reduction in the TransmissionConstrained Unit Commitment problem. Comput. Ind. Eng. 2012, 63, 702–707. [Google Scholar] [CrossRef]
 Ostrowski, J.; Anjos, M.F.; Vannelli, A. Modified orbital branching for structured symmetry with an application to unit commitment. Math. Program. 2015, 150, 99–129. [Google Scholar] [CrossRef]
 Lima, R.M.; Novais, A.Q. Symmetry breaking in MILP formulations for Unit Commitment problems. Comput. Chem. Eng. 2016, 85, 162–176. [Google Scholar] [CrossRef] [Green Version]
 Knueven, B.; Ostrowski, J.; Wang, J. The Ramping Polytope and Cut Generation for the Unit Commitment Problem. INFORMS J. Comput. 2018, 30, 739–749. [Google Scholar] [CrossRef]
 Kouyialis, G.; Misener, R. Detecting Symmetry in Designing Heat Exchanger Networks. In Proceedings of the International Conference of Foundations of ComputerAided Process OperationsFOCAPO/CPC, Tucson, AZ, USA, 8–12 January 2017. [Google Scholar]
 Letsios, D.; Kouyialis, G.; Misener, R. Heuristics with performance guarantees for the minimum number of matches problem in heat recovery network design. Comput. Chem. Eng. 2018, 113, 57–85. [Google Scholar] [CrossRef]
 Maravelias, C.T.; Grossmann, I.E. A hybrid MILP/CP decomposition approach for the continuous time scheduling of multipurpose batch plants. Comput. Chem. Eng. 2004, 28, 1921–1949. [Google Scholar] [CrossRef]
 Maravelias, C.T. MixedTime Representation for StateTask Network Models. Ind. Eng. Chem. Res. 2005, 44, 9129–9145. [Google Scholar] [CrossRef]
 Mistry, M.; Callia D’Iddio, A.; Huth, M.; Misener, R. Satisfiability modulo theories for process systems engineering. Comput. Chem. Eng. 2018, 113, 98–114. [Google Scholar] [CrossRef]
 Smith, E.M.B.; Pantelides, C.C. A symbolic reformulation/spatial branchandbound algorithm for the global optimisation of nonconvex MINLPs. Comput. Chem. Eng. 1999, 23, 457–478. [Google Scholar] [CrossRef]
 Tawarmalani, M.; Sahinidis, N.V. A polyhedral branchandcut approach to global optimization. Math. Program. 2005, 103, 225–249. [Google Scholar] [CrossRef]
 Belotti, P.; Lee, J.; Liberti, L.; Margot, F.; Wachter, A. Branching and Bounds Tightening techniques for Nonconvex MINLP. Optim. Methods Softw. 2009, 24, 597–634. [Google Scholar] [CrossRef]
 Youdong, L.; Linus, S. The global solver in the LINDO API. Optim. Methods Softw. 2009, 24, 657–668. [Google Scholar]
 Misener, R.; Floudas, C.A. ANTIGONE: Algorithms for coNTinuous/Integer Global Optimization of Nonlinear Equations. J. Glob. Optim. 2014, 59, 503–526. [Google Scholar] [CrossRef]
 Vigerske, S. Decomposition in Multistage Stochastic Programming and a Constraint Integer Programming Approach to MixedInteger Nonlinear Programming. Ph.D. Thesis, HumboldtUniversität zu Berlin, Berlin, Germany, 2013. [Google Scholar]
 Mahajan, A.; Leyffer, S.; Linderoth, J.; Luedtke, J.; Munson, T. Minotaur: A mixedinteger nonlinear optimization toolkit. Optim. Online 2017, 6275. [Google Scholar]
 Boukouvala, F.; Misener, R.; Floudas, C.A. Global optimization advances in MixedInteger Nonlinear Programming, MINLP, and Constrained DerivativeFree Optimization, CDFO. Eur. J. Oper. Res. 2016, 252, 701–727. [Google Scholar] [CrossRef] [Green Version]
 Fourer, R.; Maheshwari, C.; Neumaier, A.; Orban, D.; Schichl, H. Convexity and concavity detection in computational graphs: Tree walks for convexity assessment. INFORMS J. Comput. 2010, 22, 26–43. [Google Scholar] [CrossRef]
 Hart, W.E.; Laird, C.; Watson, J.; Woodruff, D.L. Pyomo: Modeling and solving mathematical programs in python. Math. Program. Comput. 2011, 3, 219–260. [Google Scholar] [CrossRef]
 Ceccon, F.; Siirola, J.D.; Misener, R. SUSPECT: MINLP special structure detector for Pyomo. Optim. Lett. 2019. [Google Scholar] [CrossRef]
 McKay, B.D.; Piperno, A. Practical graph isomorphism, II. J. Symb. Comput. 2014, 60, 94–112. [Google Scholar] [CrossRef]
 Ramani, A.; Aloul, F.; Markov, I.; Sakallah, K.A. Breaking instanceindependent symmetries in exact graph coloring. J. Artif. Intell. Res. 2006, 1, 324–329. [Google Scholar] [CrossRef]
 Ramani, A.; Markov, I.L. Automatically Exploiting Symmetries in Constraint Programming. In Recent Advances in Constraints; Faltings, B.V., Petcu, A., Fages, F., Rossi, F., Eds.; Springer: Berlin/Heidelberg, Germany, 2005; pp. 98–112. [Google Scholar] [Green Version]
 Anstreicher, K.M. Semidefinite programming versus the reformulationlinearization technique for nonconvex quadratically constrained quadratic programming. J. Glob. Optim. 2009, 43, 471–484. [Google Scholar] [CrossRef]
 Misener, R.; Floudas, C.A. Global optimization of mixedinteger quadraticallyconstrained quadratic programs (MIQCQP) through piecewiselinear and edgeconcave relaxations. Math. Program. 2012, 136, 155–182. [Google Scholar] [CrossRef]
 Misener, R.; Floudas, C.A. GloMIQO: Global mixedinteger quadratic optimizer. J. Glob. Optim. 2013, 57, 3–50. [Google Scholar] [CrossRef]
 Furini, F.; Traversi, E.; Belotti, P.; Frangioni, A.; Gleixner, A.; Gould, N.; Liberti, L.; Lodi, A.; Misener, R.; Mittelmann, H. QPLIB: A library of quadratic programming instances. Math. Program. Comput. 2019, 11, 237–265. [Google Scholar] [CrossRef]
 Jones, D.R. A fully general, exact algorithm for nesting irregular shapes. J. Glob. Optim. 2014, 59, 367–404. [Google Scholar] [CrossRef]
 Misener, R.; Smadbeck, J.B.; Floudas, C.A. Dynamically generated cutting planes for mixedinteger quadratically constrained quadratic programs and their incorporation into GloMIQO 2. Optim. Methods Softw. 2015, 30, 215–249. [Google Scholar] [CrossRef]
 Wang, A.; Hanselman, C.L.; Gounaris, C.E. A customized branchandbound approach for irregular shape nesting. J. Glob. Optim. 2018, 71, 935–955. [Google Scholar] [CrossRef]
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Kouyialis, G.; Wang, X.; Misener, R. Symmetry Detection for Quadratic Optimization Using Binary Layered Graphs. Processes 2019, 7, 838. https://doi.org/10.3390/pr7110838
Kouyialis G, Wang X, Misener R. Symmetry Detection for Quadratic Optimization Using Binary Layered Graphs. Processes. 2019; 7(11):838. https://doi.org/10.3390/pr7110838
Chicago/Turabian StyleKouyialis, Georgia, Xiaoyu Wang, and Ruth Misener. 2019. "Symmetry Detection for Quadratic Optimization Using Binary Layered Graphs" Processes 7, no. 11: 838. https://doi.org/10.3390/pr7110838