A matrix-free augmented Lagrangian algorithm with application to large-scale structural design optimization.

*(English)*Zbl 1364.90388Summary: In many large engineering design problems, it is not computationally feasible or realistic to store Jacobians or Hessians explicitly. Matrix-free implementations of standard optimization methods – implementations that do not explicitly form Jacobians and Hessians, and possibly use quasi-Newton approximations – circumvent those restrictions, but such implementations are virtually non-existent. We develop a matrix-free augmented-Lagrangian algorithm for nonconvex problems with both equality and inequality constraints. Our implementation is developed in the Python language, is available as an open-source package, and allows for approximating Hessian and Jacobian information. We show that our approach solves problems from the CUTEr and COPS test sets in a comparable number of iterations to state-of-the-art solvers. We report numerical results on a structural design problem that is typical in aircraft wing design optimization. The matrix-free approach makes solving problems with thousands of design variables and constraints tractable, even when function and gradient evaluations are costly.

##### MSC:

90C90 | Applications of mathematical programming |

90C06 | Large-scale problems in mathematical programming |

74P10 | Optimization of other properties in solid mechanics |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

##### Keywords:

large-scale optimization; matrix-free optimization; structural optimization; PDE-constrained optimization; augmented Lagrangian
PDF
BibTeX
XML
Cite

\textit{S. Arreckx} et al., Optim. Eng. 17, No. 2, 359--384 (2016; Zbl 1364.90388)

Full Text:
DOI

##### References:

[1] | Andreani, R; Birgin, EG; Martínez, JM; Schuverdt, ML, On augmented Lagrangian methods with general lower-level constraints, SIAM J Optim, 18, 1286-1309, (2008) · Zbl 1151.49027 |

[2] | Argyris., H, Energy theorems and structural analysis: a generalized discourse with applications on energy principles of structural analysis including the effects of temperature and non-linear stress-strain relations., Aircr Eng AerospTechnol, 26, 347-356, (1954) |

[3] | Arioli M, Orban D (2013) Iterative methods for symmetric quasi-definite linear systems—Part I: theory. Cahier du GERAD G-2013-32, GERAD, Montréal, QC, Canada |

[4] | Armand, P; Benoist, J; Orban, D, From global to local convergence of interior methods for nonlinear optimization, Optim Methods Softw, 28, 1051-1080, (2012) · Zbl 1278.90254 |

[5] | Arreckx S, Orban D (2015) A regularized SQP method for degenerate equality-constrained optimization. Technical report, GERAD, Montréal, QC, Canada In preparation · Zbl 1390.90389 |

[6] | Biros, G; Ghattas, O, Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization. part I: the Krylov-Schur solver, SIAM J Sci Comput, 27, 687-713, (2005) · Zbl 1091.65061 |

[7] | Bertsekas DP (1982) Constrained optimization and Lagrange multiplier methods. Academic Press, New York · Zbl 0572.90067 |

[8] | Byrd, RH; Nocedal, J; Waltz, RA; Pillo, G (ed.); Waltz, RA (ed.), KNITRO: an integrated package for nonlinear optimization, 35-39, (2006), Heidelberg · Zbl 1108.90004 |

[9] | Conn AR, Gould NIM, Toint PhL (1992) Lancelot: a fortran package for large-scale nonlinear optimization (release A), 1st edn. Springer Publishing Company, Incorporated, New York · Zbl 0761.90087 |

[10] | Conn, AR; Vicente, LN; Visweswariah, C, Two-step algorithms for nonlinear optimization with structured applications, SIAM J Optim, 9, 924-947, (1999) · Zbl 0997.90077 |

[11] | Conn AR, Gould NIM, Toint PhL (2000) Trust-region methods. Society for Industrial and Applied Mathematics, Philadelphia · Zbl 0958.65071 |

[12] | Simone, V; Serafino, D, A matrix-free approach to build band preconditioners for large-scale bound-constrained optimization, J Comput Appl Math, 268, 82-92, (2014) · Zbl 1293.65042 |

[13] | Dener A, Kenway GK, Lyu Z, Hicken JE, Martins JRRA (2015) Comparison of inexact- and quasi-newton algorithms for aerodynamic shape optimization. In 53rd AIAA Aerospace Sciences Meeting · Zbl 1364.49037 |

[14] | Dennis Jr, J; Walker, H, Convergence theorems for least-change secant update methods, SIAM J Numer Anal, 18, 949-987, (1981) · Zbl 0527.65032 |

[15] | Dolan ED, Moré JJ, Munson TS (2004) Benchmarking optimization software with COPS 3.0. Technical Report ANL/MCS-TM-273, Argonne national labaratory, Mathematics and Computer Science division |

[16] | Dolan, ED; Moré, JJ, Benchmarking optimization software with performance profiles, Math Program, 91, 201-213, (2002) · Zbl 1049.90004 |

[17] | Gawlik E, Munson T, Sarich J, Wild SM (2012) The TAO linearly constrained augmented Lagrangian method for PDE-constrained optimization. Preprint ANL/MCS-P2003-0112, Mathematics and Computer Science Division, January 2012. URL: http://www.mcs.anl.gov/uploads/cels/papers/P2003-0112.pdf · Zbl 1108.90004 |

[18] | Gill, PE; Murray, W; Saunders, MA, SNOPT: an SQP algorithm for large-scale constrained optimization, SIAM J Optim, 12, 979-1006, (2002) · Zbl 1027.90111 |

[19] | Gill, P; Robinson, D, A globally convergent stabilized sqp method, SIAM J Optim, 23, 1983-2010, (2013) · Zbl 1285.49023 |

[20] | Gould, N I M; Hribar, M E; Nocedal, J, On the solution of equality constrained quadratic problems arising in optimization, SIAM J Sci Comput, 23, 1375-1394, (2001) · Zbl 0999.65050 |

[21] | Gould, NIM; Orban, D; Toint, PL, \(\sf CUTEr \), a constrained and unconstrained testing environment, revisited, ACM Trans Math Softw, 28, 373-394, (2003) · Zbl 1068.90526 |

[22] | Greif, C; Moulding, E; Orban, D, Bounds on the eigenvalues of block matrices arising from interior-point methods, SIAM J Optim, 24, 49-83, (2014) · Zbl 1291.15008 |

[23] | Griewank, A; Walther, A, On constrained optimization by adjoint based quasi-Newton methods, Optim Methods Softw, 17, 869-889, (2002) · Zbl 1065.90077 |

[24] | Haftka, R T; Kamat, M P, Simultaneous nonlinear structural analysis and design, Comput Mech, 4, 409-416, (1989) · Zbl 0692.73064 |

[25] | Hicken, JE, Inexact Hessian-vector products in reduced-space differential-equation constrained optimization, Optim Eng, 15, 575-608, (2014) · Zbl 1364.49037 |

[26] | Kennedy GJ, Martins JRRA (2014) A parallel finite-element framework for large-scale gradient-based design optimization of high-performance structures. Finite Elements Anal Des 87: 56-73, September 2014. ISSN 0168874X. doi:10.1016/j.finel.2014.04.011. URL http://linkinghub.elsevier.com/retrieve/pii/S0168874X14000730 · Zbl 1274.90008 |

[27] | Kennedy GJ, Martins JRRA (2015) A parallel aerostructural optimization framework for aircraft design studies. Struct Multidiscipl Optim (In press). doi:10.1007/s00158-014-1108-9 · Zbl 1049.90004 |

[28] | Kennedy, G J; Martins, J RR A, Laminate parametrization technique for discrete ply angle problems with manufacturing constraints, Struct Multidiscipl Optim, 48, 379-393, (2013) |

[29] | Kenway, GKW; Kennedy, GJ; Martins, JRRA, Scalable parallel approach for high-fidelity steady-state aeroelastic analysis and derivative computations, AIAA J, 52, 935-951, (2014) |

[30] | Kenway, GKW; Martins, JRRA, Multipoint high-fidelity aerostructural optimization of a transport aircraft configuration, J Aircraft, 51, 144-160, (2014) |

[31] | Lin, C-J; Moré, J J, Newton’s method for large bound-constrained optimization problems, SIAM J Optim, 9, 1100-1127, (1998) · Zbl 0957.65064 |

[32] | Lyu, Z, Aerodynamic design optimization studies of a blended-wing-body aircraft, J Aircraft, 51, 1604-1617, (2014) |

[33] | Lyu Z, Kenway GK, Martins JRRA (2015) Aerodynamic shape optimization studies on the common research model wing benchmark. AIAA J (In press). doi:10.2514/1.J053318 · Zbl 1108.90004 |

[34] | Martínez HJ (1988) Local and superlinear convergence of structured secant methods for the convex class. Technical report, Rice University, Houston, USA · Zbl 1278.90254 |

[35] | Martins, JRRA; Lambe, AB, Multidisciplinary design optimization: a survey of architectures, AIAA J, 51, 2049-2075, (2013) |

[36] | Moré, JJ; Toraldo, G, Algorithms for bound constrained quadratic programming problems, Numer Math, 55, 377-400, (1989) · Zbl 0675.65061 |

[37] | Moré, JJ; Toraldo, G, On the solution of large quadratic programming problems with bound constraints, SIAM J Optim, 1, 93-113, (1991) · Zbl 0752.90053 |

[38] | Munson T, Sarich J, Wild S, Benson S, McInnes LC (2012) TAO 2.0 Users Manual. Technical Report ANL/MCS-TM-322, Mathematics and Computer Science Division, Argonne National Laboratory. http://www.mcs.anl.gov/tao |

[39] | Murtagh BA, Saunders MA (2003) \(\sf MINOS \) 5.51 user’s guide. Technical Report SOL 83-20R, Stanford University, Stanford, California, USA |

[40] | Murtagh, BA; Saunders, MA, Large-scale linearly constrained optimization, Math Program, 14, 41-72, (1978) · Zbl 0383.90074 |

[41] | Nocedal J, Wright SJ (2006) Numerical optimization, 2nd edn. Springer, New York · Zbl 1104.65059 |

[42] | Nocedal J, Yuan Y (1998) Combining trust region and line search techniques. Technical report, Advances in Nonlinear Programming, (Kluwer) · Zbl 0909.90243 |

[43] | Orban D (2014) NLPy—a large-scale optimization toolkit in Python. Cahier du GERAD G-2014-xx, GERAD, Montréal, QC, Canada. In preparation · Zbl 0383.90074 |

[44] | Perez, RE; Jansen, PW; Martins, JRRA, Pyopt: a python-based object-oriented framework for nonlinear constrained optimization, Struct Multidiscipl Optim, 45, 101-118, (2012) · Zbl 1274.90008 |

[45] | Poon, NMK; Martins, JRRA, An adaptive approach to constraint aggregation using adjoint sensitivity analysis, Struct Multidiscipl Optim, 34, 61-73, (2007) |

[46] | Rockafellar RT (1973) The multiplier method of hestenes and powell applied to convex programming. J Optim Theory Appl 12(6): 555-562. ISSN 0022-3239. doi:10.1007/BF00934777 · Zbl 0254.90045 |

[47] | Schlenkrich, S; Griewank, A; Walther, A, On the local convergence of adjoint Broyden methods, Math Program, 121, 221-247, (2010) · Zbl 1185.90207 |

[48] | Schmit LA (1960) Structural design by systematic synthesis. In 2nd Conference on Electronic Computation, ASCE, New York, NY, pp 105-132 · Zbl 0891.90153 |

[49] | Toint, PL, Non-monotone trust-region algorithms for nonlinear optimization subject to convex constraints, Math Program, 77, 69-94, (1997) · Zbl 0891.90153 |

[50] | Turner, MJ; Clough, RW; Martin, HC; Topp, LJ, Stiffness and deflection analysis of complex structures, J Aeronaut Sci, 23, 805-823, (1956) · Zbl 0072.41003 |

[51] | Wächter, A; Biegler, LT, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math Program, 106, 25-57, (2006) · Zbl 1134.90542 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.