Abstract
We study the minimization of objective functions containing non-physical jump discontinuities. These discontinuities arise when (partial) differential equations are discretized using non-constant methods and the resulting numerical solutions are used in computing the objective function. Although the functions may become discontinuous, gradient information may be computed at every point. Gradient information is computable everywhere since every point has an associated discretization for which (semi) analytical sensitivities can be calculated. Rather than the construction of global approximations using only function value information to overcome the discontinuities, we propose to use only the gradient information. We elaborate on the modifications of classical gradient based optimization algorithms for use in gradient-only approaches, and we then present gradient-only optimization strategies using both BFGS and a new spherical quadratic approximation for sequential approximate optimization (SAO). We then use the BFGS and SAO algorithms to solve three problems of practical interest, both unconstrained and constrained.
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References
Allaire G, Jouve F, Toader A-M (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393
Barthelemy J-FM, Haftka RT (1993) Approximation concepts for optimum structural design—a review. Struct Optim 5:129–144
Bazaraa MS, Sherali HD, Shetty CM (1993) Nonlinear programming—theory and algorithms, 2nd edn. Wiley, New York
Brandstatter BR, Ring W, Magele Ch, Richter KR (1998) Shape design with great geometrical deformations using continuously moving finite element nodes. IEEE Trans Magn 34(5):2877–2880
Forrester AIJ, Keane AJ (2009) Recent advances in surrogate-based optimization. Prog Aerosp Sci 45(1–3):50–79
Garcia MJ, Gonzalez CA (2004) Shape optimisation of continuum structures via evolution strategies and fixed grid finite element analysis. Struct Multidisc Optim V26(1):92–98
Gould N, Orban D, Toint P (2005) Numerical methods for Large-Scale nonlinear optimization. Acta Numer 14(1):299–361
Groenwold AA, Etman LFP, Snyman JA, Rooda JE (2007) Incomplete series expansion for function approximation. Struct Multidisc Optim 34:21–40
Haftka RT, Gürdal Z (1991) Elements of structural optimization, solid mechanics and its applications, 3rd edn, vol 11. Kluwer Academic, Dordrecht
Kocks UF (1976) Laws for work-hardening and low-temperature creep. J Eng Mater Technol Trans ASME 98 Ser H(1):76–85
Kodiyalam S, Thanedar PB (1993) Some practical aspects of shape optimization and its influence on intermediate mesh refinement. Finite Elem Anal Des 15(2):125–133
Kok S, Beaudoin AJ, Tortorelli DA (2002) On the development of stage IV hardening using a model based on the mechanical threshold. Acta Mater 50(7):1653–1667
Lui DC, Nocedal J (1989) On the limited memory BFGS method for large scale optimization. Math Program 54(1–3):503–528
Olhoff N, Rasmussen J, Lund E (1993) A method of exact numerical differentiation for error elimination in finite element based semi-analytical shape sensitivity analysis. Mechan Struct Mach 21:1–66
Persson P-O, Strang G (2004) A simple mesh generator in matlab. SIAM Rev 46(2):329–345
Potra FA, Shi Y (1995) Efficient line search algorithm for unconstrained optimization. J Optim Theory Appl 85(3):677–704
Quapp W (1996) A gradient-only algorithm for tracing a reaction path uphill to the saddle of a potential energy surface. Chem Phys Lett 253(3–4):286–292
Sacks J, Welch WJ, Mitchell TJ, Wynn HP (1989) Design and analysis of computer experiments. Stat Sci 4:409–435
Schleupen A, Maute K, Ramm E (2000) Adaptive FE-procedures in shape optimization. Struct Multidisc Optim 19(4):282302
Shor NZ, Kiwiel KC, Ruszcaynski A (1985) Minimization methods for non-differentiable functions. Springer, New York
Simpson TW, Toropov V, Balabanov V, Viana FAC (2008) Design and analysis of computer experiments in multidisciplinary design optimization: a review of how far we have come—or not. In: Proc 12th AIAA/ISSMO multidisciplinary analysis and optimization conference, Victoria, 10–12 September 2008
Snyman JA (1982) A new and dynamic method for unconstrained minimization. Appl Math Model 6(6):449–462
Snyman JA (2005a) A gradient-only line search method for the conjugate gradient method applied to constrained optimization problems with severe noise in the objective function. Int J Numer Methods Eng 62(1):72–82
Snyman JA (2005b) Practical mathematical optimization: an introduction to basic optimization theory and classical and new gradient-based algorithms. Applied optimization, 2nd edn, vol 97. Springer, New York
Snyman JA, Hay AM (2001) The spherical quadratic steepest descent (SQSD) method for unconstrained minimization with no explicit line searches. Comput Math Appl 42(1–2):169–178
Snyman JA, Hay AM (2002) The dynamic-Q optimization method: an alternative to SQP? Comput Math Appl 44(12):1589–1598
Svanberg K (2002) A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J Optim 12:555–573
Toropov VV (1989) Simulation approach to structural optimization. Struct Optim 1:37–46
Van Miegroet L, Mos N, Fleury C, Duysinx P (2005) Generalized shape optimization based on the level set method. In: 6th world congresses of structural and multidisciplinary optimization. International Society for Structural and Multidisciplinary Optimization, Daejeon, pp 1–10
Voce E (1955) A practical strain-hardening function. Metallurgica 51:219–226
Wallis J (1685) A treatise of algebra, both historical and practical. London
Wilke DN, Kok S, Groenwold AA (2006) A quadratically convergent unstructured remeshing strategy for shape optimization. Int J Numer Methods Eng 65(1):1–17
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Wilke, D.N., Kok, S. & Groenwold, A.A. The application of gradient-only optimization methods for problems discretized using non-constant methods. Struct Multidisc Optim 40, 433–451 (2010). https://doi.org/10.1007/s00158-009-0389-x
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DOI: https://doi.org/10.1007/s00158-009-0389-x