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Constrained Multifidelity Optimization Using Model Calibration

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Constrained Multifidelity Optimization Using Model Calibration Struct Multidisc Optim (2012) 46:93–109 DOI 10.1007/s00158-011-0749-1 RESEARCH PAPER Constrained multifidelity optimization using model calibration Andrew March · Karen Willcox Received: 4 April 2011 / Revised: 22 November 2011 / Accepted: 28 November 2011 / Published online: 8 January 2012 c Springer-Verlag 2012 Abstract Multifidelity optimization approaches seek to derivative-free and sequential quadratic programming meth- bring higher-fidelity analyses earlier into the design process ods. The method uses approximately the same number of by using performance estimates from lower-fidelity models high-fidelity analyses as a multifidelity trust-region algo- to accelerate convergence towards the optimum of a high- rithm that estimates the high-fidelity gradient using finite fidelity design problem. Current multifidelity optimization differences. methods generally fall into two broad categories: provably convergent methods that use either the high-fidelity gradient Keywords Multifidelity · Derivative-free · Optimization · or a high-fidelity pattern-search, and heuristic model cali- Multidisciplinary · Aerodynamic design bration approaches, such as interpolating high-fidelity data or adding a Kriging error model to a lower-fidelity function. This paper presents a multifidelity optimization method that bridges these two ideas; our method iteratively calibrates Nomenclature lower-fidelity information to the high-fidelity function in order to find an optimum of the high-fidelity design prob- A Active and violated constraint Jacobian lem. The algorithm developed minimizes a high-fidelity a Sufficient decrease parameter objective function subject to a high-fidelity constraint and B A closed and bounded set in Rn other simple constraints. The algorithm never computes the B Trust region gradient of a high-fidelity function; however, it achieves C Differentiability Class first-order optimality using sensitivity information from the c(x) Inequality constraint calibrated low-fidelity models, which are constructed to d Artificial lower bound for constraint value have negligible error in a neighborhood around the solution. e(x) Error model for objective The method is demonstrated for aerodynamic shape opti- e¯(x) Error model for constraint mization and shows at least an 80% reduction in the num- f (x) Objective function ber of high-fidelity analyses compared other single-fidelity g(x) Inequality constraint h(x) Equality constraint L Expanded level-set in Rn L Level-set in Rn A version of this paper was presented as paper AIAA-2010-9198 L Lagrangian of trust-region subproblem at the 13th AIAA/ISSMO Multidisciplinary Analysis and M Space of all fully linear models Optimization Conference, Fort Worth, TX, 13–15 September 2010. m(x) Surrogate model of the high-fidelity objective B · A. March ( ) K. Willcox function Department of Aeronautics & Astronautics, Massachusetts Institute ¯ ( ) of Technology, Cambridge, MA 02139, USA m x Surrogate model of the high-fidelity constraint e-mail: [email protected] n Number of design variables K. Willcox p Arbitrary step vector e-mail: [email protected] RBF Radial Basis Function 94 A. March, K. Willcox r Radial distance between two points 1 Introduction s Trust-region step t Constant between 0 and 1 High-fidelity numerical simulation tools can provide valu- w Constraint violation conservatism factor able insight to designers of complex systems and support x Design vector better decisions through optimal design. However, the de- α Convergence tolerance multiplier sign budget frequently prevents the use of formal optimiza- β Convergence tolerance multiplier tion methods in conjunction with high-fidelity simulation γ0 Trust-region contraction ratio tools, due to the high cost of simulations. An additional γ1 Trust-region expansion ratio challenge for optimization is obtaining accurate gradient ε Termination tolerance information for high-fidelity models, especially when the ε2 Termination tolerance for trust region size design process employs black-box and/or legacy tools. In- Trust region size stead, designers often set design aspects using crude lower- δh Linearized step size for feasibility fidelity performance estimates and later attempt to optimize δx Finite difference step size system subcomponents using higher-fidelity methods. This η0 Trust-region contraction criterion approach in many cases leads to suboptimal design per- η1,η2 Trust-region expansion criterion formance and unnecessary expenses for system operators. κ Bound related to function smoothness An alternative is to use the approximate performance esti- λ Lagrange multiplier mates of lower-fidelity simulations in a formal multifidelity λˆ Lagrange multiplier for surrogate model optimization framework, which aims to speed the design ξ Radial basis function correlation length process towards a high-fidelity optimal design using sig- ρ Ratio of actual to predicted improvement nificantly fewer costly simulations. This paper presents a σ Penalty parameter multifidelity optimization method for the design of com- τ Trust-region solution tolerance plex systems, and targets in particular, systems for which Penalty function accurate design sensitivity information is challenging to ˆ Surrogate penalty function obtain. φ Radial basis function Multifidelity optimization of systems with design con- straints can be achieved with the approximation model man- agement framework of Alexandrov et al. (1999, 2001). That Superscript technique optimizes a sequence of surrogate models of the objective function and constraints that are first-order con- ∗ Optimal sistent with their high-fidelity counterparts. The first-order + Active or violated inequality constraint consistent surrogates are constructed by correcting lower- fidelity models with either additive or multiplicative error models based on function value and gradient information of Subscript the high-fidelity models. This technique is provably conver- gent to a locally optimal high-fidelity design and in practice 0 Initial iterate can provide greater than 50% reductions in the number bhm Bound for Hessian of the surrogate model of high-fidelity function calls, translating into important blg Upper bound on the Lipschitz constant reductions in design turnaround times (Alexandrov et al. c Relating to constraint 1999). Estimating the needed gradient information is often cg Relating to constraint gradient a significant challenge, especially in the case of black-box cgm Maximum Lipschitz constant for a constraint simulation tools. For example, the high-fidelity numerical gradient simulation tool may occasionally fail to provide a solution cm Maximum Lipschitz constant for a constraint or it may contain noise in the output (e.g., due to numer- f Relating to objective function ical tolerances). In these cases, which arise frequently as fg Lipschitz constant for objective function gradient both objectives and constraints in complex system design, FCD Fraction of Cauchy Decrease if design sensitivities are unavailable then finite-difference g Relating to objective function gradient gradient estimates will likely be inaccurate and non-robust. high Relating to the high-fidelity function Therefore, there is a need for constrained multifidelity k Index of trust-region iteration optimization methods that can guarantee convergence to low Relating to a lower-fidelity function a high-fidelity optimal design while reducing the number m¯ Related to constraint surrogate model of expensive simulations, but that do not require gradient max User-set maximum value of that parameter estimates of the high-fidelity models. Constrained multifidelity optimization using model calibration 95 Constraint handling in derivative-free optimization is between the high-fidelity function and the surrogate model. challenging. Common single-fidelity approaches use lin- This ensures that the error between the high-fidelity model ear approximations (Powell 1994, 1998), an augmented gradient and surrogate model gradient is locally bounded Lagrangian (Kolda et al. 2003, 2006; Lewis and Torczon without ever calculating the high-fidelity gradient. Conn 2010), exact penalty method (Liuzzi and Lucidi 2009), or et al. (2008) showed that polynomial interpolation mod- constraint filtering (Audet and Dennis 2004) in combina- els can be made fully linear, provided the interpolating set tion with either a pattern-search or a simplex method. In the satisfies certain geometric requirements, and further devel- multifidelity setting, if the generated surrogate models are oped an unconstrained gradient-free optimization technique smooth, then surrogate sensitivity information can be used using fully linear models (Conn et al. 2009a, b). Wild to speed convergence.1 This may provide an advantage over et al. (2008) demonstrated that a radial basis function inter- single-fidelity pattern-search and simplex methods, espe- polation could satisfy the requirements for a fully linear cially when the dimension of the design space is large. For model and be used in Conn’s derivative-free optimization example, constraint-handling approaches used in conjunc- framework (Wild 2009; Wild and Shoemaker 2009). March tion with Efficient Global Optimization (EGO) (Jones et al. and Willcox (2010) generalized this method to the cases 1998) either estimate the probability that a point is both a with arbitrary low-fidelity functions or multiple low-fidelity minimum and that it is feasible, or add a smooth penalty to functions using Bayesian model calibration methods. the surrogate model
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