supergrad.utils.optimize.scipy_minimize

supergrad.utils.optimize.scipy_minimize(fun, x0, args=(), method=None, jac='jax', hess=None, hessp=None, bounds=None, constraints=(), tol=None, callback=None, options=None, *, jit: bool = True, fwd_ad: bool = False, logging: bool = False)

Minimization of scalar function of one or more variables.

SuperGrad-supported implementation of scipy.optimize._minimize.minimize().

Original docstring below.

Parameters:

• fun (callable) –

  The objective function to be minimized:

  fun(x, *args) -> float
  

  where x is a 1-D array with shape (n,) and args is a tuple of the fixed parameters needed to completely specify the function.

  Suppose the callable has signature f0(x, *my_args, **my_kwargs), where my_args and my_kwargs are required positional and keyword arguments. Rather than passing f0 as the callable, wrap it to accept only x; e.g., pass fun=lambda x: f0(x, *my_args, **my_kwargs) as the callable, where my_args (tuple) and my_kwargs (dict) have been gathered before invoking this function.

• x0 (ndarray, shape (n,)) – Initial guess. Array of real elements of size (n,), where n is the number of independent variables.

• args (tuple, optional) – Extra arguments passed to the objective function and its derivatives (fun, jac and hess functions).

• method (str or callable, optional) –

  Type of solver. Should be one of

  • ’Nelder-Mead’ (see here)

  • ’Powell’ (see here)

  • ’CG’ (see here)

  • ’BFGS’ (see here)

  • ’Newton-CG’ (see here)

  • ’L-BFGS-B’ (see here)

  • ’TNC’ (see here)

  • ’COBYLA’ (see here)

  • ’COBYQA’ (see here)

  • ’SLSQP’ (see here)

  • ’trust-constr’(see here)

  • ’dogleg’ (see here)

  • ’trust-ncg’ (see here)

  • ’trust-exact’ (see here)

  • ’trust-krylov’ (see here)

  • custom - a callable object, see below for description.

  If not given, chosen to be one of BFGS, L-BFGS-B, SLSQP, depending on whether or not the problem has constraints or bounds.

• jac ({callable, '2-point', '3-point', 'cs', bool}, optional) –

  Method for computing the gradient vector. Only for CG, BFGS, Newton-CG, L-BFGS-B, TNC, SLSQP, dogleg, trust-ncg, trust-krylov, trust-exact and trust-constr. If it is a callable, it should be a function that returns the gradient vector:

  jac(x, *args) -> array_like, shape (n,)
  

  where x is an array with shape (n,) and args is a tuple with the fixed parameters. If jac is a Boolean and is True, fun is assumed to return a tuple (f, g) containing the objective function and the gradient. Methods ‘Newton-CG’, ‘trust-ncg’, ‘dogleg’, ‘trust-exact’, and ‘trust-krylov’ require that either a callable be supplied, or that fun return the objective and gradient. If None or False, the gradient will be estimated using 2-point finite difference estimation with an absolute step size. Alternatively, the keywords {‘2-point’, ‘3-point’, ‘cs’} can be used to select a finite difference scheme for numerical estimation of the gradient with a relative step size. These finite difference schemes obey any specified bounds.

• hess ({callable, '2-point', '3-point', 'cs', HessianUpdateStrategy}, optional) –

  Method for computing the Hessian matrix. Only for Newton-CG, dogleg, trust-ncg, trust-krylov, trust-exact and trust-constr. If it is callable, it should return the Hessian matrix:

  hess(x, *args) -> {LinearOperator, spmatrix, array}, (n, n)
  

  where x is a (n,) ndarray and args is a tuple with the fixed parameters. The keywords {‘2-point’, ‘3-point’, ‘cs’} can also be used to select a finite difference scheme for numerical estimation of the hessian. Alternatively, objects implementing the HessianUpdateStrategy interface can be used to approximate the Hessian. Available quasi-Newton methods implementing this interface are:

  • BFGS

  • SR1

  Not all of the options are available for each of the methods; for availability refer to the notes.

• hessp (callable, optional) –

  Hessian of objective function times an arbitrary vector p. Only for Newton-CG, trust-ncg, trust-krylov, trust-constr. Only one of hessp or hess needs to be given. If hess is provided, then hessp will be ignored. hessp must compute the Hessian times an arbitrary vector:

  hessp(x, p, *args) ->  ndarray shape (n,)
  

  where x is a (n,) ndarray, p is an arbitrary vector with dimension (n,) and args is a tuple with the fixed parameters.

• bounds (sequence or Bounds, optional) –

  Bounds on variables for Nelder-Mead, L-BFGS-B, TNC, SLSQP, Powell, trust-constr, COBYLA, and COBYQA methods. There are two ways to specify the bounds:

  1. Instance of Bounds class.

  2. Sequence of (min, max) pairs for each element in x. None is used to specify no bound.

• constraints ({Constraint, dict} or List of {Constraint, dict}, optional) –

  Constraints definition. Only for COBYLA, COBYQA, SLSQP and trust-constr.

  Constraints for ‘trust-constr’ and ‘cobyqa’ are defined as a single object or a list of objects specifying constraints to the optimization problem. Available constraints are:

  • LinearConstraint

  • NonlinearConstraint

  Constraints for COBYLA, SLSQP are defined as a list of dictionaries. Each dictionary with fields:

  typestr

  Constraint type: ‘eq’ for equality, ‘ineq’ for inequality.

  funcallable

  The function defining the constraint.

  jaccallable, optional

  The Jacobian of fun (only for SLSQP).

  argssequence, optional

  Extra arguments to be passed to the function and Jacobian.

  Equality constraint means that the constraint function result is to be zero whereas inequality means that it is to be non-negative. Note that COBYLA only supports inequality constraints.

• tol (float, optional) – Tolerance for termination. When tol is specified, the selected minimization algorithm sets some relevant solver-specific tolerance(s) equal to tol. For detailed control, use solver-specific options.

• options (dict, optional) –

  A dictionary of solver options. All methods except TNC accept the following generic options:

  maxiterint

  Maximum number of iterations to perform. Depending on the method each iteration may use several function evaluations.

  For TNC use maxfun instead of maxiter.

  dispbool

  Set to True to print convergence messages.

  For method-specific options, see show_options().

• callback (callable, optional) –

  A callable called after each iteration.

  All methods except TNC, SLSQP, and COBYLA support a callable with the signature:

  callback(intermediate_result: OptimizeResult)
  

  where intermediate_result is a keyword parameter containing an OptimizeResult with attributes x and fun, the present values of the parameter vector and objective function. Note that the name of the parameter must be intermediate_result for the callback to be passed an OptimizeResult. These methods will also terminate if the callback raises StopIteration.

  All methods except trust-constr (also) support a signature like:

  callback(xk)
  

  where xk is the current parameter vector.

  Introspection is used to determine which of the signatures above to invoke.

Returns:

res – The optimization result represented as a OptimizeResult object. Important attributes are: x the solution array, success a Boolean flag indicating if the optimizer exited successfully and message which describes the cause of the termination. See OptimizeResult for a description of other attributes.

Return type:

OptimizeResult

References

[1]

Nelder, J A, and R Mead. 1965. A Simplex Method for Function Minimization. The Computer Journal 7: 308-13.

[2]

Wright M H. 1996. Direct search methods: Once scorned, now respectable, in Numerical Analysis 1995: Proceedings of the 1995 Dundee Biennial Conference in Numerical Analysis (Eds. D F Griffiths and G A Watson). Addison Wesley Longman, Harlow, UK. 191-208.

[3]

Powell, M J D. 1964. An efficient method for finding the minimum of a function of several variables without calculating derivatives. The Computer Journal 7: 155-162.

[4]

Press W, S A Teukolsky, W T Vetterling and B P Flannery. Numerical Recipes (any edition), Cambridge University Press.

[5]

Nocedal, J, and S J Wright. 2006. Numerical Optimization. Springer New York.

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Byrd, R H and P Lu and J. Nocedal. 1995. A Limited Memory Algorithm for Bound Constrained Optimization. SIAM Journal on Scientific and Statistical Computing 16 (5): 1190-1208.

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Zhu, C and R H Byrd and J Nocedal. 1997. L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization. ACM Transactions on Mathematical Software 23 (4): 550-560.

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Nash, S G. Newton-Type Minimization Via the Lanczos Method. 1984. SIAM Journal of Numerical Analysis 21: 770-778.

[9]

Powell, M J D. A direct search optimization method that models the objective and constraint functions by linear interpolation. 1994. Advances in Optimization and Numerical Analysis, eds. S. Gomez and J-P Hennart, Kluwer Academic (Dordrecht), 51-67.

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Powell M J D. Direct search algorithms for optimization calculations. 1998. Acta Numerica 7: 287-336.

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Powell M J D. A view of algorithms for optimization without derivatives. 2007.Cambridge University Technical Report DAMTP 2007/NA03

[12]

Kraft, D. A software package for sequential quadratic programming. 1988. Tech. Rep. DFVLR-FB 88-28, DLR German Aerospace Center – Institute for Flight Mechanics, Koln, Germany.

[13]

Conn, A. R., Gould, N. I., and Toint, P. L. Trust region methods. 2000. Siam. pp. 169-200.

[14]

F. Lenders, C. Kirches, A. Potschka: “trlib: A vector-free implementation of the GLTR method for iterative solution of the trust region problem”, arXiv:1611.04718

[15]

N. Gould, S. Lucidi, M. Roma, P. Toint: “Solving the Trust-Region Subproblem using the Lanczos Method”, SIAM J. Optim., 9(2), 504–525, (1999).

[16]

Byrd, Richard H., Mary E. Hribar, and Jorge Nocedal. 1999. An interior point algorithm for large-scale nonlinear programming. SIAM Journal on Optimization 9.4: 877-900.

[17]

Lalee, Marucha, Jorge Nocedal, and Todd Plantenga. 1998. On the implementation of an algorithm for large-scale equality constrained optimization. SIAM Journal on Optimization 8.3: 682-706.

[18]

Ragonneau, T. M. Model-Based Derivative-Free Optimization Methods and Software. PhD thesis, Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China, 2022. URL: https://theses.lib.polyu.edu.hk/handle/200/12294.