naginterfaces.library.opt.bounds_bobyqa_func¶

naginterfaces.library.opt.bounds_bobyqa_func(objfun, npt, x, bl, bu, rhobeg, rhoend, maxcal, monfun=None, data=None)[source]¶

bounds_bobyqa_func is an easy-to-use algorithm that uses methods of quadratic approximation to find a minimum of an objective function $F$ over $x \in R^{n}$ , subject to fixed lower and upper bounds on the independent variables $x_{1}, x_{2}, \dots, x_{n}$ . Derivatives of $F$ are not required.

The function is intended for functions that are continuous and that have continuous first and second derivatives (although it will usually work even if the derivatives have occasional discontinuities). Efficiency is maintained for large $n$ .

Deprecated since version 27.0.0.0: bounds_bobyqa_func is deprecated. Please use handle_solve_dfno() and handle_solve_dfno_rcomm() instead. See also the Replacement Calls document.

For full information please refer to the NAG Library document for e04jc

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/e04/e04jcf.html

Parameters

objfuncallable f = objfun(x, data=None)

$o b j f u n$ must evaluate the objective function $F$ at a specified vector $x$ .

Parameters

xfloat, ndarray, shape $(n)$: $x$ , the vector at which the objective function is to be evaluated.
dataarbitrary, optional, modifiable in place: User-communication data for callback functions.

Returns

ffloat: Must be set to the value of the objective function at $x$ .

nptint

$m$ , the number of interpolation conditions imposed on the quadratic approximation at each iteration.

Suggested value: $n p t = 2 \times n_{r} + 1$ , where $n_{r}$ denotes the number of non-fixed variables.

xfloat, array-like, shape $(n)$

An estimate of the position of the minimum. If any component is out-of-bounds it is replaced internally by the bound it violates.

blfloat, array-like, shape $(n)$

$b l$ must contain the fixed vector of lower bounds, $ℓ$ .

bufloat, array-like, shape $(n)$

$b u$ must contain the fixed vector of upper bounds, $u$ .

rhobegfloat

An initial lower bound on the value of the trust region radius.

Suggested value: $r h o b e g$ should be about one tenth of the greatest expected overall change to a variable: the initial quadratic model will be constructed by taking steps from the initial $x$ of length $r h o b e g$ along each coordinate direction.

rhoendfloat

A final lower bound on the value of the trust region radius.

Suggested value: $r h o e n d$ should indicate the absolute accuracy that is required in the final values of the variables.

maxcalint

The maximum permitted number of calls to $o b j f u n$ .

monfunNone or callable monfun(nf, x, f, rho, data=None), optional

Note: if this argument is None then a NAG-supplied facility will be used.

$m o n f u n$ may be used to monitor the optimization process.

It is invoked every time a new trust region radius is chosen.

If no monitoring is required, $m o n f u n$ may be None.

Parameters

nfint: The cumulative number of calls made to $o b j f u n$ .
xfloat, ndarray, shape $(n)$: The current best point.
ffloat: The value of $o b j f u n$ at $x$ .
rhofloat: A lower bound on the current trust region radius.
dataarbitrary, optional, modifiable in place: User-communication data for callback functions.

dataarbitrary, optional

User-communication data for callback functions.

Returns

xfloat, ndarray, shape $(n)$: The lowest point found during the calculations. Thus, if no exception or warning is raised on exit, $x$ is the position of the minimum.
ffloat: The function value at the lowest point found ( $x$ ).
nfint: Unless $e r r n o$ = 1 or NagMemoryError on exit, the total number of calls made to $o b j f u n$ .

Raises

NagValueError

(errno $1$ )

There were $n_{r} = ⟨ v a l u e ⟩$ unequal bounds and $n p t = ⟨ v a l u e ⟩$ on entry.

Constraint: $n_{r} + 2 \leq n p t \leq \frac{(n_{r} + 1) \times (n_{r} + 2)}{2}$ .

(errno $1$ )

There were $n_{r} = ⟨ v a l u e ⟩$ unequal bounds.

Constraint: $n_{r} \geq 2$ .

(errno $1$ )

On entry, $r h o b e g = ⟨ v a l u e ⟩$ , $b l [i - 1] = ⟨ v a l u e ⟩$ , $b u [i - 1] = ⟨ v a l u e ⟩$ and $i = ⟨ v a l u e ⟩$ .

Constraint: if $b l [i - 1] \neq b u [i - 1]$ in coordinate $i$ , $b u [i - 1] - b l [i - 1] \geq 2 \times r h o b e g$ .

(errno $1$ )

On entry, $r h o b e g = ⟨ v a l u e ⟩$ .

Constraint: $r h o b e g > 0.0$ .

(errno $1$ )

On entry, $r h o e n d = ⟨ v a l u e ⟩$ .

Constraint: $r h o e n d \geq macheps$ , where $macheps = machine.precision ()$ , the machine precision.

(errno $1$ )

On entry, $r h o b e g = ⟨ v a l u e ⟩$ and $r h o e n d = ⟨ v a l u e ⟩$ .

Constraint: $r h o e n d \leq r h o b e g$ .

(errno $1$ )

On entry, $m a x c a l = ⟨ v a l u e ⟩$ .

Constraint: $m a x c a l \geq 1$ .

Warns

NagAlgorithmicMajorWarning

(errno $2$ ): The function evaluations limit was reached: $o b j f u n$ has been called $m a x c a l$ times.
(errno $3$ ): The predicted reduction in a trust region step was non-positive. Check your specification of $o b j f u n$ and whether the function needs rescaling. Try a different initial $x$ .
(errno $4$ ): A rescue procedure has been called in order to correct damage from rounding errors when computing an update to a quadratic approximation of $F$ , but no further progess could be made. Check your specification of $o b j f u n$ and whether the function needs rescaling. Try a different initial $x$ .

NagCallbackTerminateWarning

(errno $5$ ): User-supplied objective function requested termination.
(errno $5$ ): User-supplied monitoring function requested termination.

Notes

bounds_bobyqa_func is applicable to problems of the form:

{minimize}_{x \in R^{n}} F (x) subject to ℓ \leq x \leq u and ℓ \leq u,

where $F$ is a nonlinear scalar function whose derivatives may be unavailable, and where the bound vectors are elements of $R^{n}$ . Relational operators between vectors are interpreted elementwise.

Fixing variables (that is, setting $ℓ_{i} = u_{i}$ for some $i$ ) is allowed in bounds_bobyqa_func.

You must supply a function to calculate the value of $F$ at any given point $x$ .

The method used by bounds_bobyqa_func is based on BOBYQA, the method of Bound Optimization BY Quadratic Approximation described in Powell (2009). In particular, each iteration of bounds_bobyqa_func generates a quadratic approximation $Q$ to $F$ that agrees with $F$ at $m$ automatically chosen interpolation points. The value of $m$ is a constant prescribed by you. Updates to the independent variables mostly occur from approximate solutions to trust region subproblems, using the current quadratic model.

References: Powell, M J D, 2009, The BOBYQA algorithm for bound constrained optimization without derivatives, Report DAMTP 2009/NA06, University of Cambridge, https://www.damtp.cam.ac.uk/user/na/NA_papers/NA2009_06.pdf

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naginterfaces.library.opt.bounds_bobyqa_func¶

naginterfaces.library.opt.bounds_​bobyqa_​func¶

naginterfaces.library.opt.bounds_bobyqa_func¶