STK_SAMPCRIT_AKG_EVAL computes the Approximate KG criterion

 CALL: AKG = stk_sampcrit_akg_eval (ZC_MEAN, ZC_STD, ZR_MEAN, ZR_STD, ZCR_COV)

    computes the value AKG of the Approximate KG criterion for a set of
    candidates points, with respect to a certain reference grid.  The
    predictive distributions of the objective function (to be minimized) at
    the candidates and reference points is assumed to be jointly Gaussian,
    with mean ZC_MEAN and standard deviation ZC_STD for the candidate points,
    mean ZR_MEAN and satandard deviation ZR_STD on the reference points, and
    covariance matrix ZCR_COV between the candidate and reference points.
    The input argument must have the following sizes:

       * ZC_MEAN    M x 1,
       * ZC_STD     M x 1,
       * ZR_MEAN    L x 1,
       * ZR_STD     L x 1,
       * ZCR_COV    M x L,

    where M is the number of candidate points and L the number of reference
    points.  The output has size M x 1.

 NOTE ABOUT THE "KNOWLEDGE GRADIENT" CRITERION

    The "Knowlegde Gradient" (KG) criterion is the one-step look-ahead (a.k.a
    myopic) sampling criterion associated to the problem of estimating the
    minimizer of the objective function under the L^1 loss (equivalently,
    under the linear loss/utility).

    This sampling strategy was proposed for the first time in the work of
    Mockus and co-authors in the 70's (see [1] and refs therein), for the case
    of noiseless evaluations, but only applied to particular Brownian-like
    processes for which the minimum of the posterior mean coincides with the
    best evaluations so far (in which case the KG criterion coincides with the
    EI criterion introduced later by Jones et al [2]).

    It was later discussed for the case of a finite space with independent
    Gaussian priors first by Gupta and Miescke [3] and then by Frazier et al
    [4] who named it "knowledge gradient".  It was extended to the case of
    correlated priors by Frazier et al [5].

 NOTE ABOUT THE REFERENCE SET

    For the case of continuous input spaces, there is no exact expression of
    the KG criterion.  The approximate KG criterion proposed in this function
    is an approximation of the KG criterion where the continuous 'min' in the
    expression of the criterion at the i^th candidate point are replaced by
    discrete mins over some reference grid *augmented* with the i^th candidate
    point.

    This type of approximation has been proposed by Scott et al [6] under the
    name "knowledge gradient for continuous parameters" (KGCP).  In [6], the
    reference grid is composed of the current set of evaluation points.  The
    implementation proposed in STK leaves this choice to the user.

    Note that, with the reference grid proposed in [6], the complexity of one
    evaluation of the AKG (KGCP) criterion increases as O(N log N), where N
    denotes the number of evaluation points.

 NOTE ABOUT THE NOISELESS CASE

    Simplified formulas are available for the noiseless case (see [7]) but not
    currenly implemented in STK.

 REFERENCES

   [1] J. Mockus, V. Tiesis and A. Zilinskas. The application of Bayesian
       methods for seeking the extremum. In L.C.W. Dixon and G.P. Szego, eds,
       Towards Global Optimization, 2:117-129, North Holland NY, 1978.

   [2] D. R. Jones, M. Schonlau and William J. Welch. Efficient global
       optimization of expensive black-box functions.  Journal of Global
       Optimization, 13(4):455-492, 1998.

   [3] S. Gupta and K. Miescke,  Bayesian look ahead one-stage sampling
       allocations for selection of the best population,  J. Statist. Plann.
       Inference, 54:229-244, 1996.

   [4] P. I. Frazier, W. B. Powell, S. Dayanik,  A knowledge gradient policy
       for sequential information collection,  SIAM J. Control Optim.,
       47(5):2410-2439, 2008.

   [5] P. I. Frazier, W. B. Powell, and S. Dayanik.  The Knowledge-Gradient
       Policy for Correlated Normal Beliefs.  INFORMS Journal on Computing
       21(4):599-613, 2009.

   [6] W. Scott, P. I. Frazier and W. B. Powell.  The correlated knowledge
       gradient for simulation optimization of continuous parameters using
       Gaussian process regression.  SIAM J. Optim, 21(3):996-1026, 2011.

   [7] J. van der Herten, I. Couckuyt, D. Deschrijver, T. Dhaene,  Fast
       Calculation of the Knowledge Gradient for Optimization of Deterministic
       Engineering Simulations,  arXiv preprint arXiv:1608.04550

 See also: STK_SAMPCRIT_EI_EVAL

Package: stk