Machine learning techniques, specifically Gradient-Enhanced Kriging (GEK), has been implemented for molecular geometry optimization.<br>GEK has many advantages as compared to conventional -- step-restricted second-order truncated -- molecular optimization methods.<br>In particular, the surrogate model associated with GEK can have multiple stationary points, will smoothly converge to the<br>exact model as the size of the data set increases, and contains an explicit expression for the expected average error of the model function<br>at an arbitrary point in space.<br>In this respect GEK can be of interest for methods used in molecular geometry optimizations.<br>GEK is usually, however, associated with abundance of data, contrary to the situation desired for<br>efficient geometry optimizations.<br>In the paper we will demonstrate how the GEK procedure can be utilized in a fashion such that in the presence of few data points, the<br>surrogate surface will in a robust way guide the optimization to a minimum of a molecular structure.<br>In this respect the GEK procedure will be used to mimic the behavior of a conventional second-order scheme, but retaining the<br>flexibility of the superior machine learning approach -- GEK is an exact interpolator.<br>Moreover, the expected variance will be used in the optimization to facilitate restricted-variance rational function optimizations (RV-RFO).<br>A procedure which relates the eigenvalues of the Hessian-model-function Hessian with the individual characteristic<br>lengths, used in the GEK, reduces the number of empirical parameters to two -- the value of the trend function and the<br>maximum allowed variance. These parameters are determined using the extended Baker (e-Baker) test suite, at the Hartree-Fock level of approximation,<br>and a single reaction of the Baker transition-state (Baker-TS) test suite as a training set. The so-created optimization<br>procedure -- RV-RFO-GEK -- is tested using the e-Baker, the full Baker-TS, and the S22 test suites, at the density-functional-theory level for the two Baker test suites<br>and at the second order Møller-Plesset level of approximation for the S22 test suite, respectively.<br>The tests show that the new method is generally on par with a state-of-the-art conventional method, while for difficult cases it exhibits a definite advantage.<br>