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Journal articles on the topic 'Directional derivatives'

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Published: 4 June 2021

Last updated: 1 February 2022

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1

Craven, B. D. "On quasidifferentiable optimization." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 41, no. 1 (August 1986): 64–78. http://dx.doi.org/10.1017/s1446788700028081.

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AbstractLagrangian necessary conditions for optimality, of both Fritz John and Kuhn Tucker types, are obtained for a constrained minimization problem, where the functions are locally Lipschitz and have directional derivatives, but need not have linear Gâteaux derivatives; the variable may be constrained to lie in a nonconvex set. The directional derivatives are assumed to have some convexity properties as functions of direction; this generalizes the concept of quasidifferentiable function. The convexity is not required when directional derivatives are replaced by Clarke generalized derivatives. Sufficient Kuhn Tucker conditions, and a criterion for the locally solvable constraint qualification, are obtained for directionally differentiable functions.

2

Li, Meng, and Yi Zhan. "Integrating Feature Direction Information with a Level Set Formulation for Image Segmentation." East Asian Journal on Applied Mathematics 6, no. 1 (January 27, 2016): 1–22. http://dx.doi.org/10.4208/eajam.231114.240915a.

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AbstractA feature-dependent variational level set formulation is proposed for image segmentation. Two second order directional derivatives act as the external constraint in the level set evolution, with the directional derivative across the image features direction playing a key role in contour extraction and another only slightly contributes. To overcome the local gradient limit, we integrate the information from the maximal (in magnitude) second-order directional derivative into a common variational framework. It naturally encourages the level set function to deform (up or down) in opposite directions on either side of the image edges, and thus automatically generates object contours. An additional benefit of this proposed model is that it does not require manual initial contours, and our method can capture weak objects in noisy or intensity-inhomogeneous images. Experiments on infrared and medical images demonstrate its advantages.

3

Chen, W., and Z. Ditzian. "Mixed and directional derivatives." Proceedings of the American Mathematical Society 108, no. 1 (January 1, 1990): 177. http://dx.doi.org/10.1090/s0002-9939-1990-0994773-0.

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4

Bednařík, Dušan, and Karel Pastor. "A characterization of $C^{1,1}$ functions via lower directional derivatives." Mathematica Bohemica 134, no. 2 (2009): 217–21. http://dx.doi.org/10.21136/mb.2009.140656.

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5

Preiss, D., and L. Zajíček. "Directional derivatives of Lipschitz functions." Israel Journal of Mathematics 125, no. 1 (December 2001): 1–27. http://dx.doi.org/10.1007/bf02773371.

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6

Korablev, A. I. "Directional derivatives of quasiconvex functionals." Journal of Soviet Mathematics 41, no. 6 (June 1988): 1425–28. http://dx.doi.org/10.1007/bf01097072.

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7

Ben-Tal, A., and J. Zowe. "Directional derivatives in nonsmooth optimization." Journal of Optimization Theory and Applications 47, no. 4 (December 1985): 483–90. http://dx.doi.org/10.1007/bf00942193.

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8

Yang, X. Q. "On second-order directional derivatives." Nonlinear Analysis: Theory, Methods & Applications 26, no. 1 (January 1996): 55–66. http://dx.doi.org/10.1016/0362-546x(94)00209-z.

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9

Apalak, MK, and MD Demirbas. "Thermal stress analysis of in-plane two-directional functionally graded plates subjected to in-plane edge heat fluxes." Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications 232, no. 8 (April 11, 2016): 693–716. http://dx.doi.org/10.1177/1464420716643857.

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This study investigates the thermal stress and deformation states of bi-directional functionally graded clamped plates subjected to constant in-plane heat fluxes along two ceramic edges. The material properties of the functionally graded plates were assumed to vary with a power law along two in-plane directions not through the plate thickness direction. The spatial derivatives of thermal and mechanical properties of the material composition were considered, and the effects of the bi-directional composition variations and spatial derivative terms on the displacement, strain and stress distributions were also investigated. The heat conduction and Navier equations describing the two-dimensional thermo-elastic problem were discretized using finite-difference method, and the set of linear equations were solved using the pseudo singular value method. The compositional gradient exponents and the spatial derivatives of thermal and mechanical properties of the material composition were observed to play an important role especially on the heat transfer durations, the displacement and strain distributions, but had a minor effect on the temperature and stress distributions.

10

Ojha, Bhuwan Prasad. "Different Concepts of Derivatives." Journal of Advanced College of Engineering and Management 3 (January 10, 2018): 11. http://dx.doi.org/10.3126/jacem.v3i0.18809.

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<p>In this paper, different concept of derivatives with some properties has been introduced. In differential calculus, the partial derivative, directional derivative and total derivative are studied. Their generalization for Banach spaces are the Gateaux differential and Freshet derivative.</p><p><strong>Journal of Advanced College of Engineering and Management,</strong> Vol.3, 2017, Page: 11-14</p>

11

Qu, Tao, Andrew J. P. White, and Anthony G. M. Barrett. "Four-directional synthesis of adamantane derivatives." Arkivoc 2021, no. 4 (June 11, 2020): 18–50. http://dx.doi.org/10.24820/ark.5550190.p011.237.

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12

Sherstov, Alexander A. "Communication Lower Bounds Using Directional Derivatives." Journal of the ACM 61, no. 6 (December 17, 2014): 1–71. http://dx.doi.org/10.1145/2629334.

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13

Mandelbaum, Avi, and Kavita Ramanan. "Directional Derivatives of Oblique Reflection Maps." Mathematics of Operations Research 35, no. 3 (August 2010): 527–58. http://dx.doi.org/10.1287/moor.1100.0453.

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14

Borisenko, O. F., and L. I. Minchenko. "Directional derivatives of the maximum function." Cybernetics and Systems Analysis 28, no. 2 (1992): 309–12. http://dx.doi.org/10.1007/bf01126219.

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15

Lü, Gui-xia, Hao Wu, and Long-jun Shen. "Extremum of second-order directional derivatives." Applied Mathematics-A Journal of Chinese Universities 26, no. 4 (December 2011): 379–89. http://dx.doi.org/10.1007/s11766-011-2535-7.

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16

Kharchenko, V. K. "On directional derivatives of semiprime rings." Siberian Mathematical Journal 32, no. 6 (1992): 1045–51. http://dx.doi.org/10.1007/bf00971213.

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17

Zhang, Xia, Zun-Quan Xia, and Yan Gao. "Exponential Stabilizability of Switched Systems with Polytopic Uncertainties." Journal of Applied Mathematics 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/853170.

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The exponential stabilizability of switched nonlinear systems with polytopic uncertainties is explored by employing the methods of nonsmooth analysis and the minimum quadratic Lyapunov function. The switchings among subsystems are dependent on the directional derivative along the vertex directions of subsystems. In particular, a sufficient condition for exponential stabilizability of the switched nonlinear systems is established considering the sliding modes and the directional derivatives along sliding modes. Furthermore, the matrix conditions of exponential stabilizability are derived for the case of switched linear system and the numerical example is given to show the validity of the synthesis results.

18

Bowden, Roger J. "Generalising Interest Rate Duration with Directional Derivatives: Direction X and Applications." Management Science 43, no. 5 (May 1997): 586–95. http://dx.doi.org/10.1287/mnsc.43.5.586.

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19

Hu, Guanghui, Wen Dai, Sijin Li, Liyang Xiong, and Guoan Tang. "A Vector Operation to Extract Second-Order Terrain Derivatives from Digital Elevation Models." Remote Sensing 12, no. 19 (September 24, 2020): 3134. http://dx.doi.org/10.3390/rs12193134.

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Terrain derivatives exhibit surface morphology in various aspects. However, existing spatial change calculation methods for terrain derivatives are based on a mathematical scalar operating system, which may disregard the directional property of the original data to a certain extent. This situation is particularly true in second-order terrain derivatives, in which original data can be terrain derivatives with clear directional properties, such as slope or aspect. Thus, this study proposes a mathematical vector operation method for the calculation of second-order terrain derivatives. Given the examples of the first-order terrain derivatives of slope and aspect, their second-order terrain derivatives are calculated using the proposed vector method. Directional properties are considered and vectorized using the following steps: rotation-type judgment, standardization of initial direction, and vector representation. The proposed vector method is applied to one mathematical Gaussian surface and three different ground landform areas using digital elevation models (DEMs) with 5 and 1 m resolutions. Comparison analysis results between the vector and scalar methods show that the former achieves more reasonable and accurate second-order terrain derivatives than the latter. Moreover, the vector method avoids overexpression or even exaggeration errors. This vector operation concept and its expanded methods can be applied in calculating other terrain derivatives in geomorphometry.

20

Singhal, Martand, Alejandro G. Marchetti, Timm Faulwasser, and Dominique Bonvin. "Improved Directional Derivatives for Modifier-Adaptation Schemes." IFAC-PapersOnLine 50, no. 1 (July 2017): 5718–23. http://dx.doi.org/10.1016/j.ifacol.2017.08.1124.

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21

ThÄmelt, W. "Directional derivatives and generalized gradients on manifolds." Optimization 25, no. 2-3 (January 1992): 97–115. http://dx.doi.org/10.1080/02331939208843813.

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22

Treiman, Jay S. "Generalized Gradients, Lipschitz behavior and Directional Derivatives." Canadian Journal of Mathematics 37, no. 6 (December 1, 1985): 1074–84. http://dx.doi.org/10.4153/cjm-1985-058-1.

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In the study of optimization problems it is necessary to consider functions that are not differentiable. This has led to the consideration of generalized gradients and a corresponding calculus for certain classes of functions. Rockafellar [16] and others have developed a very strong and elegant theory of subgradients for convex functions. This convex theory gives point-wise criteria for the existence of extrema in optimization problems.There are however many optimization problems that involve functions which are neither differentiable nor convex. Such functions arise in many settings including optimal value functions [15]. In order to deal with such problems Clarke [3] defined a type of subgradient for nonconvex functions. This definition was initially for Lipschitz functions on R”. Clarke extended this definition to include lower semicontinuous (l.s.c.) functions on Banach spaces through the use of a directional derivative, the distance function from a closed set and tangent and normal cones to closed sets.

23

Li, S. J., K. L. Teo, and X. Q. Yang. "Second-order directional derivatives of spectral functions." Computers & Mathematics with Applications 50, no. 5-6 (September 2005): 947–55. http://dx.doi.org/10.1016/j.camwa.2004.11.021.

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24

Correa, R., and A. Jofre. "Tangentially continuous directional derivatives in nonsmooth analysis." Journal of Optimization Theory and Applications 61, no. 1 (April 1989): 1–21. http://dx.doi.org/10.1007/bf00940840.

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25

Chaney, Robin W. "Second-order directional derivatives for nonsmooth functions." Journal of Mathematical Analysis and Applications 128, no. 2 (December 1987): 495–511. http://dx.doi.org/10.1016/0022-247x(87)90202-2.

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26

Lang, Nguyen Duc, Tran Gia Lich, and Le Duc. "Two approximation methods of spatial derivatives on unstructured triangular meshes and their application in computing two dimensional flows." Vietnam Journal of Mechanics 28, no. 4 (December 31, 2006): 230–40. http://dx.doi.org/10.15625/0866-7136/28/4/5584.

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Two approximation methods (the Green's theorem technique and the directional derivative technique) of spatial derivatives have been proposed for finite differences on unstructured triangular meshes. Both methods have the first order accuracy. A semi-implicit time matching methods beside the third order Adams-Bashforth method are used in integrating the water shallow equations written in both non-conservative and conservative forms. To remove spurious waves, a smooth procedure has been used. The model is tested on rectangular grids triangulari2jed after the 8-neighbours strategy. In the context of the semi-implicit time matching methods, the directional Derivative technique is more accurate than Green's theorem technique. The results from the third order Adams-Bashforth scheme are the most accurate, especially for discontinuous problems. In this case, there is a minor difference between two approximation techniques of spatial derivatives.

27

Apalak, M. Kemal, and M. Didem Demirbas. "Improved Mathematical Models of Thermal Residual Stresses in Functionally Graded Adhesively Bonded Joints: A Critical Review." Reviews of Adhesion and Adhesives 7, no. 4 (December 1, 2019): 367–416. http://dx.doi.org/10.7569/raa.2019.097313.

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Functionally graded material (FGM) concept has been applied successfully in order to improve/design heat transfer, electric and electronic conductivity, static and dynamic strengths of adhesive joints by reliving stress distributions in both adhesive and adherend materials. This new approach relies on tailoring material composition of adhesive and adherends along one or more coordinate directions. Thermal residual stresses in adhesive joints are a vital issue in terms of the joint strength. FGM concept also allows to relieve/control thermal residual stresses encountered in adhesive joints due to mismatches between coefficients of thermal expansion of adhesive and adherend materials. Mathematical models and solutions on the thermal residual stress analysis have been continuously improved. This paper reviews the current status of mathematical models, and offers an improved mathematical model and numerical solution method by considering two-dimensional thermal stress and deformation states of adhesively bonded bi-directional functionally graded clamped plates subjected to an in-plane heat flux along one of the ceramic edges. This mathematical model assumes the material properties of the functionally graded plates to vary with a power law along two in-plane directions and not through the plate thickness direction, in particular, considers the spatial derivatives of thermal and mechanical properties of the material, and enables the investigation of the effects of the bi-directional composition variations and spatial derivative terms on the displacement, strain and stress distributions. The heat conduction and Navier equations describing the twodimensional thermo-elastic problem are discretized using finite-difference method, and the set of linear equations are solved using the pseudo singular value method. The functionally graded plates relieve both stress and strain distributions and levels in the adhesive layer and in the plates even though the adhesive layer is still ungraded. The spatial derivatives of mechanical and thermal properties of the local material become more effective on the strain and stress distributions of the plates and adhesive layer. The model, disregarding these derivative terms, exhibits sensitivity to small changes in the compositional gradients (n, m) by adjusting the variations of ceramic volume fraction along the x - and y-directions, respectively, and instability in the calculation of stress and strain distributions and levels. However, the improved model with material derivatives, which considers the effects of these derivative terms, predicts stress and strain distributions and levels complying with changes in the compositional gradient exponents.

28

Beiki, Majid. "Analytic signals of gravity gradient tensor and their application to estimate source location." GEOPHYSICS 75, no. 6 (November 2010): I59—I74. http://dx.doi.org/10.1190/1.3493639.

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The analytic signal concept can be applied to gravity gradient tensor data in three dimensions. Within the gravity gradient tensor, the horizontal and vertical derivatives of gravity vector components are Hilbert transform pairs. Three analytic signal functions then are introduced along [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-directions. The amplitude of the first vertical derivative of the analytic signals in [Formula: see text]- and [Formula: see text]-directions enhances the edges of causative bodies. The directional analytic signals are homogenous and satisfy Euler’s homogeneity equation. The application of directional analytic signals to Euler deconvolution on generic models demonstrates their ability to locate causative bodies. One of the advantages of this method is that it allows the automatic identification of the structural index from solving three Euler equations derived from the gravity gradient tensor for a collection of data points in a window. The other advantage is a reduction of interference effects from neighboring sources by differentiation of the directional analytic signals in [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-directions. Application of the method is demonstrated on gravity gradient tensor data in the Vredefort impact structure, South Africa.

29

Kuczumow, Rzynowski, and Stachura. "LIPSCHITZIAN HOMEOMORPHISMS WITH LARGE SETS OF DIRECTIONAL DERIVATIVES." Real Analysis Exchange 15, no. 2 (1989): 696. http://dx.doi.org/10.2307/44152044.

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30

Yang, X. Q. "Generalized second-order directional derivatives and optimality conditions." Bulletin of the Australian Mathematical Society 51, no. 1 (February 1995): 175–76. http://dx.doi.org/10.1017/s0004972700014015.

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31

ZHANG, LiWei, XianTao XIAO, and Ning ZHANG. "The second-order directional derivatives of singular values." SCIENTIA SINICA Mathematica 43, no. 2 (January 1, 2013): 121–36. http://dx.doi.org/10.1360/012011-668.

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32

Minchenko, L., and A. Tarakanov. "On second-order directional derivatives of value functions." Optimization 64, no. 2 (January 23, 2013): 389–407. http://dx.doi.org/10.1080/02331934.2012.754441.

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33

Seierstad, A. "Directional derivatives and the control of nonsmooth systems." Optimization 35, no. 1 (January 1995): 61–75. http://dx.doi.org/10.1080/02331939508844127.

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34

Legarda-Sáenz, Ricardo, Mariano Rivera, Ramón Rodríguez-Vera, and Gerardo Trujillo-Schiaffino. "Robust wave-front estimation from multiple directional derivatives." Optics Letters 25, no. 15 (August 1, 2000): 1089. http://dx.doi.org/10.1364/ol.25.001089.

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35

Seeger, Alberto. "Second Order Directional Derivatives in Parametric Optimization Problems." Mathematics of Operations Research 13, no. 1 (February 1988): 124–39. http://dx.doi.org/10.1287/moor.13.1.124.

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36

Sach, Pham Huu, and Jean-Paul Penot. "Characterizations of generalized convexities via generalized directional derivatives." Numerical Functional Analysis and Optimization 19, no. 5-6 (January 1998): 615–34. http://dx.doi.org/10.1080/01630569808816849.

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37

A. Nekvinda and L. Zajíček. "Gâteaux Differentiability of Lipschitz Functions via Directional Derivatives." Real Analysis Exchange 28, no. 2 (2003): 287. http://dx.doi.org/10.14321/realanalexch.28.2.0287.

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38

Audet, Charles, and Warren Hare. "Algorithmic Construction of the Subdifferential from Directional Derivatives." Set-Valued and Variational Analysis 26, no. 3 (October 4, 2016): 431–47. http://dx.doi.org/10.1007/s11228-016-0388-1.

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39

Yang, X. Q. "Directional derivatives for set-valued mappings and applications." Mathematical Methods of Operations Research 48, no. 2 (November 1998): 273–85. http://dx.doi.org/10.1007/s001860050028.

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40

Xie, Ting, Zengtai Gong, and Dapeng Li. "Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions." Open Mathematics 18, no. 1 (December 7, 2020): 1451–77. http://dx.doi.org/10.1515/math-2020-0081.

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Abstract In this paper, we present the concepts of generalized derivative, directional generalized derivative, subdifferential and conjugate for n-dimensional fuzzy-number-valued functions and discuss the characterizations of generalized derivative and directional generalized derivative by, respectively, using the derivative and directional derivative of crisp functions that are determined by the fuzzy mapping. Furthermore, the relations among generalized derivative, directional generalized derivative, subdifferential and convexity for n-dimensional fuzzy-number-valued functions are investigated. Finally, under two kinds of partial orderings defined on the set of all n-dimensional fuzzy numbers, the duality theorems and saddle point optimality criteria in fuzzy optimization problems with constraints are discussed.

41

Maricic, N. L. "Numerical estimation of aircrafts' unsteady lateral-directional stability derivatives." Theoretical and Applied Mechanics 33, no. 4 (2006): 311–37. http://dx.doi.org/10.2298/tam0604311m.

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A technique for predicting steady and oscillatory aerodynamic loads on general configuration has been developed. The prediction is based on the Doublet-Lattice Method, Slender Body Theory and Method of Images. The chord and span wise loading on lifting surfaces and longitudinal bodies (in horizontal and vertical plane) load distributions are determined. The configuration may be composed of an assemblage of lifting surfaces (with control surfaces) and bodies (with circular cross sections and a longitudinal variation of radius). Loadings predicted by this method are used to calculate (estimate) steady and unsteady (dynamic) lateral-directional stability derivatives. The short outline of the used methods is given in [1], [2], [3], [4] and [5]. Applying the described methodology software DERIV is developed. The obtained results from DERIV are compared to NASTRAN examples HA21B and HA21D from [4]. In the first example (HA21B), the jet transport wing (BAH wing) is steady rolling and lateral stability derivatives are determined. In the second example (HA21D), lateral-directional stability derivatives are calculated for forward- swept-wing (FSW) airplane in antisymmetric quasi-steady maneuvers. Acceptable agreement is achieved comparing the results from [4] and DERIV.

42

Bartlett, Michael O., John R. Giles, and Jon D. Vanderwerff. "Structural implications of norms with Hölder right-hand derivatives." Bulletin of the Australian Mathematical Society 57, no. 3 (June 1998): 415–25. http://dx.doi.org/10.1017/s000497270003183x.

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We study a nonsmooth extension of Gateaux differentiability satisfying a directional Hölder condition. In particular, we show that a Banach space is an Asplund space if it has an equivalent norm with a directionally Hölder right-hand derivative at each point of its sphere.

43

Wu, Xinming. "Directional structure-tensor-based coherence to detect seismic faults and channels." GEOPHYSICS 82, no. 2 (March 1, 2017): A13—A17. http://dx.doi.org/10.1190/geo2016-0473.1.

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Seismic coherence is widely used in seismic interpretation and reservoir characterization to highlight (with low values) faults and stratigraphic features from a seismic image. A coherence image can be computed from the eigenvalues of conventional structure tenors, which are outer products of gradients of a seismic image. I have developed a simple but effective method to improve such a coherence image by using directional structure tensors, which are different from the conventional structure tensors in only two aspects. First, instead of using image gradients with vertical and horizontal derivatives, I use directional derivatives, computed in directions perpendicular and parallel to seismic structures (reflectors), to construct directional structure tensors. With these directional derivatives, lateral seismic discontinuities, especially those subtle stratigraphic features aligned within dipping structures, can be better captured in the structure tensors. Second, instead of applying Gaussian smoothing to each element of the constructed structure tensors, I apply approximately fault- and stratigraphy-oriented smoothing to enhance the lateral discontinuities corresponding to faults and stratigraphic features in the structure tensors. Real 3D examples show that the new coherence images computed from such structure tensors display much cleaner and more continuous faults and stratigraphic features compared with those computed from conventional structure tensors and covariance matrices.

44

Kim, Ji-Eun. "Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System." Axioms 10, no. 3 (August 30, 2021): 206. http://dx.doi.org/10.3390/axioms10030206.

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The step derivative of a complex function can be defined with various methods. The step direction defines a basis that is distinct from that of a complex number; the derivative can then be treated by using Taylor series expansion in this direction. In this study, we define step derivatives based on complex numbers and quaternions that are orthogonal to the complex basis while simultaneously being distinct from it. Considering previous studies, the step derivative defined using quaternions was insufficient for applying the properties of quaternions by setting a quaternion basis distinct from the complex basis or setting the step direction to which only a part of the quaternion basis was applied. Therefore, in this study, we examine the definition of quaternions and define the step derivative in the direction of a generalized quaternion basis including a complex basis. We find that the step derivative based on the definition of a quaternion has a relative error in some domains; however, it can be used as a substitute derivative in specific domains.

45

Zhang, W. Y., S. Xu, and S. J. Li. "Necessary Conditions for Weak Sharp Minima in Cone-Constrained Optimization Problems." Abstract and Applied Analysis 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/909520.

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We study weak sharp minima for optimization problems with cone constraints. Some necessary conditions for weak sharp minima of higher order are established by means of upper Studniarski or Dini directional derivatives. In particular, when the objective and constrained functions are strict derivative, a necessary condition is obtained by a normal cone.

46

Bandura, A. I. "Some weaker sufficient conditions of $L$-index boundedness in direction for functions analytic in the unit ball." Carpathian Mathematical Publications 11, no. 1 (June 30, 2019): 14–25. http://dx.doi.org/10.15330/cmp.11.1.14-25.

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We partially reinforce some criteria of $L$-index boundedness in direction for functions analytic in the unit ball. These results describe local behavior of directional derivatives on the circle, estimates of maximum modulus, minimum modulus of analytic function, distribution of its zeros and modulus of directional logarithmic derivative of analytic function outside some exceptional set. Replacement of universal quantifier on existential quantifier gives new weaker sufficient conditions of $L$-index boundedness in direction for functions analytic in the unit ball. The results are also new for analytic functions in the unit disc. The logarithmic criterion has applications in analytic theory of differential equations. This is convenient to investigate index boundedness for entire solutions of linear differential equations. It is also apllicable to infinite products.Auxiliary class of positive continuous functions in the unit ball (so-denoted $Q_{\mathbf{b}}(\mathbb{B}^n)$) is also considered. There are proved some characterizing properties of these functions. The properties describe local behavior of these functions in the polydisc neighborhood of every point from the unit ball.

47

Pastor, Karel. "On relations among the generalized second-order directional derivatives." Discussiones Mathematicae. Differential Inclusions, Control and Optimization 21, no. 2 (2001): 235. http://dx.doi.org/10.7151/dmdico.1026.

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48

Joy, Tinu Theckel, Santu Rana, Sunil Gupta, and Svetha Venkatesh. "Fast hyperparameter tuning using Bayesian optimization with directional derivatives." Knowledge-Based Systems 205 (October 2020): 106247. http://dx.doi.org/10.1016/j.knosys.2020.106247.

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49

Khachatryan, R. A. "On directional derivatives of selections of set-valued mappings." Journal of Contemporary Mathematical Analysis 51, no. 3 (May 2016): 148–59. http://dx.doi.org/10.3103/s1068362316030055.

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50

Legarda-Saenz, Ricardo. "Wavefront reconstruction using multiple directional derivatives and Fourier transform." Optical Engineering 50, no. 4 (April 1, 2011): 040501. http://dx.doi.org/10.1117/1.3560540.

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