# Styblinski-Tang Function Styblinski-Tang function is defined by [REMOVE_ME]$$ f_{\rm styblinski}(x_1, ..., x_d) = \frac{1}{2} \sum_{k = 1}^{d} (x_k^4 - 16x_k^2 + 5x_k) [REMOVE_ME_2]$$ with $x_k \in [-5, 5]$ for $k = 1, ..., d$. UTF-8 ## Description Styblinski-Tang function is defined by $$ f_{\rm styblinski}(x_1, ..., x_d) = \frac{1}{2} \sum_{k = 1}^{d} (x_k^4 - 16x_k^2 + 5x_k) $$ with $x_k \in [-5, 5]$ for $k = 1, ..., d$. ```r styblinski(x) styblinskiGrad(x) ``` ## Arguments - `x`: a numeric `vector` or a numeric `matrix` with `n` rows and `d` columns. If a `vector` is passed, the 1-dimensional version of the Rastrigin function is calculated. ## Details The gradient of Styblinski-Tang function is $$ \nabla f_{\rm styblinski}(x_1, ..., x_d) = \begin{pmatrix} 2x_1^3 - 16 x_1 + 2.5\\ \vdots \\ 2 x_d^3 - 16 x_d + 2.5 \end{pmatrix}. $$ Styblinski-Tang function has one global minimum $f_{\rm styblinski}(x^{\star}) = -39.16599d$ at $x^{\star} = (-2.903534,\dots, -2.903534)$. ## Returns `styblinski` returns the function value of Styblinski-Tang function at `x`. `styblinskiGrad` returns the gradient of Styblinski-Tang function at `x`. ## References Jamil, M. and Yang, X.-S. (2013). A Literature Survey of Benchmark Functions for Global Optimization Problems. **International Journal of Mathematical Modelling and Numerical Optimisation**, 4 (2):150-–194. tools:::Rd_expr_doi("10.1504/IJMMNO.2013.055204") . Surjanovic, S. and Bingham, D. (2013). Virtual Library of Simulation Experiments: Test Functions and Datasets. [https://www.sfu.ca/~ssurjano/](https://www.sfu.ca/~ssurjano/) (retrieved January 19, 2024). ## Author(s) Carmen van Meegen ## See Also `tangents` for drawing tangent lines. ## Examples ```r # 1-dimensional Styblinski-Tang function with tangents curve(styblinski(x), from = -5, to = 5) x <- seq(-4.5, 4.5, length = 5) y <- styblinski(x) dy <- styblinskiGrad(x) tangents(x, y, dy, length = 2, lwd = 2, col = "red") points(x, y, pch = 16) # Contour plot of Styblinski-Tang function n.grid <- 50 x1 <- seq(-5, 5, length.out = n.grid) x2 <- seq(-5, 5, length.out = n.grid) y <- outer(x1, x2, function(x1, x2) styblinski(cbind(x1, x2))) contour(x1, x2, y, xaxs = "i", yaxs = "i", nlevels = 25, xlab = "x1", ylab = "x2") # Perspective plot of Styblinski-Tang function col.pal <- colorRampPalette(c("#00007F", "blue", "#007FFF", "cyan", "#7FFF7F", "yellow", "#FF7F00", "red", "#7F0000")) colors <- col.pal(100) y.facet.center <- (y[-1, -1] + y[-1, -n.grid] + y[-n.grid, -1] + y[-n.grid, -n.grid])/4 y.facet.range <- cut(y.facet.center, 100) persp(x1, x2, y, phi = 30, theta = -315, expand = 0.75, ticktype = "detailed", col = colors[y.facet.range]) ```

styblinski function