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Add the Rosenbrock function for optimization and metamodeling
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| Original file line number | Diff line number | Diff line change |
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| --- | ||
| jupytext: | ||
| formats: ipynb,md:myst | ||
| text_representation: | ||
| extension: .md | ||
| format_name: myst | ||
| format_version: 0.13 | ||
| jupytext_version: 1.14.1 | ||
| kernelspec: | ||
| display_name: Python 3 (ipykernel) | ||
| language: python | ||
| name: python3 | ||
| --- | ||
|
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| (test-functions:rosenbrock)= | ||
| # Rosenbrock Function | ||
|
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| The Rosenbrock function, originally introduced in {cite}`Rosenbrock1960` | ||
| as a two-dimensional scalar-valued test function for global optimization, | ||
| was later generalized to $M$ dimensions. | ||
| It has since become a widely used benchmark for global optimization methods | ||
| (e.g., {cite}`Dixon1978, Picheny2013`). | ||
| In {cite}`Tan2015`, the function was employed in a metamodeling exercise. | ||
|
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| The function is also known as the valley function or the banana function. | ||
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| ```{code-cell} ipython3 | ||
| import numpy as np | ||
| import matplotlib.pyplot as plt | ||
| import uqtestfuns as uqtf | ||
| ``` | ||
|
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| The surface and contour plots for the two-dimensional Rosenbrock function are | ||
| shown below for the default parameter set and for $x \in [-2, 2] \times [-1, 3]$. | ||
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| As shown, the function features a curved, non-convex valley. | ||
| While it is relatively easy to reach the valley, the convergence to the global | ||
| minimum is difficult. | ||
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| ```{code-cell} ipython3 | ||
| :tags: [remove-input] | ||
| from mpl_toolkits.axes_grid1 import make_axes_locatable | ||
| # --- Create 2D data | ||
| x1_1d = np.arange(-2, 2, 0.05) | ||
| x2_1d = np.arange(-1, 3, 0.05) | ||
| my_fun = uqtf.Rosenbrock(input_dimension=2) | ||
| mesh_2d = np.meshgrid(x1_1d, x2_1d) | ||
| xx_2d = np.array(mesh_2d).T.reshape(-1, 2) | ||
| yy_2d = my_fun(xx_2d) | ||
| # --- Create two-dimensional plots | ||
| fig = plt.figure(figsize=(10, 10)) | ||
| # Surface | ||
| axs_1 = plt.subplot(121, projection='3d') | ||
| axs_1.plot_surface( | ||
| mesh_2d[0], | ||
| mesh_2d[1], | ||
| yy_2d.reshape(len(x1_1d), len(x2_1d)).T, | ||
| linewidth=0, | ||
| cmap="plasma", | ||
| antialiased=False, | ||
| alpha=0.5 | ||
| ) | ||
| axs_1.set_xlabel("$x_1$", fontsize=14) | ||
| axs_1.set_ylabel("$x_2$", fontsize=14) | ||
| axs_1.set_title("Surface plot of 2D Rosenbrock", fontsize=14) | ||
| # Contour | ||
| axs_2 = plt.subplot(122) | ||
| cf = axs_2.contour( | ||
| mesh_2d[0], mesh_2d[1], yy_2d.reshape(len(x1_1d), len(x2_1d)).T, | ||
| levels=1000, cmap="plasma", | ||
| ) | ||
| axs_2.set_xlabel("$x_1$", fontsize=14) | ||
| axs_2.set_ylabel("$x_2$", fontsize=14) | ||
| axs_2.set_title("Contour plot of 2D Rosenbrock", fontsize=14) | ||
| divider = make_axes_locatable(axs_2) | ||
| cax = divider.append_axes('right', size='5%', pad=0.05) | ||
| fig.colorbar(cf, cax=cax, orientation='vertical') | ||
| axs_2.axis('scaled') | ||
| fig.tight_layout(pad=3.0) | ||
| plt.gcf().set_dpi(75); | ||
| ``` | ||
|
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| ## Test function instance | ||
|
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| To create a default instance of the test function, type: | ||
|
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| ```{code-cell} ipython3 | ||
| my_testfun = uqtf.Rosenbrock() | ||
| ``` | ||
|
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| Check if it has been correctly instantiated: | ||
|
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| ```{code-cell} ipython3 | ||
| print(my_testfun) | ||
| ``` | ||
|
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| By default, the input dimension is set to $2$[^default_dimension]. | ||
| To create an instance with another value of input dimension, | ||
| pass an integer to the parameter `input_dimension` (keyword only). | ||
| For example, to create an instance of 10-dimensional Rosenbrock function, type: | ||
|
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| ```python | ||
| my_testfun = uqtf.Ackley(input_dimension=10) | ||
| ``` | ||
|
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| ## Description | ||
|
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| The generalized Rosenbrock function is defined as follows: | ||
|
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| $$ | ||
| \mathcal{M}(\boldsymbol{x}; \boldsymbol{p}) = \frac{1}{d} \left[ | ||
| \left( \sum_{i = 1}^{M - 1} (x_i - a)^2 + b \, (x_{i + 1} - x_i^2)^2 \right) | ||
| - c \right] | ||
| $$ | ||
|
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||
| where $\boldsymbol{x} = \{ x_1, \ldots, x_M \}$ is the $M$-dimensional vector | ||
| of input variables, as defined below, and | ||
| $\boldsymbol{p} = \{ a, b, c, d \}$ is the set of parameters also defined | ||
| further below. | ||
|
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| ```{info} | ||
| For $M < 2$, the Rosenbrock function returns a constant $0$ for any $boldsymbol{x}$. | ||
| ``` | ||
|
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| ## Input | ||
|
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| The Rosenbrock function is defined on $\mathbb{R}^M$, but in the context | ||
| of optimization test function, the search space is often limited as shown | ||
| in the table below. | ||
|
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| ```{code-cell} ipython3 | ||
| :tags: [hide-input] | ||
| print(my_testfun.prob_input) | ||
| ``` | ||
|
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| ## Parameters | ||
|
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| The Rosenbrock function requires four additional parameters | ||
| to complete the specification. | ||
| The default values are shown below. | ||
|
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| ```{code-cell} ipython3 | ||
| :tags: [hide-input] | ||
| print(my_testfun.parameters) | ||
| ``` | ||
|
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| The parameter $a$ controls the location and value of the global optimum. | ||
| The parameter $b$ controls the steepness and width of the valley. | ||
| In particular, it determines the scale of variation of the function; large | ||
| value of $b$ creates a steeper and tight valley. | ||
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| The parameters $c$ and $d$ are used to shift and scale the function such that | ||
| its mean and standard deviation become $0.0$ and $1.0$, respectively. | ||
|
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| Below are some contour plots of the function in two dimensions with different | ||
| values of $a$ and $b$ along with the location of the global optimum. | ||
|
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| ```{code-cell} ipython3 | ||
| :tags: [remove-input] | ||
| from mpl_toolkits.axes_grid1 import make_axes_locatable | ||
| # --- Create 2D data | ||
| x1_1d = np.arange(-2, 2, 0.05) | ||
| x2_1d = np.arange(-1, 3, 0.05) | ||
| mesh_2d = np.meshgrid(x1_1d, x2_1d) | ||
| xx_2d = np.array(mesh_2d).T.reshape(-1, 2) | ||
| # --- Create contour plots | ||
| fig, axs = plt.subplots(2, 2, figsize=(10, 10)) | ||
| # a = 1.0, b = 100.0 | ||
| my_fun = uqtf.Rosenbrock(input_dimension=2) | ||
| yy_2d = my_fun(xx_2d) | ||
| a = my_fun.parameters["a"] | ||
| b = my_fun.parameters["b"] | ||
| cf = axs[0, 0].contour( | ||
| mesh_2d[0], mesh_2d[1], yy_2d.reshape(len(x1_1d), len(x2_1d)).T, | ||
| levels=1000, cmap="plasma", | ||
| ) | ||
| axs[0, 0].set_xlabel("$x_1$", fontsize=14) | ||
| axs[0, 0].set_ylabel("$x_2$", fontsize=14) | ||
| axs[0, 0].set_title(f"a = {a}, b = {b}", fontsize=14) | ||
| divider = make_axes_locatable(axs[0, 0]) | ||
| cax = divider.append_axes('right', size='5%', pad=0.05) | ||
| fig.colorbar(cf, cax=cax, orientation='vertical') | ||
| axs[0, 0].axis('scaled') | ||
| axs[0, 0].scatter(a, a**2, marker="x", s=150, color="k") | ||
| # a = 1.6, b = 100.0 | ||
| my_fun = uqtf.Rosenbrock(input_dimension=2) | ||
| my_fun.parameters["a"] = 1.6 | ||
| yy_2d = my_fun(xx_2d) | ||
| a = my_fun.parameters["a"] | ||
| b = my_fun.parameters["b"] | ||
| cf = axs[0, 1].contour( | ||
| mesh_2d[0], mesh_2d[1], yy_2d.reshape(len(x1_1d), len(x2_1d)).T, | ||
| levels=1000, cmap="plasma", | ||
| ) | ||
| axs[0, 1].set_xlabel("$x_1$", fontsize=14) | ||
| axs[0, 1].set_ylabel("$x_2$", fontsize=14) | ||
| axs[0, 1].set_title(f"a = {a}, b = {b}", fontsize=14) | ||
| divider = make_axes_locatable(axs[0, 1]) | ||
| cax = divider.append_axes('right', size='5%', pad=0.05) | ||
| fig.colorbar(cf, cax=cax, orientation='vertical') | ||
| axs[0, 1].axis('scaled') | ||
| axs[0, 1].scatter(a, a**2, marker="x", s=150, color="k") | ||
| # a = 1.0, b = 500.0 | ||
| my_fun = uqtf.Rosenbrock(input_dimension=2) | ||
| my_fun.parameters["b"] = 500.0 | ||
| yy_2d = my_fun(xx_2d) | ||
| a = my_fun.parameters["a"] | ||
| b = my_fun.parameters["b"] | ||
| cf = axs[1, 0].contour( | ||
| mesh_2d[0], mesh_2d[1], yy_2d.reshape(len(x1_1d), len(x2_1d)).T, | ||
| levels=1000, cmap="plasma", | ||
| ) | ||
| axs[1, 0].set_xlabel("$x_1$", fontsize=14) | ||
| axs[1, 0].set_ylabel("$x_2$", fontsize=14) | ||
| axs[1, 0].set_title(f"a = {a}, b = {b}", fontsize=14) | ||
| divider = make_axes_locatable(axs[1, 0]) | ||
| cax = divider.append_axes('right', size='5%', pad=0.05) | ||
| fig.colorbar(cf, cax=cax, orientation='vertical') | ||
| axs[1, 0].axis('scaled') | ||
| axs[1, 0].scatter(a, a**2, marker="x", s=150, color="k") | ||
| # a = 1.6, b = 500.0 | ||
| my_fun = uqtf.Rosenbrock(input_dimension=2) | ||
| my_fun.parameters["a"] = 1.6 | ||
| my_fun.parameters["b"] = 500.0 | ||
| yy_2d = my_fun(xx_2d) | ||
| a = my_fun.parameters["a"] | ||
| b = my_fun.parameters["b"] | ||
| cf = axs[1, 1].contour( | ||
| mesh_2d[0], mesh_2d[1], yy_2d.reshape(len(x1_1d), len(x2_1d)).T, | ||
| levels=1000, cmap="plasma", | ||
| ) | ||
| axs[1, 1].set_xlabel("$x_1$", fontsize=14) | ||
| axs[1, 1].set_ylabel("$x_2$", fontsize=14) | ||
| axs[1, 1].set_title(f"a = {a}, b = {b}", fontsize=14) | ||
| divider = make_axes_locatable(axs[1, 1]) | ||
| cax = divider.append_axes('right', size='5%', pad=0.05) | ||
| fig.colorbar(cf, cax=cax, orientation='vertical') | ||
| axs[1, 1].axis('scaled') | ||
| axs[1, 1].scatter(a, a**2, marker="x", s=150, color="k") | ||
| fig.tight_layout() | ||
| plt.gcf().set_dpi(75); | ||
| ``` | ||
|
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| As mentioned earlier, the changing the value of $a$ modifies the location | ||
| of the global optimum (and in $M > 2$ modifies the function value at the | ||
| optimum). Changing the value of $b$, on the other hand, modifies the scale | ||
| of the function variation and the steepness of the valley (see the color bar). | ||
|
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| ## Reference results | ||
|
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| This section provides several reference results related to the test function. | ||
|
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| ### Optimum values | ||
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| The global optimum of the Rosenbrock function is located at | ||
|
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| $$ | ||
| \begin{aligned} | ||
| \boldsymbol{x}^* & = \left( x^*_1, \ldots, x^*_M \right), \\ | ||
| x_i^* & = a^{2^{i - 1}}. | ||
| \end{aligned} | ||
| $$ | ||
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| In dimension two, the function value at the optimum location is always $0.0$ | ||
| (but not in higher dimension!). | ||
|
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| ## References | ||
|
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| ```{bibliography} | ||
| :style: unsrtalpha | ||
| :filter: docname in docnames | ||
| ``` | ||
|
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| [^default_dimension]: This default dimension applies to all variable dimension | ||
| test functions. It will be used if the `input_dimension` argument is not given. | ||
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| [^location]: The original two-dimensional expression can be found in Eq. (10), | ||
| Section 3.2, in {cite}`Rosenbrock1960`. |
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