.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_numerical_methods/optimization/plot_optimization_rosenbrock.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:`here ` to download the full example code .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_numerical_methods_optimization_plot_optimization_rosenbrock.py: Quick start guide to optimization ================================= .. GENERATED FROM PYTHON SOURCE LINES 6-26 In this example, we perform the optimization of the Rosenbrock test function. Let :math:`a, b\in\mathbb{R}` be parameters. The Rosenbrock function is defined by .. math:: f(x_1, x_2) = (a-x_1)^2 + b(x_2 - x_1^2)^2 for any :math:`\mathbf{x}\in\mathbb{R}^2`. This function is often used with :math:`a=1` and :math:`b=100`. In this case, the function has a single global minimum at: .. math:: \mathbf{x}^\star = (1, 1)^T. This function has a nonlinear least squares structure. *References* * Rosenbrock, H.H. (1960). "An automatic method for finding the greatest or least value of a function". The Computer Journal. 3 (3): 175–184. .. GENERATED FROM PYTHON SOURCE LINES 29-31 Definition of the function -------------------------- .. GENERATED FROM PYTHON SOURCE LINES 33-38 .. code-block:: default import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE) .. GENERATED FROM PYTHON SOURCE LINES 39-42 .. code-block:: default rosenbrock = ot.SymbolicFunction(['x1', 'x2'], ['(1-x1)^2+100*(x2-x1^2)^2']) .. GENERATED FROM PYTHON SOURCE LINES 43-45 .. code-block:: default x0 = [-1.0, 1.0] .. GENERATED FROM PYTHON SOURCE LINES 46-48 .. code-block:: default xexact = ot.Point([1.0, 1.0]) .. GENERATED FROM PYTHON SOURCE LINES 49-52 .. code-block:: default lowerbound = [-2.0, -2.0] upperbound = [2.0, 2.0] .. GENERATED FROM PYTHON SOURCE LINES 53-55 Plot the iso-values of the objective function --------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 57-59 .. code-block:: default rosenbrock = ot.MemoizeFunction(rosenbrock) .. GENERATED FROM PYTHON SOURCE LINES 60-64 .. code-block:: default graph = rosenbrock.draw(lowerbound, upperbound, [100]*2) graph.setTitle("Rosenbrock function") view = viewer.View(graph) .. image-sg:: /auto_numerical_methods/optimization/images/sphx_glr_plot_optimization_rosenbrock_001.png :alt: Rosenbrock function :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_optimization_rosenbrock_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 65-66 We see that the minimum is on the top right of the picture and the starting point is on the top left of the picture. Since the function has a long valley following the curve :math:`x_2 - x^2=0`, the algorithm generally have to follow the bottom of the valley. .. GENERATED FROM PYTHON SOURCE LINES 68-70 Create and solve the optimization problem ----------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 72-74 .. code-block:: default problem = ot.OptimizationProblem(rosenbrock) .. GENERATED FROM PYTHON SOURCE LINES 75-81 .. code-block:: default algo = ot.Cobyla(problem) algo.setMaximumRelativeError(1.e-1) # on x algo.setMaximumEvaluationNumber(50000) algo.setStartingPoint(x0) algo.run() .. GENERATED FROM PYTHON SOURCE LINES 82-84 .. code-block:: default result = algo.getResult() .. GENERATED FROM PYTHON SOURCE LINES 85-88 .. code-block:: default xoptim = result.getOptimalPoint() xoptim .. raw:: html

[0.99251,0.985022]



.. GENERATED FROM PYTHON SOURCE LINES 89-93 .. code-block:: default delta = xexact - xoptim absoluteError = delta.norm() absoluteError .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 0.016745946097259285 .. GENERATED FROM PYTHON SOURCE LINES 94-95 We see that the algorithm found an accurate approximation of the solution. .. GENERATED FROM PYTHON SOURCE LINES 97-99 .. code-block:: default result.getOptimalValue() # f(x*) .. raw:: html

[5.6392e-05]



.. GENERATED FROM PYTHON SOURCE LINES 100-102 .. code-block:: default result.getEvaluationNumber() .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 10520 .. GENERATED FROM PYTHON SOURCE LINES 103-111 .. code-block:: default graph = rosenbrock.draw(lowerbound, upperbound, [100]*2) cloud = ot.Cloud(ot.Sample([x0, xoptim])) cloud.setColor("black") cloud.setPointStyle("bullet") graph.add(cloud) graph.setTitle("Rosenbrock function") view = viewer.View(graph) .. image-sg:: /auto_numerical_methods/optimization/images/sphx_glr_plot_optimization_rosenbrock_002.png :alt: Rosenbrock function :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_optimization_rosenbrock_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 112-113 We see that the algorithm had to start from the top left of the banana and go to the top right. .. GENERATED FROM PYTHON SOURCE LINES 115-118 .. code-block:: default graph = result.drawOptimalValueHistory() view = viewer.View(graph) .. image-sg:: /auto_numerical_methods/optimization/images/sphx_glr_plot_optimization_rosenbrock_003.png :alt: Optimal value history :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_optimization_rosenbrock_003.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 119-120 The function value history make the path of the algorithm clear. In the first step, the algorithm went in the valley, which made the function value decrease rapidly. Once there, the algorithm had to follow the bottom of the valley so that the function decreased but slowly. In the final steps, the algorithm found the neighbourhood of the minimum so that the local convergence could take place. .. GENERATED FROM PYTHON SOURCE LINES 122-123 In order to see where the function was evaluated, we use the `getInputSample` method. .. GENERATED FROM PYTHON SOURCE LINES 125-127 .. code-block:: default inputSample = result.getInputSample() .. GENERATED FROM PYTHON SOURCE LINES 128-134 .. code-block:: default graph = rosenbrock.draw(lowerbound, upperbound, [100]*2) graph.setTitle("Rosenbrock function solved with Cobyla") cloud = ot.Cloud(inputSample) graph.add(cloud) view = viewer.View(graph) .. image-sg:: /auto_numerical_methods/optimization/images/sphx_glr_plot_optimization_rosenbrock_004.png :alt: Rosenbrock function solved with Cobyla :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_optimization_rosenbrock_004.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 135-136 We see that the algorithm made lots of evaluations in the bottom of the valley before getting in the neighbourhood of the minimum. .. GENERATED FROM PYTHON SOURCE LINES 138-142 Solving the problem with NLopt ------------------------------ We see that the `Cobyla` algorithm required lots of function evaluations. This is why we now use the `NLopt` class with the LBFGS algorithm. However, the algorithm may use input points which are far away from the input domain we used so far. This is why we had bounds to the problem, so that the algorithm never goes to far away from the valley. .. GENERATED FROM PYTHON SOURCE LINES 144-146 .. code-block:: default bounds = ot.Interval(lowerbound, upperbound) .. GENERATED FROM PYTHON SOURCE LINES 147-150 .. code-block:: default problem = ot.OptimizationProblem(rosenbrock) problem.setBounds(bounds) .. GENERATED FROM PYTHON SOURCE LINES 151-155 .. code-block:: default algo = ot.NLopt(problem, 'LD_LBFGS') algo.setStartingPoint(x0) algo.run() .. GENERATED FROM PYTHON SOURCE LINES 156-158 .. code-block:: default result = algo.getResult() .. GENERATED FROM PYTHON SOURCE LINES 159-162 .. code-block:: default xoptim = result.getOptimalPoint() xoptim .. raw:: html

[1,1]



.. GENERATED FROM PYTHON SOURCE LINES 163-167 .. code-block:: default delta = xexact - xoptim absoluteError = delta.norm() absoluteError .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 7.740583643426769e-12 .. GENERATED FROM PYTHON SOURCE LINES 168-169 We see that the algorithm found an extremely accurate approximation of the solution. .. GENERATED FROM PYTHON SOURCE LINES 171-173 .. code-block:: default result.getOptimalValue() # f(x*) .. raw:: html

[1.77616e-23]



.. GENERATED FROM PYTHON SOURCE LINES 174-176 .. code-block:: default result.getEvaluationNumber() .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 44 .. GENERATED FROM PYTHON SOURCE LINES 177-178 This number of iterations is much less than the previous experiment. .. GENERATED FROM PYTHON SOURCE LINES 180-182 .. code-block:: default inputSample = result.getInputSample() .. GENERATED FROM PYTHON SOURCE LINES 183-190 .. code-block:: default graph = rosenbrock.draw(lowerbound, upperbound, [100]*2) graph.setTitle("Rosenbrock function solved with NLopt/LD_LBFGS") cloud = ot.Cloud(inputSample) graph.add(cloud) view = viewer.View(graph) plt.show() .. image-sg:: /auto_numerical_methods/optimization/images/sphx_glr_plot_optimization_rosenbrock_005.png :alt: Rosenbrock function solved with NLopt/LD_LBFGS :srcset: /auto_numerical_methods/optimization/images/sphx_glr_plot_optimization_rosenbrock_005.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 0.559 seconds) .. _sphx_glr_download_auto_numerical_methods_optimization_plot_optimization_rosenbrock.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_optimization_rosenbrock.py ` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_optimization_rosenbrock.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_