{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Global optimization using simplicial homology global optimization " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "1" }, { "__type": "Text", "__tag": 4046, "value": ". Appropriate for solving general purpose NLP and blackbox optimization problems to global optimality (low-dimensional problems)." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "In general, the optimization problems are of the form " } ] }, { "__type": "Code", "__tag": 4050, "value": "minimize f(x) subject to\n\ng_i(x) >= 0, i = 1,...,m\nh_j(x) = 0, j = 1,...,p", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where x is a vector of one or more variables. 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The default " }, { "__type": "InlineCode", "__tag": 4051, "value": "simplicial" }, { "__type": "Text", "__tag": 4046, "value": " provides the theoretical guarantee of convergence to the global minimum in finite time. " }, { "__type": "InlineCode", "__tag": 4051, "value": "halton" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineCode", "__tag": 4051, "value": "sobol" }, { "__type": "Text", "__tag": 4046, "value": " method are faster in terms of sampling point generation at the cost of the loss of guaranteed convergence. It is more appropriate for most \"easier\" problems where the convergence is relatively fast. 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Supply -1 to use all available CPU cores, or an int to use that many Processes (uses " }, { "__type": "InlineRole", "__tag": 4003, "value": "multiprocessing.Pool ", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ")." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Alternatively supply a map-like callable, such as " }, { "__type": "InlineRole", "__tag": 4003, "value": "multiprocessing.Pool.map", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " for parallel evaluation. This evaluation is carried out as " }, { "__type": "InlineCode", "__tag": 4051, "value": "workers(func, iterable)" }, { "__type": "Text", "__tag": 4046, "value": ". Requires that " }, { "__type": "ParamRef", "__tag": 4071, "name": "func" }, { "__type": "Text", "__tag": 4046, "value": " be pickleable." } ] }, { "__type": "Admonition", "__tag": 4056, "kind": "versionadded", "base_type": "neutral", "children": [ { "__type": "AdmonitionTitle", "__tag": 4055, "children": [ { "__type": "Text", "__tag": 4046, "value": "versionadded 1.11.0" } ] } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "SHGO stands for \"simplicial homology global optimization\"." } ] } ], "title": [], "level": 0, "target": null }, "Other Parameters": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null } }, "_ordered_sections": [ "Summary", "Extended Summary", "Parameters", "Attributes", "Methods", "Returns", "Yields", "Receives", "Other Parameters", "Raises", "Warns", "Warnings", "Notes" ], "item_file": "/scipy/optimize/_shgo.py", "item_line": 22, "item_type": "function", "aliases": [ "scipy.optimize.shgo" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "First consider the problem of minimizing the Rosenbrock function, `rosen`:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "from scipy.optimize import rosen, shgo\nbounds = [(0,2), (0, 2), (0, 2), (0, 2), (0, 2)]\nresult = shgo(rosen, bounds)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "result.x, result.fun\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nNote that bounds determine the dimensionality of the objective\nfunction and is therefore a required input, however you can specify\nempty bounds using ``None`` or objects like ``np.inf`` which will be\nconverted to large float numbers.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "bounds = [(None, None), ]*4\nresult = shgo(rosen, bounds)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "result.x\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nNext, we consider the Eggholder function, a problem with several local\nminima and one global minimum. We will demonstrate the use of arguments and\nthe capabilities of `shgo`.\n(https://en.wikipedia.org/wiki/Test_functions_for_optimization)\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\ndef eggholder(x):\n return (-(x[1] + 47.0)\n * np.sin(np.sqrt(abs(x[0]/2.0 + (x[1] + 47.0))))\n - x[0] * np.sin(np.sqrt(abs(x[0] - (x[1] + 47.0))))\n )\nbounds = [(-512, 512), (-512, 512)]\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\n`shgo` has built-in low discrepancy sampling sequences. First, we will\ninput 64 initial sampling points of the *Sobol'* sequence:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "result = shgo(eggholder, bounds, n=64, sampling_method='sobol')\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "result.x, result.fun\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\n`shgo` also has a return for any other local minima that was found, these\ncan be called using:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "result.xl\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\n" }, { "__type": "Code", "__tag": 4050, "value": "result.funl\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nThese results are useful in applications where there are many global minima\nand the values of other global minima are desired or where the local minima\ncan provide insight into the system (for example morphologies\nin physical chemistry [4]_).\n\nIf we want to find a larger number of local minima, we can increase the\nnumber of sampling points or the number of iterations. We'll increase the\nnumber of sampling points to 64 and the number of iterations from the\ndefault of 1 to 3. Using ``simplicial`` this would have given us\n64 x 3 = 192 initial sampling points.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "result_2 = shgo(eggholder,\n bounds, n=64, iters=3, sampling_method='sobol')\nlen(result.xl), len(result_2.xl)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nNote the difference between, e.g., ``n=192, iters=1`` and ``n=64,\niters=3``.\nIn the first case the promising points contained in the minimiser pool\nare processed only once. In the latter case it is processed every 64\nsampling points for a total of 3 times.\n\nTo demonstrate solving problems with non-linear constraints consider the\nfollowing example from Hock and Schittkowski problem 73 (cattle-feed)\n[3]_::\n\n minimize: f = 24.55 * x_1 + 26.75 * x_2 + 39 * x_3 + 40.50 * x_4\n\n subject to: 2.3 * x_1 + 5.6 * x_2 + 11.1 * x_3 + 1.3 * x_4 - 5 >= 0,\n\n 12 * x_1 + 11.9 * x_2 + 41.8 * x_3 + 52.1 * x_4 - 21\n -1.645 * sqrt(0.28 * x_1**2 + 0.19 * x_2**2 +\n 20.5 * x_3**2 + 0.62 * x_4**2) >= 0,\n\n x_1 + x_2 + x_3 + x_4 - 1 == 0,\n\n 1 >= x_i >= 0 for all i\n\nThe approximate answer given in [3]_ is::\n\n f([0.6355216, -0.12e-11, 0.3127019, 0.05177655]) = 29.894378\n\n" }, { "__type": "Code", "__tag": 4050, "value": "def f(x): # (cattle-feed)\n return 24.55*x[0] + 26.75*x[1] + 39*x[2] + 40.50*x[3]\ndef g1(x):\n return 2.3*x[0] + 5.6*x[1] + 11.1*x[2] + 1.3*x[3] - 5 # >=0\ndef g2(x):\n return (12*x[0] + 11.9*x[1] +41.8*x[2] + 52.1*x[3] - 21\n - 1.645 * np.sqrt(0.28*x[0]**2 + 0.19*x[1]**2\n + 20.5*x[2]**2 + 0.62*x[3]**2)\n ) # >=0\ndef h1(x):\n return x[0] + x[1] + x[2] + x[3] - 1 # == 0\ncons = ({'type': 'ineq', 'fun': g1},\n {'type': 'ineq', 'fun': g2},\n {'type': 'eq', 'fun': h1})\nbounds = [(0, 1.0),]*4\nres = shgo(f, bounds, n=150, constraints=cons)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "res\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\n" }, { "__type": "Code", "__tag": 4050, "value": "g1(res.x), g2(res.x), h1(res.x)\n", "execution_status": "failure" } ], "title": [], "level": 0, "target": null }, "see_also": [], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "func", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "bounds", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "args", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "()" }, { "__type": "SigParam", "__tag": 4030, "name": "constraints", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "n", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "100" }, { "__type": "SigParam", "__tag": 4030, "name": "iters", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "1" }, { "__type": "SigParam", "__tag": 4030, "name": "callback", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "minimizer_kwargs", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "options", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "sampling_method", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "simplicial" }, { "__type": "SigParam", "__tag": 4030, "name": "workers", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "1" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "shgo" }, "references": [ ".. [1] Endres, SC, Sandrock, C, Focke, WW (2018) \"A simplicial homology", " algorithm for lipschitz optimisation\", Journal of Global", " Optimization.", ".. [2] Joe, SW and Kuo, FY (2008) \"Constructing Sobol' sequences with", " better two-dimensional projections\", SIAM J. Sci. Comput. 30,", " 2635-2654.", ".. [3] Hock, W and Schittkowski, K (1981) \"Test examples for nonlinear", " programming codes\", Lecture Notes in Economics and Mathematical", " Systems, 187. Springer-Verlag, New York.", " http://www.ai7.uni-bayreuth.de/test_problem_coll.pdf", ".. [4] Wales, DJ (2015) \"Perspective: Insight into reaction coordinates and", " dynamics from the potential energy landscape\",", " Journal of Chemical Physics, 142(13), 2015.", ".. 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