{ "__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, "out": "" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where x is a vector of one or more variables. " }, { "__type": "InlineCode", "__tag": 4051, "value": "f(x)" }, { "__type": "Text", "__tag": 4046, "value": " is the objective function " }, { "__type": "InlineCode", "__tag": 4051, "value": "R^n -> R" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineCode", "__tag": 4051, "value": "g_i(x)" }, { "__type": "Text", "__tag": 4046, "value": " are the inequality constraints, and " }, { "__type": "InlineCode", "__tag": 4051, "value": "h_j(x)" }, { "__type": "Text", "__tag": 4046, "value": " are the equality constraints." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Optionally, the lower and upper bounds for each element in x can also be specified using the " }, { "__type": "ParamRef", "__tag": 4071, "name": "bounds" }, { "__type": "Text", "__tag": 4046, "value": " argument." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "While most of the theoretical advantages of SHGO are only proven for when " }, { "__type": "InlineCode", "__tag": 4051, "value": "f(x)" }, { "__type": "Text", "__tag": 4046, "value": " is a Lipschitz smooth function, the algorithm is also proven to converge to the global optimum for the more general case where " }, { "__type": "InlineCode", "__tag": 4051, "value": "f(x)" }, { "__type": "Text", "__tag": 4046, "value": " is non-continuous, non-convex and non-smooth, if the default sampling method is used " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "1" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The local search method may be specified using the " }, { "__type": "InlineCode", "__tag": 4051, "value": "minimizer_kwargs" }, { "__type": "Text", "__tag": 4046, "value": " parameter which is passed on to " }, { "__type": "InlineCode", "__tag": 4051, "value": "scipy.optimize.minimize" }, { "__type": "Text", "__tag": 4046, "value": ". By default, the " }, { "__type": "InlineCode", "__tag": 4051, "value": "SLSQP" }, { "__type": "Text", "__tag": 4046, "value": " method is used. In general, it is recommended to use the " }, { "__type": "InlineCode", "__tag": 4051, "value": "SLSQP" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineCode", "__tag": 4051, "value": "COBYLA" }, { "__type": "Text", "__tag": 4046, "value": ", or " }, { "__type": "InlineCode", "__tag": 4051, "value": "COBYQA" }, { "__type": "Text", "__tag": 4046, "value": " local minimization if inequality constraints are defined for the problem since the other methods do not use constraints." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The " }, { "__type": "InlineCode", "__tag": 4051, "value": "halton" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineCode", "__tag": 4051, "value": "sobol" }, { "__type": "Text", "__tag": 4046, "value": " method points are generated using " }, { "__type": "CrossRef", "__tag": 4002, "value": "scipy.stats.qmc", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.stats.qmc" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": ". Any other QMC method could be used." } ] } ], "title": [], "level": 0, "target": null }, "Warns": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Raises": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Yields": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Methods": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Returns": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "res", "annotation": "OptimizeResult", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The optimization result represented as a " }, { "__type": "CrossRef", "__tag": 4002, "value": "OptimizeResult", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.optimize._optimize:OptimizeResult" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " object. Important attributes are: " }, { "__type": "InlineCode", "__tag": 4051, "value": "x" }, { "__type": "Text", "__tag": 4046, "value": " the solution array corresponding to the global minimum, " }, { "__type": "InlineCode", "__tag": 4051, "value": "fun" }, { "__type": "Text", "__tag": 4046, "value": " the function output at the global solution, " }, { "__type": "InlineCode", "__tag": 4051, "value": "xl" }, { "__type": "Text", "__tag": 4046, "value": " an ordered list of local minima solutions, " }, { "__type": "InlineCode", "__tag": 4051, "value": "funl" }, { "__type": "Text", "__tag": 4046, "value": " the function output at the corresponding local solutions, " }, { "__type": "InlineCode", "__tag": 4051, "value": "success" }, { "__type": "Text", "__tag": 4046, "value": " a Boolean flag indicating if the optimizer exited successfully, " }, { "__type": "InlineCode", "__tag": 4051, "value": "message" }, { "__type": "Text", "__tag": 4046, "value": " which describes the cause of the termination, " }, { "__type": "InlineCode", "__tag": 4051, "value": "nfev" }, { "__type": "Text", "__tag": 4046, "value": " the total number of objective function evaluations including the sampling calls, " }, { "__type": "InlineCode", "__tag": 4051, "value": "nlfev" }, { "__type": "Text", "__tag": 4046, "value": " the total number of objective function evaluations culminating from all local search optimizations, " }, { "__type": "InlineCode", "__tag": 4051, "value": "nit" }, { "__type": "Text", "__tag": 4046, "value": " number of iterations performed by the global routine." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Finds the global minimum of a function using SHG optimization." } ] } ], "title": [], "level": 0, "target": null }, "Receives": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Warnings": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Attributes": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Parameters": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "func", "annotation": "callable", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The objective function to be minimized. 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(Defining a " }, { "__type": "InlineCode", "__tag": 4051, "value": "constraints" }, { "__type": "Text", "__tag": 4046, "value": " sequence in " }, { "__type": "InlineCode", "__tag": 4051, "value": "minimizer_kwargs" }, { "__type": "Text", "__tag": 4046, "value": " means that " }, { "__type": "InlineCode", "__tag": 4051, "value": "constraints" }, { "__type": "Text", "__tag": 4046, "value": " will not be added so if equality constraints and so forth need to be added then the inequality functions in " }, { "__type": "InlineCode", "__tag": 4051, "value": "constraints" }, { "__type": "Text", "__tag": 4046, "value": " need to be added to " }, { "__type": "InlineCode", "__tag": 4051, "value": "minimizer_kwargs" }, { "__type": "Text", "__tag": 4046, "value": " too). COBYLA only supports inequality constraints." } ] } ] }, { "__type": "Admonition", "__tag": 4056, "kind": "versionchanged", "base_type": "neutral", "children": [ { "__type": "AdmonitionTitle", "__tag": 4055, "children": [ { "__type": "Text", "__tag": 4046, "value": "versionchanged 1.11.0" } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineCode", "__tag": 4051, "value": "constraints" }, { "__type": "Text", "__tag": 4046, "value": " accepts " }, { "__type": "CrossRef", "__tag": 4002, "value": "NonlinearConstraint", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.optimize._constraints:NonlinearConstraint" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "CrossRef", "__tag": 4002, "value": "LinearConstraint", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.optimize._constraints:LinearConstraint" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "n", "annotation": "int, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Number of sampling points used in the construction of the simplicial complex. For the default " }, { "__type": "InlineCode", "__tag": 4051, "value": "simplicial" }, { "__type": "Text", "__tag": 4046, "value": " sampling method 2**dim + 1 sampling points are generated instead of the default " }, { "__type": "InlineCode", "__tag": 4051, "value": "n=100" }, { "__type": "Text", "__tag": 4046, "value": ". For all other specified values " }, { "__type": "ParamRef", "__tag": 4071, "name": "n" }, { "__type": "Text", "__tag": 4046, "value": " sampling points are generated. For " }, { "__type": "InlineCode", "__tag": 4051, "value": "sobol" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineCode", "__tag": 4051, "value": "halton" }, { "__type": "Text", "__tag": 4046, "value": " and other arbitrary " }, { "__type": "InlineRole", "__tag": 4003, "value": "sampling_methods", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " " }, { "__type": "InlineCode", "__tag": 4051, "value": "n=100" }, { "__type": "Text", "__tag": 4046, "value": " or another specified number of sampling points are generated." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "iters", "annotation": "int, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Number of iterations used in the construction of the simplicial complex. 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Some important options could be:" } ] }, { "__type": "DefList", "__tag": 4033, "children": [ { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "method" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The minimization method. If not given, chosen to be one of BFGS, L-BFGS-B, SLSQP, depending on whether or not the problem has constraints or bounds." } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "args" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Extra arguments passed to the objective function (" }, { "__type": "InlineCode", "__tag": 4051, "value": "func" }, { "__type": "Text", "__tag": 4046, "value": ") and its derivatives (Jacobian, Hessian)." } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "options" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Note that by default the tolerance is specified as " }, { "__type": "InlineCode", "__tag": 4051, "value": "{ftol: 1e-12}" } ] } ] } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "options", "annotation": "dict, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "A dictionary of solver options. 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(Note only methods that support this option will terminate the routine at precisely exact specified value. Otherwise the criterion will only terminate during a global iteration)" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "f_min" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Specify the minimum objective function value, if it is known." } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "f_tol" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Precision goal for the value of f in the stopping criterion. Note that the global routine will also terminate if a sampling point in the global routine is within this tolerance." } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "maxiter" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Maximum number of iterations to perform." } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "maxev" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Maximum number of sampling evaluations to perform (includes searching in infeasible points)." } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "maxtime" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Maximum processing runtime allowed" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "minhgrd" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Minimum homology group rank differential. The homology group of the objective function is calculated (approximately) during every iteration. The rank of this group has a one-to-one correspondence with the number of locally convex subdomains in the objective function (after adequate sampling points each of these subdomains contain a unique global minimum). If the difference in the hgr is 0 between iterations for " }, { "__type": "InlineCode", "__tag": 4051, "value": "maxhgrd" }, { "__type": "Text", "__tag": 4046, "value": " specified iterations the algorithm will terminate." } ] } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Objective function knowledge:" } ] }, { "__type": "DefList", "__tag": 4033, "children": [ { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "symmetry" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Specify if the objective function contains symmetric variables. The search space (and therefore performance) is decreased by up to O(n!) times in the fully symmetric case. 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Default is set to False." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "E.g. f(x) = (x_1 + x_2 + x_3) + (x_4)**2 + (x_5)**2 + (x_6)**2" } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "In this equation x_2 and x_3 are symmetric to x_1, while x_5 and x_6 are symmetric to x_4, this can be specified to the solver as " } ] }, { "__type": "Code", "__tag": 4050, "value": "symmetry = [0, # Variable 1\n 0, # symmetric to variable 1\n 0, # symmetric to variable 1\n 3, # Variable 4\n 3, # symmetric to variable 4\n 3, # symmetric to variable 4\n ]", "execution_status": null, "out": "" } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "jac" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Jacobian (gradient) of objective function. 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" }, { "__type": "InlineCode", "__tag": 4051, "value": "hessp" }, { "__type": "Text", "__tag": 4046, "value": " must compute the Hessian times an arbitrary vector. 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Defaults to True." } ] } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Feedback:" } ] }, { "__type": "DefList", "__tag": 4033, "children": [ { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "disp" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Set to True to print convergence messages." } ] } ] } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "sampling_method", "annotation": "str or function, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Current built in sampling method options are " }, { "__type": "InlineCode", "__tag": 4051, "value": "halton" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineCode", "__tag": 4051, "value": "sobol" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineCode", "__tag": 4051, "value": "simplicial" }, { "__type": "Text", "__tag": 4046, "value": ". 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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", "out": "" }, { "__type": "Code", "__tag": 4050, "value": "result.x, result.fun\n", "execution_status": "failure", "out": "(array([1., 1., 1., 1., 1.]), np.float64(2.920392374190081e-18))\n" }, { "__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", "out": "" }, { "__type": "Code", "__tag": 4050, "value": "result.x\n", "execution_status": "failure", "out": "array([1., 1., 1., 1.])\n" }, { "__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", "out": "" }, { "__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", "out": "" }, { "__type": "Code", "__tag": 4050, "value": "result.x, result.fun\n", "execution_status": "failure", "out": "(array([512. , 404.232]), np.float64(-959.6406627208507))\n" }, { "__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", "out": "array([[ 512. , 404.232],\n [ 283.076, -487.126],\n [-294.668, -462.02 ],\n [-105.877, 423.153],\n [-242.979, 274.38 ],\n [-506.258, 6.313],\n [-408.72 , -156.101],\n [ 150.232, 301.314],\n [ 91.009, -391.284],\n [ 202.897, -269.38 ],\n [ 361.666, -106.965],\n [-219.406, -244.06 ]])\n" }, { "__type": "Text", "__tag": 4046, "value": "\n" }, { "__type": "Code", "__tag": 4050, "value": "result.funl\n", "execution_status": "failure", "out": "array([-959.641, -718.167, -704.807, -565.998, -559.787, -557.369,\n -507.874, -493.961, -426.488, -421.156, -419.312, -410.985])\n" }, { "__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", "out": "(12, 23)\n" }, { "__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", "out": "" }, { "__type": "Code", "__tag": 4050, "value": "res\n", "execution_status": "failure", "out": " message: Optimization terminated successfully.\n success: True\n fun: 29.894378159143\n funl: [ 2.989e+01]\n x: [ 6.355e-01 2.695e-14 3.127e-01 5.178e-02]\n xl: [[ 6.355e-01 2.695e-14 3.127e-01 5.178e-02]]\n nit: 1\n nfev: 198\n nlfev: 91\n nljev: 11\n nlhev: 0\n" }, { "__type": "Text", "__tag": 4046, "value": "\n" }, { "__type": "Code", "__tag": 4050, "value": "g1(res.x), g2(res.x), h1(res.x)\n", "execution_status": "failure", "out": "(np.float64(1.1546319456101628e-13), np.float64(-1.0409451078885468e-12), np.float64(0.0))\n" } ], "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.", ".. [5] https://docs.scipy.org/doc/scipy/tutorial/optimize.html#constrained-minimization-of-multivariate-scalar-functions-minimize" ], "qa": "scipy.optimize._shgo:shgo", "arbitrary": [], "local_refs": [ "args", "bounds", "callback", "constraints", "func", "iters", "minimizer_kwargs", "n", "options", "res", "sampling_method", "workers" ] }