{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Loop API Example on Hartmann6\n",
"\n",
"The loop API is the most lightweight way to do optimization in Ax. The user makes one call to `optimize`, which performs all of the optimization under the hood and returns the optimized parameters.\n",
"\n",
"For more customizability of the optimization procedure, consider the Service or Developer API."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"execution": {
"iopub.execute_input": "2022-08-17T19:11:10.534242Z",
"iopub.status.busy": "2022-08-17T19:11:10.533717Z",
"iopub.status.idle": "2022-08-17T19:11:12.798395Z",
"shell.execute_reply": "2022-08-17T19:11:12.797709Z"
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"outputs": [
{
"data": {
"text/html": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:12] ax.utils.notebook.plotting: Injecting Plotly library into cell. Do not overwrite or delete cell.\n"
]
},
{
"data": {
"text/html": [
" \n",
" "
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"import numpy as np\n",
"\n",
"from ax.plot.contour import plot_contour\n",
"from ax.plot.trace import optimization_trace_single_method\n",
"from ax.service.managed_loop import optimize\n",
"from ax.metrics.branin import branin\n",
"from ax.utils.measurement.synthetic_functions import hartmann6\n",
"from ax.utils.notebook.plotting import render, init_notebook_plotting\n",
"\n",
"init_notebook_plotting()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 1. Define evaluation function\n",
"\n",
"First, we define an evaluation function that is able to compute all the metrics needed for this experiment. This function needs to accept a set of parameter values and can also accept a weight. It should produce a dictionary of metric names to tuples of mean and standard error for those metrics."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": true,
"execution": {
"iopub.execute_input": "2022-08-17T19:11:12.860615Z",
"iopub.status.busy": "2022-08-17T19:11:12.859836Z",
"iopub.status.idle": "2022-08-17T19:11:12.864571Z",
"shell.execute_reply": "2022-08-17T19:11:12.863973Z"
}
},
"outputs": [],
"source": [
"def hartmann_evaluation_function(parameterization):\n",
" x = np.array([parameterization.get(f\"x{i+1}\") for i in range(6)])\n",
" # In our case, standard error is 0, since we are computing a synthetic function.\n",
" return {\"hartmann6\": (hartmann6(x), 0.0), \"l2norm\": (np.sqrt((x ** 2).sum()), 0.0)}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If there is only one metric in the experiment – the objective – then evaluation function can return a single tuple of mean and SEM, in which case Ax will assume that evaluation corresponds to the objective. It can also return only the mean as a float, in which case Ax will treat SEM as unknown and use a model that can infer it. For more details on evaluation function, refer to the \"Trial Evaluation\" section in the docs."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 2. Run optimization\n",
"The setup for the loop is fully compatible with JSON. The optimization algorithm is selected based on the properties of the problem search space."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"execution": {
"iopub.execute_input": "2022-08-17T19:11:12.867987Z",
"iopub.status.busy": "2022-08-17T19:11:12.867411Z",
"iopub.status.idle": "2022-08-17T19:13:25.753357Z",
"shell.execute_reply": "2022-08-17T19:13:25.752690Z"
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"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:12] ax.service.utils.instantiation: Inferred value type of ParameterType.FLOAT for parameter x2. If that is not the expected value type, you can explicity specify 'value_type' ('int', 'float', 'bool' or 'str') in parameter dict.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:12] ax.service.utils.instantiation: Inferred value type of ParameterType.FLOAT for parameter x3. If that is not the expected value type, you can explicity specify 'value_type' ('int', 'float', 'bool' or 'str') in parameter dict.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:12] ax.service.utils.instantiation: Inferred value type of ParameterType.FLOAT for parameter x4. If that is not the expected value type, you can explicity specify 'value_type' ('int', 'float', 'bool' or 'str') in parameter dict.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:12] ax.service.utils.instantiation: Inferred value type of ParameterType.FLOAT for parameter x5. If that is not the expected value type, you can explicity specify 'value_type' ('int', 'float', 'bool' or 'str') in parameter dict.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:12] ax.service.utils.instantiation: Inferred value type of ParameterType.FLOAT for parameter x6. If that is not the expected value type, you can explicity specify 'value_type' ('int', 'float', 'bool' or 'str') in parameter dict.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:12] ax.service.utils.instantiation: Created search space: SearchSpace(parameters=[RangeParameter(name='x1', parameter_type=FLOAT, range=[0.0, 1.0]), RangeParameter(name='x2', parameter_type=FLOAT, range=[0.0, 1.0]), RangeParameter(name='x3', parameter_type=FLOAT, range=[0.0, 1.0]), RangeParameter(name='x4', parameter_type=FLOAT, range=[0.0, 1.0]), RangeParameter(name='x5', parameter_type=FLOAT, range=[0.0, 1.0]), RangeParameter(name='x6', parameter_type=FLOAT, range=[0.0, 1.0])], parameter_constraints=[ParameterConstraint(1.0*x1 + 1.0*x2 <= 20.0)]).\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:12] ax.modelbridge.dispatch_utils: Using Bayesian optimization since there are more ordered parameters than there are categories for the unordered categorical parameters.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:12] ax.modelbridge.dispatch_utils: Using Bayesian Optimization generation strategy: GenerationStrategy(name='Sobol+GPEI', steps=[Sobol for 12 trials, GPEI for subsequent trials]). Iterations after 12 will take longer to generate due to model-fitting.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:12] ax.service.managed_loop: Started full optimization with 30 steps.\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:12] ax.service.managed_loop: Running optimization trial 1...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:12] ax.service.managed_loop: Running optimization trial 2...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:12] ax.service.managed_loop: Running optimization trial 3...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:12] ax.service.managed_loop: Running optimization trial 4...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:12] ax.service.managed_loop: Running optimization trial 5...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:12] ax.service.managed_loop: Running optimization trial 6...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:13] ax.service.managed_loop: Running optimization trial 7...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:13] ax.service.managed_loop: Running optimization trial 8...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:13] ax.service.managed_loop: Running optimization trial 9...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:13] ax.service.managed_loop: Running optimization trial 10...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:13] ax.service.managed_loop: Running optimization trial 11...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:13] ax.service.managed_loop: Running optimization trial 12...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:13] ax.service.managed_loop: Running optimization trial 13...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"/opt/hostedtoolcache/Python/3.7.13/x64/lib/python3.7/site-packages/gpytorch/lazy/lazy_tensor.py:1811: UserWarning:\n",
"\n",
"torch.triangular_solve is deprecated in favor of torch.linalg.solve_triangularand will be removed in a future PyTorch release.\n",
"torch.linalg.solve_triangular has its arguments reversed and does not return a copy of one of the inputs.\n",
"X = torch.triangular_solve(B, A).solution\n",
"should be replaced with\n",
"X = torch.linalg.solve_triangular(A, B). (Triggered internally at ../aten/src/ATen/native/BatchLinearAlgebra.cpp:2189.)\n",
"\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"/opt/hostedtoolcache/Python/3.7.13/x64/lib/python3.7/site-packages/botorch/optim/optimize.py:287: RuntimeWarning:\n",
"\n",
"Optimization failed in `gen_candidates_scipy` with the following warning(s):\n",
"[OptimizationWarning('Optimization failed within `scipy.optimize.minimize` with status 9.')]\n",
"Trying again with a new set of initial conditions.\n",
"\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"/opt/hostedtoolcache/Python/3.7.13/x64/lib/python3.7/site-packages/botorch/optim/optimize.py:301: RuntimeWarning:\n",
"\n",
"Optimization failed on the second try, after generating a new set of initial conditions.\n",
"\n",
"[INFO 08-17 19:11:24] ax.service.managed_loop: Running optimization trial 14...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"/opt/hostedtoolcache/Python/3.7.13/x64/lib/python3.7/site-packages/botorch/optim/optimize.py:287: RuntimeWarning:\n",
"\n",
"Optimization failed in `gen_candidates_scipy` with the following warning(s):\n",
"[OptimizationWarning('Optimization failed within `scipy.optimize.minimize` with status 9.')]\n",
"Trying again with a new set of initial conditions.\n",
"\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"/opt/hostedtoolcache/Python/3.7.13/x64/lib/python3.7/site-packages/botorch/optim/optimize.py:301: RuntimeWarning:\n",
"\n",
"Optimization failed on the second try, after generating a new set of initial conditions.\n",
"\n",
"[INFO 08-17 19:11:36] ax.service.managed_loop: Running optimization trial 15...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"/opt/hostedtoolcache/Python/3.7.13/x64/lib/python3.7/site-packages/botorch/optim/optimize.py:287: RuntimeWarning:\n",
"\n",
"Optimization failed in `gen_candidates_scipy` with the following warning(s):\n",
"[OptimizationWarning('Optimization failed within `scipy.optimize.minimize` with status 9.')]\n",
"Trying again with a new set of initial conditions.\n",
"\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"/opt/hostedtoolcache/Python/3.7.13/x64/lib/python3.7/site-packages/botorch/optim/optimize.py:301: RuntimeWarning:\n",
"\n",
"Optimization failed on the second try, after generating a new set of initial conditions.\n",
"\n",
"[INFO 08-17 19:11:47] ax.service.managed_loop: Running optimization trial 16...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"/opt/hostedtoolcache/Python/3.7.13/x64/lib/python3.7/site-packages/botorch/optim/optimize.py:287: RuntimeWarning:\n",
"\n",
"Optimization failed in `gen_candidates_scipy` with the following warning(s):\n",
"[OptimizationWarning('Optimization failed within `scipy.optimize.minimize` with status 9.')]\n",
"Trying again with a new set of initial conditions.\n",
"\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:11:58] ax.service.managed_loop: Running optimization trial 17...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"/opt/hostedtoolcache/Python/3.7.13/x64/lib/python3.7/site-packages/botorch/optim/optimize.py:287: RuntimeWarning:\n",
"\n",
"Optimization failed in `gen_candidates_scipy` with the following warning(s):\n",
"[OptimizationWarning('Optimization failed within `scipy.optimize.minimize` with status 9.')]\n",
"Trying again with a new set of initial conditions.\n",
"\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"/opt/hostedtoolcache/Python/3.7.13/x64/lib/python3.7/site-packages/botorch/optim/optimize.py:301: RuntimeWarning:\n",
"\n",
"Optimization failed on the second try, after generating a new set of initial conditions.\n",
"\n",
"[INFO 08-17 19:12:09] ax.service.managed_loop: Running optimization trial 18...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:12:14] ax.service.managed_loop: Running optimization trial 19...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:12:19] ax.service.managed_loop: Running optimization trial 20...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:12:24] ax.service.managed_loop: Running optimization trial 21...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"/opt/hostedtoolcache/Python/3.7.13/x64/lib/python3.7/site-packages/botorch/optim/optimize.py:287: RuntimeWarning:\n",
"\n",
"Optimization failed in `gen_candidates_scipy` with the following warning(s):\n",
"[OptimizationWarning('Optimization failed within `scipy.optimize.minimize` with status 9.')]\n",
"Trying again with a new set of initial conditions.\n",
"\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"/opt/hostedtoolcache/Python/3.7.13/x64/lib/python3.7/site-packages/botorch/optim/optimize.py:301: RuntimeWarning:\n",
"\n",
"Optimization failed on the second try, after generating a new set of initial conditions.\n",
"\n",
"[INFO 08-17 19:12:35] ax.service.managed_loop: Running optimization trial 22...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"/opt/hostedtoolcache/Python/3.7.13/x64/lib/python3.7/site-packages/botorch/optim/optimize.py:287: RuntimeWarning:\n",
"\n",
"Optimization failed in `gen_candidates_scipy` with the following warning(s):\n",
"[OptimizationWarning('Optimization failed within `scipy.optimize.minimize` with status 9.')]\n",
"Trying again with a new set of initial conditions.\n",
"\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"/opt/hostedtoolcache/Python/3.7.13/x64/lib/python3.7/site-packages/botorch/optim/optimize.py:301: RuntimeWarning:\n",
"\n",
"Optimization failed on the second try, after generating a new set of initial conditions.\n",
"\n",
"[INFO 08-17 19:12:48] ax.service.managed_loop: Running optimization trial 23...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:12:53] ax.service.managed_loop: Running optimization trial 24...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:12:58] ax.service.managed_loop: Running optimization trial 25...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:13:03] ax.service.managed_loop: Running optimization trial 26...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:13:07] ax.service.managed_loop: Running optimization trial 27...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:13:11] ax.service.managed_loop: Running optimization trial 28...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:13:16] ax.service.managed_loop: Running optimization trial 29...\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"[INFO 08-17 19:13:21] ax.service.managed_loop: Running optimization trial 30...\n"
]
}
],
"source": [
"best_parameters, values, experiment, model = optimize(\n",
" parameters=[\n",
" {\n",
" \"name\": \"x1\",\n",
" \"type\": \"range\",\n",
" \"bounds\": [0.0, 1.0],\n",
" \"value_type\": \"float\", # Optional, defaults to inference from type of \"bounds\".\n",
" \"log_scale\": False, # Optional, defaults to False.\n",
" },\n",
" {\n",
" \"name\": \"x2\",\n",
" \"type\": \"range\",\n",
" \"bounds\": [0.0, 1.0],\n",
" },\n",
" {\n",
" \"name\": \"x3\",\n",
" \"type\": \"range\",\n",
" \"bounds\": [0.0, 1.0],\n",
" },\n",
" {\n",
" \"name\": \"x4\",\n",
" \"type\": \"range\",\n",
" \"bounds\": [0.0, 1.0],\n",
" },\n",
" {\n",
" \"name\": \"x5\",\n",
" \"type\": \"range\",\n",
" \"bounds\": [0.0, 1.0],\n",
" },\n",
" {\n",
" \"name\": \"x6\",\n",
" \"type\": \"range\",\n",
" \"bounds\": [0.0, 1.0],\n",
" },\n",
" ],\n",
" experiment_name=\"test\",\n",
" objective_name=\"hartmann6\",\n",
" evaluation_function=hartmann_evaluation_function,\n",
" minimize=True, # Optional, defaults to False.\n",
" parameter_constraints=[\"x1 + x2 <= 20\"], # Optional.\n",
" outcome_constraints=[\"l2norm <= 1.25\"], # Optional.\n",
" total_trials=30, # Optional.\n",
")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"And we can introspect optimization results:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"execution": {
"iopub.execute_input": "2022-08-17T19:13:25.756925Z",
"iopub.status.busy": "2022-08-17T19:13:25.756527Z",
"iopub.status.idle": "2022-08-17T19:13:25.767110Z",
"shell.execute_reply": "2022-08-17T19:13:25.766570Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"{'x1': 0.19524173623547994,\n",
" 'x2': 0.15186164835050317,\n",
" 'x3': 0.48196199677256235,\n",
" 'x4': 0.2694201848346438,\n",
" 'x5': 0.3100010075843861,\n",
" 'x6': 0.6595043991274802}"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"best_parameters"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"execution": {
"iopub.execute_input": "2022-08-17T19:13:25.770157Z",
"iopub.status.busy": "2022-08-17T19:13:25.769757Z",
"iopub.status.idle": "2022-08-17T19:13:25.773869Z",
"shell.execute_reply": "2022-08-17T19:13:25.773322Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"{'l2norm': 0.9471377627558926, 'hartmann6': -3.320339296175767}"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"means, covariances = values\n",
"means"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"For comparison, minimum of Hartmann6 is:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"execution": {
"iopub.execute_input": "2022-08-17T19:13:25.776779Z",
"iopub.status.busy": "2022-08-17T19:13:25.776380Z",
"iopub.status.idle": "2022-08-17T19:13:25.780253Z",
"shell.execute_reply": "2022-08-17T19:13:25.779726Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"-3.32237"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"hartmann6.fmin"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 3. Plot results\n",
"Here we arbitrarily select \"x1\" and \"x2\" as the two parameters to plot for both metrics, \"hartmann6\" and \"l2norm\"."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"execution": {
"iopub.execute_input": "2022-08-17T19:13:25.782978Z",
"iopub.status.busy": "2022-08-17T19:13:25.782761Z",
"iopub.status.idle": "2022-08-17T19:13:26.557799Z",
"shell.execute_reply": "2022-08-17T19:13:26.557178Z"
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hartmann6: -0.0178151 (SEM: 0)
x1: 0.975964
x2: 0.586741
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x4: 0.403315
x5: 0.393716
x6: 0.0759413",
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hartmann6: -1.30808 (SEM: 0)
x1: 0.217709
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hartmann6: -0.164584 (SEM: 0)
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hartmann6: -0.0294028 (SEM: 0)
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x6: 0.219928",
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hartmann6: -0.00237808 (SEM: 0)
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x6: 0.528179",
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hartmann6: -0.177833 (SEM: 0)
x1: 0.777396
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x4: 0.32362
x5: 0.05389
x6: 0.740995",
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hartmann6: -0.0545647 (SEM: 0)
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x4: 0.122442
x5: 0.664571
x6: 0.733239",
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hartmann6: -1.84581 (SEM: 0)
x1: 0.415137
x2: 0.47656
x3: 0.627314
x4: 0.259632
x5: 0.289654
x6: 0.665181",
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hartmann6: -0.301289 (SEM: 0)
x1: 0.409604
x2: 0.615204
x3: 0.214551
x4: 0.313589
x5: 0.544114
x6: 0.903365",
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hartmann6: -0.142023 (SEM: 0)
x1: 0.00211063
x2: 0.609144
x3: 0.208627
x4: 0.750822
x5: 0.29935
x6: 0.646879",
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hartmann6: -0.0955188 (SEM: 0)
x1: 0.0451019
x2: 0.486235
x3: 0.0151449
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x5: 0.00415688
x6: 0.190821",
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hartmann6: -2.1942 (SEM: 0)
x1: 0.351075
x2: 0.428514
x3: 0.549949
x4: 0.307319
x5: 0.262745
x6: 0.624945",
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hartmann6: -2.38157 (SEM: 0)
x1: 0.319733
x2: 0.35794
x3: 0.577248
x4: 0.310015
x5: 0.232613
x6: 0.636048",
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hartmann6: -2.47194 (SEM: 0)
x1: 0.273517
x2: 0.335423
x3: 0.571746
x4: 0.228221
x5: 0.22766
x6: 0.62438",
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hartmann6: -2.75533 (SEM: 0)
x1: 0.304272
x2: 0.268285
x3: 0.569762
x4: 0.232054
x5: 0.274794
x6: 0.593846",
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hartmann6: -2.77786 (SEM: 0)
x1: 0.349128
x2: 0.198171
x3: 0.523766
x4: 0.206189
x5: 0.28411
x6: 0.601962",
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hartmann6: -3.00295 (SEM: 0)
x1: 0.293149
x2: 0.179272
x3: 0.568715
x4: 0.243326
x5: 0.325758
x6: 0.602037",
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hartmann6: -2.95002 (SEM: 0)
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x2: 0.0810094
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x4: 0.268964
x5: 0.339166
x6: 0.583872",
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hartmann6: -3.02058 (SEM: 0)
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x2: 0.152584
x3: 0.549924
x4: 0.254275
x5: 0.37414
x6: 0.63086",
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hartmann6: -2.71806 (SEM: 0)
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x2: 0.1342
x3: 0.607576
x4: 0.221403
x5: 0.384661
x6: 0.621581",
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hartmann6: -3.21446 (SEM: 0)
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x3: 0.525
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x6: 0.614527",
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hartmann6: -3.29515 (SEM: 0)
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x3: 0.506709
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x5: 0.310522
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hartmann6: -3.1194 (SEM: 0)
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hartmann6: -3.22333 (SEM: 0)
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hartmann6: -3.30061 (SEM: 0)
x1: 0.194044
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hartmann6: -0.0178151 (SEM: 0)
x1: 0.975964
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hartmann6: -0.164584 (SEM: 0)
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hartmann6: -0.0294028 (SEM: 0)
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hartmann6: -0.00237808 (SEM: 0)
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x6: 0.528179",
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hartmann6: -0.177833 (SEM: 0)
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hartmann6: -0.0545647 (SEM: 0)
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x5: 0.664571
x6: 0.733239",
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hartmann6: -1.84581 (SEM: 0)
x1: 0.415137
x2: 0.47656
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x4: 0.259632
x5: 0.289654
x6: 0.665181",
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hartmann6: -0.301289 (SEM: 0)
x1: 0.409604
x2: 0.615204
x3: 0.214551
x4: 0.313589
x5: 0.544114
x6: 0.903365",
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hartmann6: -0.142023 (SEM: 0)
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x2: 0.609144
x3: 0.208627
x4: 0.750822
x5: 0.29935
x6: 0.646879",
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hartmann6: -0.0955188 (SEM: 0)
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x4: 0.368306
x5: 0.00415688
x6: 0.190821",
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hartmann6: -2.1942 (SEM: 0)
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x2: 0.428514
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x4: 0.307319
x5: 0.262745
x6: 0.624945",
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hartmann6: -2.38157 (SEM: 0)
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x6: 0.636048",
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hartmann6: -2.47194 (SEM: 0)
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x5: 0.22766
x6: 0.62438",
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hartmann6: -2.75533 (SEM: 0)
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x6: 0.593846",
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hartmann6: -2.77786 (SEM: 0)
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x5: 0.28411
x6: 0.601962",
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x5: 0.339166
x6: 0.583872",
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hartmann6: -3.02058 (SEM: 0)
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x4: 0.254275
x5: 0.37414
x6: 0.63086",
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hartmann6: -2.71806 (SEM: 0)
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hartmann6: -3.21446 (SEM: 0)
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x6: 0.599013",
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"# `plot_single_method` expects a 2-d array of means, because it expects to average means from multiple \n",
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