.. include:: headings.inc .. role:: boldred .. role:: boldgreen .. role:: backred .. role:: backyellow .. role:: backgreen .. role:: red .. role:: green .. _NLSE: |infinity| NLSE =============== | .. table:: =================== =================== ==================== F\ :sub:`opt` Known X\ :sub:`opt` Known Difficulty =================== =================== ==================== :green:`Yes` :green:`Yes` :backgreen:`Easy` =================== =================== ==================== | The realm of nonlinear systems of equations is not 100% tailored to global optimization algorithms; nevertheless, nothing stops you from applying a global solver to try and optimize a system of nonlinear equations, assuming the formulation is appropriate for the solvers. | |methodology| Methodology ------------------------- In order to build this dataset I have drawn from many different sources (`Publications `_, `ALIAS/COPRIN `_ and many others) to create 44 systems of nonlinear equations with dimensionality ranging from 2 to 8. A complete set of formulas giving the actual equations is presented in the section :ref:`NLSE Datasets`, and specifically in `Table 14.4`_ . The approach for building a single-valued objective function out of a system of nonlinear equations: .. math:: \begin{cases} f_1 (x) = ... \\ f_2 (x) = ... \\ ... \\ f_n (x) = ... \end{cases} Is to treat it kind of like a nonlinear least squares problem, where the actual (scalar-valued) objective function is easily formulated as: .. math:: F(x) = \sum_{i=1}^n f_i (x)^2 Then, of course, the global minimum is attained at :math:`F(x) = 0`. A few examples of 2D benchmark functions created with the NLSE test suite can be seen in `Figure 14.1`_. .. _Figure 14.1: +------------------------------------------------------------------------------+------------------------------------------------------------------------------+---------------------------------------------------------------------+ | .. figure:: ../benchmarks/NLSE/figures/docs/NLSE_1.png | .. figure:: ../benchmarks/NLSE/figures/docs/NLSE_2.png | .. figure:: ../benchmarks/NLSE/figures/docs/NLSE_3.png | | :align: center | :align: center | :align: center | | | | | | **NLSE Function Article92191** | **NLSE Function CaseStudy7** | **NLSE Function EffatiGrosan1** | +------------------------------------------------------------------------------+------------------------------------------------------------------------------+---------------------------------------------------------------------+ | .. figure:: ../benchmarks/NLSE/figures/docs/NLSE_4.png | .. figure:: ../benchmarks/NLSE/figures/docs/NLSE_5.png | .. figure:: ../benchmarks/NLSE/figures/docs/NLSE_6.png | | :align: center | :align: center | :align: center | | | | | | **NLSE Function Kincox** | **NLSE Function Merlet** | **NLSE Function Pinter** | +------------------------------------------------------------------------------+------------------------------------------------------------------------------+---------------------------------------------------------------------+ | |test_functions| General Solvers Performances --------------------------------------------- `Table 14.1`_ below shows the overall success of all Global Optimization algorithms, considering every benchmark function, for a maximum allowable budget of :math:`NF = 2,000`. The NLSE benchmark suite is an easy test suite: the best solver for :math:`NF = 2,000` is :ref:`MCS`, with a success rate of 79.6%, but many solvers are able to correctly optimize more than 60% of objectiv functions:: **BiteOpt**, **AMPGO**, **SHGO**, **BasinHopping** and **SCE**. .. note:: The reported number of functions evaluations refers to **successful optimizations only**. | .. _Table 14.1: .. cssclass:: pretty-table benchmark_dimensionality_table .. table:: **Table 14.1**: Solvers performances on the NLSE benchmark suite at NF = 2,000 +---------------------+---------------------+-----------------------+ | Optimization Method | Overall Success (%) | Functions Evaluations | +=====================+=====================+=======================+ | AMPGO | 63.64% | 321 | +---------------------+---------------------+-----------------------+ | BasinHopping | 63.64% | 342 | +---------------------+---------------------+-----------------------+ | BiteOpt | 75.00% | 490 | +---------------------+---------------------+-----------------------+ | CMA-ES | 52.27% | 622 | +---------------------+---------------------+-----------------------+ | CRS2 | 38.64% | 801 | +---------------------+---------------------+-----------------------+ | DE | 45.45% | 1,234 | +---------------------+---------------------+-----------------------+ | DIRECT | 47.73% | 433 | +---------------------+---------------------+-----------------------+ | DualAnnealing | 56.82% | 132 | +---------------------+---------------------+-----------------------+ | LeapFrog | 54.55% | 412 | +---------------------+---------------------+-----------------------+ | MCS | 79.55% | 195 | +---------------------+---------------------+-----------------------+ | PSWARM | 27.27% | 1,215 | +---------------------+---------------------+-----------------------+ | SCE | 61.36% | 811 | +---------------------+---------------------+-----------------------+ | SHGO | 65.91% | 257 | +---------------------+---------------------+-----------------------+ These results are also depicted in `Figure 14.2`_, which shows that :ref:`MCS` is the better-performing optimization algorithm, followed by very many other solvers with similar performances. .. _Figure 14.2: .. figure:: figures/NLSE/performances_NLSE_2000.png :alt: Optimization algorithms performances on the NLSE test suite at :math:`NF = 2,000` :align: center **Figure 14.2**: Optimization algorithms performances on the NLSE test suite at :math:`NF = 2,000` | Pushing the available budget to a very generous :math:`NF = 10,000`, the results show :ref:`MCS` basically solving all the problems at a 95.5% success rate, with **BiteOpt** now much closer and **AMPGO** trailing in third place. The results are also shown visually in `Figure 14.3`_. .. _Table 14.2: .. cssclass:: pretty-table benchmark_dimensionality_table .. table:: **Table 14.2**: Solvers performances on the NLSE benchmark suite at NF = 10,000 +---------------------+---------------------+-----------------------+ | Optimization Method | Overall Success (%) | Functions Evaluations | +=====================+=====================+=======================+ | AMPGO | 75.00% | 999 | +---------------------+---------------------+-----------------------+ | BasinHopping | 68.18% | 683 | +---------------------+---------------------+-----------------------+ | BiteOpt | 90.91% | 1,061 | +---------------------+---------------------+-----------------------+ | CMA-ES | 59.09% | 1,146 | +---------------------+---------------------+-----------------------+ | CRS2 | 52.27% | 1,645 | +---------------------+---------------------+-----------------------+ | DE | 65.91% | 2,306 | +---------------------+---------------------+-----------------------+ | DIRECT | 52.27% | 974 | +---------------------+---------------------+-----------------------+ | DualAnnealing | 61.36% | 373 | +---------------------+---------------------+-----------------------+ | LeapFrog | 65.91% | 1,234 | +---------------------+---------------------+-----------------------+ | MCS | 95.45% | 910 | +---------------------+---------------------+-----------------------+ | PSWARM | 54.55% | 2,423 | +---------------------+---------------------+-----------------------+ | SCE | 68.18% | 970 | +---------------------+---------------------+-----------------------+ | SHGO | 65.91% | 257 | +---------------------+---------------------+-----------------------+ .. _Figure 14.3: .. figure:: figures/NLSE/performances_NLSE_10000.png :alt: Optimization algorithms performances on the NLSE test suite at :math:`NF = 10,000` :align: center **Figure 14.3**: Optimization algorithms performances on the NLSE test suite at :math:`NF = 10,000` | |results| Sensitivities on Functions Evaluations Budget ------------------------------------------------------- It is also interesting to analyze the success of an optimization algorithm based on the fraction (or percentage) of problems solved given a fixed number of allowed function evaluations, let’s say 100, 200, 300,... 2000, 5000, 10000. In order to do that, we can present the results using two different types of visualizations. The first one is some sort of "small multiples" in which each solver gets an individual subplot showing the improvement in the number of solved problems as a function of the available number of function evaluations - on top of a background set of grey, semi-transparent lines showing all the other solvers performances. This visual gives an indication of how good/bad is a solver compared to all the others as function of the budget available. Results are shown in `Figure 14.4`_. .. _Figure 14.4: .. figure:: figures/NLSE/sm_maxfun_NLSE.png :alt: Percentage of problems solved given a fixed number of function evaluations on the NLSE test suite :align: center **Figure 14.4**: Percentage of problems solved given a fixed number of function evaluations on the NLSE test suite | The second type of visualization is sometimes referred as "Slopegraph" and there are many variants on the plot layout and appearance that we can implement. The version shown in `Figure 14.5`_ aggregates all the solvers together, so it is easier to spot when a solver overtakes another or the overall performance of an algorithm while the available budget of function evaluations changes. .. _Figure 14.5: .. figure:: figures/NLSE/sg_maxfun_NLSE.png :alt: Percentage of problems solved given a fixed number of function evaluations on the NLSE test suite :align: center **Figure 14.5**: Percentage of problems solved given a fixed number of function evaluations on the NLSE test suite | A few obvious conclusions we can draw from these pictures are: 1. For this specific benchmark test suite, :ref:`MCS` is the best solver no matter the budgets of function evaluations. For very limited budgets, **DualAnnealing** an **SHGO** are also good choices. 2. For medium to large number of functions evaluations, :ref:`MCS` and **BiteOpt** are by far the best solvers, both achieving more than 90% success rate. | .. _NLSE Datasets: |description| NLSE Datasets =========================== .. _Table 14.4: .. cssclass:: nist-table .. table:: **Table 14.4**: NLSE dataset summary +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | **Name** | **Dimension** | **Function** | +===============+===============+====================================================================================================================================================================================================================================================================================================================================================================================================================================================================+ | AOLcosh1 | 3 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = x \cosh{\left (y + 1.0 / x \right )} + z - 1.0 \\ | | | | y_2 = x \cosh{\left (y + 2.0 / x \right )} + z - 4.0 \\ | | | | y_3 = x \cosh{\left (y + 3.0 / x \right )} + z - 9.0 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Article89741 | 3 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = - x + 0.111111111111111 y^{2} + \sin{\left (z \right )} / 3 - 0.0123456790123457 \cos{\left (x \right )} \\ | | | | y_2 = - y + 0.333333333333333 \sin{\left (x \right )} + 0.333333333333333 \cos{\left (z \right )} \\ | | | | y_3 = 0.333333333333333 y - z + 0.166666666666667 \sin{\left (z \right )} - 0.111111111111111 \cos{\left (x \right )} \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Article90897 | 2 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = x_{1} - 1.7929755 \cdot 10^{-5} x_{2}^{2.0} \\ | | | | y_2 = x_{2} - \frac{90.0}{- x_{1} + 1.0} \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Article92191 | 2 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = -32.0 + 0.3 e^{- 13.0 b} \cos{\left (13.0 a \right )} + 0.7 e^{- 12.0 b} \cos{\left (12.0 a \right )} \\ | | | | y_2 = 0.3 e^{- 13.0 b} \sin{\left (13.0 a \right )} + 0.7 e^{- 12.0 b} \sin{\left (12.0 a \right )} \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Auto2Fit1 | 3 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = - x y \sin{\left (z \right )} + x / y z + \left(x + y - z\right)^{\cos{\left (x - 1 \right )}} + \left(\left(x - 0.3\right)^{y}\right)^{z} - 1.0 \\ | | | | y_2 = - y z \sin{\left (x \right )} + \left(- x + y + z\right)^{\cos{\left (y - 2 \right )}} + \left(\left(y - 0.2\right)^{z}\right)^{x} - 2.0 + y / x z \\ | | | | y_3 = - x z \sin{\left (y \right )} + \left(x - y + z\right)^{\cos{\left (z - 3 \right )}} + \left(\left(z - 0.1\right)^{x}\right)^{y} - 3.0 + z / x y \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Auto2Fit2 | 3 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = 2.0^{\sin{\left (y \right )}} + x^{2.23} - z - 1.25 \\ | | | | y_2 = 2 x + y + 5.5 z - 7 \\ | | | | y_3 = x - y + 0.2 z \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Bronstein | 3 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = x^{2} + y^{2} + z^{2} - 36.0 \\ | | | | y_2 = x + y - z \\ | | | | y_3 = x y + z^{2.0} - 1.0 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Bullard | 2 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = 10000.0 x_{1} x_{2} - 1.0 \\ | | | | y_2 = -1.001 + e^{- x_{2}} + e^{- x_{1}} \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Butcher8 | 8 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = - a - b + b_{1} + b_{2} + b_{3} \\ | | | | y_2 = a b - b^{2} - 0.5 b + b_{2} c_{2} + b_{3} c_{3} - 0.5 \\ | | | | y_3 = - a \left(b^{2} + 0.333333333333333\right) + b^{3} + b^{2} + 1.33333333333333 b + b_{2} c_{2}^{2} + b_{3} c_{3}^{2} \\ | | | | y_4 = - a \left(b^{2} + 0.5 b + 0.166666666666667\right) + a_{32} b_{3} c_{2} + b^{3} + b^{2} + 0.666666666666667 b \\ | | | | y_5 = a \left(b^{3} + b\right) - b^{4} - 1.5 b^{3} - 2.5 b^{2} - 0.25 b + 83.0 b_{2} c_{2} + b_{3} c_{3}^{3} - 0.25 \\ | | | | y_6 = a \left(b^{3} + 0.5 b^{2} + 0.5 b\right) + a_{32} b_{3} c_{2} c_{3} - b^{4} - 1.5 b^{3} - 1.75 b^{2} - 0.375 b - 0.125 \\ | | | | y_7 = a \left(b^{3} + b^{2} + 0.666666666666667 b\right) + a_{32} b_{3} c_{2}^{2} - b^{4} - 1.5 b^{3} - 1.16666666666667 b^{2} - 0.0833333333333333 b - 0.0833333333333333 \\ | | | | y_8 = - a \left(b^{3} + b^{2} + 0.333333333333333 b\right) + b^{4} + 1.5 b^{3} + 1.08333333333333 b^{2} + 0.291666666666667 b + 0.0416666666666667 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | CSTR | 2 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = x_{1} - \left(- R + 1.0\right) \left(\frac{D}{10.0 \beta_{1} + 10.0} - x_{1}\right) e^{\frac{10.0 x_{1}}{1.0 + 10.0 x_{1} / \gamma_{1}}} \\ | | | | y_2 = x_{1} - x_{2} \left(\beta_{2} + 1\right) + \left(- R + 1.0\right) \left(0.1 D - \beta_{1} x_{1} - x_{2} \left(- \beta_{2} + 1.0\right)\right) e^{\frac{10.0 x_{2}}{1.0 + 10.0 x_{2} / \gamma_{1}}} \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | CaseStudy3 | 6 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = x_{1} + 0.25 x_{2}^{4.0} x_{4} x_{6} + 0.75 \\ | | | | y_2 = x_{2} + 1.10090414052591 e^{x_{1} x_{2}} - 1.405 \\ | | | | y_3 = x_{3} - 0.5 x_{4} x_{6} + 1.5 \\ | | | | y_4 = x_{4} - 0.605 e^{- x_{3}^{2.0} + 1} - 0.395 \\ | | | | y_5 = - 0.5 x_{2} x_{6} + x_{5} + 1.5 \\ | | | | y_6 = - x_{1} x_{5} + x_{6} \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | CaseStudy4 | 3 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = - 5.0 x_{1} x_{2} x_{3} + x_{1}^{x_{2}} + x_{2}^{x_{1}} - 85.0 \\ | | | | y_2 = x_{1}^{3.0} - x_{2}^{x_{3}} - x_{3}^{x_{2}} - 60.0 \\ | | | | y_3 = x_{1}^{x_{3}} - x_{2} + x_{3}^{x_{1}} - 2 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | CaseStudy5 | 3 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = - 8.0 x_{1} \sin{\left (x_{2} \right )} + e^{x_{1}^{2.0}} \\ | | | | y_2 = x_{1} + x_{2} - 1.0 \\ | | | | y_3 = \left(x_{3} - 1.0\right)^{3.0} \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | CaseStudy6 | 3 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = 3.0 x_{1} - \cos{\left (x_{2} x_{3} \right )} - 0.5 \\ | | | | y_2 = x_{1}^{2.0} - 625.0 x_{2}^{2.0} - 0.25 \\ | | | | y_3 = 20.0 x_{3} - 1.0 + 3.33333333333333 \pi + e^{- x_{1} x_{2}} \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | CaseStudy7 | 2 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = - 3.0 x_{1} x_{2}^{2.0} + x_{1}^{3.0} - 1.0 \\ | | | | y_2 = 3.0 x_{1}^{2.0} x_{2} - x_{2}^{3.0} + 1.0 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Celestial | 3 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = 4 p^{3} \phi^{3} - 6 p^{3} - 12 p^{2} \phi^{3} s + 15 p \phi^{3} s^{3} - 3 p \phi^{3} s - 3 \phi^{3} s^{5} + \phi^{3} s^{3} \\ | | | | y_2 = - 6 p^{3} s - 9 p \phi^{3} s^{2} + 5 p \phi^{3} + 3 \phi^{3} s^{4} - 5 \phi^{3} s^{2} \\ | | | | y_3 = - 12 p s^{2} + 12 p + 4 \phi^{2} + 3 s^{4} - 6 s^{2} + 3 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Chem | 5 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = - x_{1} \left(x_{2} + 1\right) + 3 x_{5} \\ | | | | y_2 = - r x_{5} + x_{1} + x_{2} \left(2 r_{10} x_{2} + r_{7} x_{3} + r_{8} + r_{9} x_{4} + 2 x_{1} + x_{3}^{2}\right) \\ | | | | y_3 = x_{3} \left(2 r_{5} x_{3} + r_{6} + r_{7} x_{2} + 2 x_{2} x_{3}\right) - 8 x_{5} \\ | | | | y_4 = - 4 r x_{5} + x_{4} \left(r_{9} x_{2} + 2 x_{4}\right) \\ | | | | y_5 = r_{5} x_{3}^{2} + r_{6} x_{3} + x_{1} + x_{2} \left(r_{10} x_{2} + r_{7} x_{3} + r_{8} + r_{9} x_{4} + x_{1} + x_{3}^{2}\right) + x_{4}^{2} - 1 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Chemk | 4 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = x_{1}^{2} - x_{2} \\ | | | | y_2 = - x_{3} + x_{4}^{2} \\ | | | | y_3 = 0.55 x_{1} x_{4} + 0.45 x_{1} - 16970000.0 x_{2} x_{4} + 21770000.0 x_{2} - x_{4} \\ | | | | y_4 = 41260000.0 x_{1} x_{3} - 8282000.0 x_{1} x_{4} + 158500000000000.0 x_{2} x_{4} + 22840000.0 x_{3} x_{4} - 19180000.0 x_{3} + 48.4 x_{4} - 27.73 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Cyclo | 3 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = - y^{2} z^{2} - y^{2} + 24 y z - z^{2} - 13 \\ | | | | y_2 = - x^{2} z^{2} - x^{2} + 24 x z - z^{2} - 13 \\ | | | | y_3 = - x^{2} y^{2} - x^{2} + 24 x y - y^{2} - 13 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | DiGregorio | 3 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = \left(40 \sin{\left (p \right )} - 40.0\right)^{2.0} + \left(q - 40.0 \cos{\left (p \right )} + 10.0\right)^{2.0} - 1000.0 \\ | | | | y_2 = \left(- 40 \cos{\left (p \right )} + 20.0\right)^{2.0} - \left(- 35.0 \cos{\left (t \right )} + 20.0\right)^{2.0} + 1600.0 \sin^{2.0}{\left (p \right )} - 1225.0 \sin^{2.0}{\left (t \right )} \\ | | | | y_3 = \left(- q + 10\right)^{2.0} - \left(- 35 \cos{\left (t \right )} - 10\right)^{2.0} - 1225.0 \sin^{2.0}{\left (t \right )} + 1600.0 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Dipole | 8 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = a + b - 0.63254 \\ | | | | y_2 = c + d + 1.34534 \\ | | | | y_3 = a t + b u - c v - d w + 0.8365348 \\ | | | | y_4 = a v + b w + c t + d u - 1.7345334 \\ | | | | y_5 = a t^{2} - a v^{2} + b u^{2} - b w^{2} - 2 c t v - 2 d u w - 1.352352 \\ | | | | y_6 = 2 a t v + 2 b u w + c t^{2} - c v^{2} + d u^{2} - d w^{2} + 0.843453 \\ | | | | y_7 = a t^{3} - 3 a t v^{2} + b u^{3} - 3 b u w^{2} - 3 c t^{2} v + c v^{3} - 3 d u^{2} w + d w^{3} + 0.9563453 \\ | | | | y_8 = 3 a t^{2} v - a v^{3} + 3 b u^{2} w - b w^{3} + c t^{3} - 3 c t v^{2} + d u^{3} - 3 d u w^{2} - 1.2342523 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Eco9 | 8 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = x_{1} + x_{2} \left(x_{1} + x_{3}\right) + x_{4} \left(x_{3} + x_{5}\right) + x_{6} \left(x_{5} + x_{7}\right) - x_{8} \left(- x_{7} + 0.125\right) \\ | | | | y_2 = x_{2} + x_{3} \left(x_{1} + x_{5}\right) + x_{4} \left(x_{2} + x_{6}\right) + x_{5} x_{7} - x_{8} \left(- x_{6} + 0.25\right) \\ | | | | y_3 = x_{2} x_{5} + x_{3} \left(x_{6} + 1\right) + x_{4} \left(x_{1} + x_{7}\right) - x_{8} \left(- x_{5} + 0.375\right) \\ | | | | y_4 = x_{1} x_{5} + x_{2} x_{6} + x_{3} x_{7} + x_{4} - x_{8} \left(- x_{4} + 0.5\right) \\ | | | | y_5 = x_{1} x_{6} + x_{2} x_{7} + x_{5} - x_{8} \left(- x_{3} + 0.625\right) \\ | | | | y_6 = x_{1} x_{7} + x_{6} - x_{8} \left(- x_{2} + 75.0\right) \\ | | | | y_7 = x_{7} - x_{8} \left(- x_{1} + 0.875\right) \\ | | | | y_8 = x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} + x_{8} + 1.0 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | EffatiGrosan1 | 2 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = \cos{\left (2 x_{1} \right )} - \cos{\left (2 x_{2} \right )} - 0.4 \\ | | | | y_2 = - 2 x_{1} + 2 x_{2} - \sin{\left (2 x_{1} \right )} + \sin{\left (2 x_{2} \right )} - 1.2 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | EffatiGrosan2 | 2 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = x_{1} x_{2} + e^{x_{1}} - 1.0 \\ | | | | y_2 = x_{1} + x_{2} + \sin{\left (x_{1} x_{2} \right )} - 1.0 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | F3 | 2 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = 3.0 x_{1} x_{2} + 2.0 x_{1} - 1.0 \\ | | | | y_2 = x_{1} - x_{2} + x_{2} e^{- x_{1}} - e^{-1} \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Ferrais | 2 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = 0.5 x_{1} + 0.25 x_{2} / \pi - 0.5 \sin{\left (x_{1} x_{2} \right )} \\ | | | | y_2 = - 5.43656365691809 x_{1} + 2.71828182845905 x_{2} / \pi + \left(- 0.25 / \pi + 1.0\right) \left(e^{2.0 x_{1}} - 2.71828182845905\right) \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Gaussian | 3 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = 0.0009 \\ | | | | y_2 = 0.0044 \\ | | | | y_3 = 0.0175 \\ | | | | y_1 = x_{1} e^{- 0.5 x_{2} \left(t_{1} - x_{3}\right)^{2.0}} - y_{11} \\ | | | | y_2 = x_{2} e^{- 0.5 x_{2} \left(t_{2} - x_{3}\right)^{2.0}} - y_{22} \\ | | | | y_3 = x_{3} e^{- 0.5 x_{2} \left(t_{3} - x_{3}\right)^{2.0}} - y_{33} \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | GenEig | 6 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = - 10 x_{1} x_{6}^{2} + x_{1} x_{6} + 10 x_{1} + 2 x_{2} x_{6}^{2} + 2 x_{2} x_{6} + 2 x_{2} - x_{3} x_{6}^{2} + x_{3} x_{6} - x_{3} + x_{4} x_{6}^{2} + 2 x_{4} x_{6} + 2 x_{4} + 3 x_{5} x_{6}^{2} + x_{5} x_{6} - 2 x_{5} \\ | | | | y_2 = 2 x_{1} x_{6}^{2} + 2 x_{1} x_{6} + 2 x_{1} - 11 x_{2} x_{6}^{2} + x_{2} x_{6} + 9 x_{2} + 2 x_{3} x_{6}^{2} + 2 x_{3} x_{6} + 3 x_{3} - 2 x_{4} x_{6}^{2} + x_{4} x_{6} - x_{4} + x_{5} x_{6}^{2} + 3 x_{5} x_{6} - 2 x_{5} \\ | | | | y_3 = - x_{1} x_{6}^{2} + x_{1} x_{6} - x_{1} + 2 x_{2} x_{6}^{2} + 2 x_{2} x_{6} + 3 x_{2} - 12 x_{3} x_{6}^{2} + 10 x_{3} - x_{4} x_{6}^{2} - 2 x_{4} x_{6} + 2 x_{4} + x_{5} x_{6}^{2} - 2 x_{5} x_{6} - x_{5} \\ | | | | y_4 = x_{1} x_{6}^{2} + 2 x_{1} x_{6} + 2 x_{1} - 2 x_{2} x_{6}^{2} + x_{2} x_{6} - x_{2} - x_{3} x_{6}^{2} - 2 x_{3} x_{6} + 2 x_{3} - 10 x_{4} x_{6}^{2} + 2 x_{4} x_{6} + 12 x_{4} + 2 x_{5} x_{6}^{2} + 3 x_{5} x_{6} + x_{5} \\ | | | | y_5 = 3 x_{1} x_{6}^{2} + x_{1} x_{6} - 2 x_{1} + x_{2} x_{6}^{2} + 3 x_{2} x_{6} - 2 x_{2} + x_{3} x_{6}^{2} - 2 x_{3} x_{6} - x_{3} + 2 x_{4} x_{6}^{2} + 3 x_{4} x_{6} + x_{4} - 11 x_{5} x_{6}^{2} + 3 x_{5} x_{6} + 10 x_{5} \\ | | | | y_6 = x_{1} + x_{2} + x_{3} + x_{4} + x_{5} - 1 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Helical | 2 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = - 100.0 \arctan{\left (x_{2} / x_{1} \right )} / \pi + 0.5 \\ | | | | y_2 = 10.0 \sqrt{x_{1}^{2.0} + x_{2}^{2.0}} - 10.0 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Helical1 | 3 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = 10 x_{3} + 100.0 \arctan{\left (x_{2} / x_{1} \right )} / \pi - 50.0 \\ | | | | y_2 = 10.0 \sqrt{x_{1}^{2.0} + x_{2}^{2.0}} - 10.0 \\ | | | | y_3 = x_{1} - \sin{\left (x_{3} \right )} \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Kearl11 | 8 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = x_{1}^{2} + x_{2}^{2} - 1 \\ | | | | y_2 = x_{3}^{2} + x_{4}^{2} - 1 \\ | | | | y_3 = x_{5}^{2} + x_{6}^{2} - 1 \\ | | | | y_4 = x_{7}^{2} + x_{8}^{2} - 1 \\ | | | | y_5 = 0.004731 x_{1} x_{3} - 0.1238 x_{1} - 0.3578 x_{2} x_{3} - 0.001637 x_{2} - 0.9338 x_{4} + x_{7} - 0.3571 \\ | | | | y_6 = 0.2238 x_{1} x_{3} + 0.2638 x_{1} + 0.7623 x_{2} x_{3} - 0.07745 x_{2} - 0.6734 x_{4} - 0.6022 \\ | | | | y_7 = 0.3578 x_{1} + 0.004731 x_{2} + x_{6} x_{8} \\ | | | | y_8 = - 0.7623 x_{1} + 0.2238 x_{2} + 0.3461 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Kin1 | 6 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = \sin{\left (t_{2} \right )} \sin{\left (t_{6} \right )} \cos{\left (t_{5} \right )} - \sin{\left (t_{3} \right )} \sin{\left (t_{6} \right )} \cos{\left (t_{5} \right )} - \sin{\left (t_{4} \right )} \sin{\left (t_{6} \right )} \cos{\left (t_{5} \right )} + \cos{\left (t_{2} \right )} \cos{\left (t_{6} \right )} + \cos{\left (t_{3} \right )} \cos{\left (t_{6} \right )} + \cos{\left (t_{4} \right )} \cos{\left (t_{6} \right )} - 0.4077 \\ | | | | y_2 = \sin{\left (t_{1} \right )} \cos{\left (t_{5} \right )} + \sin{\left (t_{5} \right )} \cos{\left (t_{1} \right )} \cos{\left (t_{2} \right )} + \sin{\left (t_{5} \right )} \cos{\left (t_{1} \right )} \cos{\left (t_{3} \right )} + \sin{\left (t_{5} \right )} \cos{\left (t_{1} \right )} \cos{\left (t_{4} \right )} - 1.9115 \\ | | | | y_3 = \sin{\left (t_{2} \right )} \sin{\left (t_{5} \right )} + \sin{\left (t_{3} \right )} \sin{\left (t_{5} \right )} + \sin{\left (t_{4} \right )} \sin{\left (t_{5} \right )} - 1.9791 \\ | | | | y_4 = 3 \cos{\left (t_{1} \right )} \cos{\left (t_{2} \right )} + 2 \cos{\left (t_{1} \right )} \cos{\left (t_{3} \right )} + \cos{\left (t_{1} \right )} \cos{\left (t_{4} \right )} - 4.0616 \\ | | | | y_5 = 3 \sin{\left (t_{1} \right )} \cos{\left (t_{2} \right )} + 2 \sin{\left (t_{1} \right )} \cos{\left (t_{3} \right )} + \sin{\left (t_{1} \right )} \cos{\left (t_{4} \right )} - 1.7172 \\ | | | | y_6 = 3 \sin{\left (t_{2} \right )} + 2 \sin{\left (t_{3} \right )} + \sin{\left (t_{4} \right )} - 3.9701 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Kincox | 2 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = - 6.0 \sin{\left (t_{1} \right )} \sin{\left (t_{2} \right )} + 6.0 \cos{\left (t_{1} \right )} \cos{\left (t_{2} \right )} + 10.0 \cos{\left (t_{1} \right )} - 1.0 \\ | | | | y_2 = 6.0 \sin{\left (t_{1} \right )} \cos{\left (t_{2} \right )} + 10.0 \sin{\left (t_{1} \right )} + 6.0 \sin{\left (t_{2} \right )} \cos{\left (t_{1} \right )} - 4.0 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Merlet | 2 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = - \sin{\left (x_{1} \right )} \cos{\left (x_{2} \right )} - 2.0 \sin{\left (x_{2} \right )} \cos{\left (x_{1} \right )} \\ | | | | y_2 = - 2.0 \sin{\left (x_{1} \right )} \cos{\left (x_{2} \right )} - \sin{\left (x_{2} \right )} \cos{\left (x_{1} \right )} \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Neuro | 6 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = x_{1}^{2.0} + x_{3}^{2.0} - 1.0 \\ | | | | y_2 = x_{2}^{2.0} + x_{4}^{2.0} - 1.0 \\ | | | | y_3 = x_{3}^{3.0} x_{5} + x_{4}^{3.0} x_{6} \\ | | | | y_4 = x_{1}^{3.0} x_{5} + x_{2}^{3.0} x_{6} \\ | | | | y_5 = x_{1} x_{3}^{2.0} x_{5} + x_{2} x_{4}^{2.0} x_{6} \\ | | | | y_6 = x_{1}^{2.0} x_{3} x_{5} + x_{2}^{2.0} x_{4} x_{6} \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Pinter | 2 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = x_{1} - \sin{\left (2.0 x_{1} + 3.0 x_{2} \right )} - \cos{\left (3.0 x_{1} - 5.0 x_{2} \right )} \\ | | | | y_2 = x_{2} - \sin{\left (x_{1} - 2.0 x_{2} \right )} + \cos{\left (x_{1} + 3.0 x_{2} \right )} \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Puma | 4 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = - 0.3578 \sin{\left (t_{1} \right )} \cos{\left (t_{2} \right )} - 0.001637 \sin{\left (t_{1} \right )} - 0.9338 \sin{\left (t_{2} \right )} + 0.004731 \cos{\left (t_{1} \right )} \cos{\left (t_{2} \right )} - 0.1238 \cos{\left (t_{1} \right )} + \cos{\left (t_{4} \right )} - 0.3571 \\ | | | | y_2 = 0.7623 \sin{\left (t_{1} \right )} \cos{\left (t_{2} \right )} - 0.07745 \sin{\left (t_{1} \right )} - 0.6734 \sin{\left (t_{2} \right )} + 0.2238 \cos{\left (t_{1} \right )} \cos{\left (t_{2} \right )} + 0.2638 \cos{\left (t_{1} \right )} - 0.6022 \\ | | | | y_3 = 0.004731 \sin{\left (t_{1} \right )} + \sin{\left (t_{3} \right )} \sin{\left (t_{4} \right )} + 0.3578 \cos{\left (t_{1} \right )} \\ | | | | y_4 = 0.2238 \sin{\left (t_{1} \right )} - 0.7623 \cos{\left (t_{1} \right )} + 0.3461 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Semiconductor | 6 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = x_{2} \\ | | | | y_2 = x_{3} \\ | | | | y_3 = - V + x_{5} \\ | | | | y_4 = - V + x_{6} \\ | | | | y_5 = - DDD / ni + e^{a \left(- x_{1} + x_{3}\right)} - e^{a \left(x_{1} - x_{2}\right)} \\ | | | | y_6 = DDD / ni + e^{a \left(- x_{4} + x_{6}\right)} - e^{a \left(x_{4} - x_{5}\right)} \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | SixBody | 6 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = B + D - 6 F + \left(B - D\right) \left(3 b - 3 d\right) + 4 \\ | | | | y_2 = 5 B - 5 D + \left(3 b - 3 d\right) \left(B + D - 2 F\right) \\ | | | | y_3 = - 6 b - 6 d + 4 f + 3 \left(b - d\right)^{2} + 3 \\ | | | | y_4 = B^{2} b^{3} - 1 \\ | | | | y_5 = D^{2} d^{3} - 1 \\ | | | | y_6 = F^{2} f^{3} - 1 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | SjirkBoon | 4 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = C_{1} \cos{\left (3.0 \phi_{1} \right )} + C_{2} \cos{\left (3.0 \phi_{2} \right )} + 5.0 \\ | | | | y_2 = C_{1} \cos{\left (\phi_{1} \right )} + C_{2} \cos{\left (\phi_{2} \right )} - 3.0 \\ | | | | y_3 = C_{1} \sin{\left (3.0 \phi_{1} \right )} + C_{2} \sin{\left (3.0 \phi_{2} \right )} \\ | | | | y_4 = C_{1} \sin{\left (\phi_{1} \right )} + C_{2} \sin{\left (\phi_{2} \right )} - 4.0 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | ThinWall | 3 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = b h + \left(b - 2.0 t\right) \left(h - 2.0 t\right) - 165.0 \\ | | | | y_2 = 0.0833333333333333 b h^{3.0} - 0.0833333333333333 \left(b - 2.0 t\right) \left(h - 2.0 t\right)^{3.0} - 9369.0 \\ | | | | y_3 = \frac{2.0 t \left(b - t\right)^{2.0} \left(h - t\right)^{2.0}}{b + h - 2.0 t} - 6835.0 \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | Xu | 2 | .. math:: | | | | | | | | \begin{cases} | | | | y_1 = - x_{1} + 2.0 \sin{\left (x_{1} \right )} + 7 \sin{\left (x_{2} \right )} + 0.8 \cos{\left (2 x_{1} \right )} \\ | | | | y_2 = - x_{2} + 4.0 \sin{\left (2 x_{1} \right )} + 1.4 \sin{\left (3 x_{2} \right )} + 3.1 \cos{\left (2 x_{2} \right )} \\ | | | | \end{cases} | +---------------+---------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+