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163 lines
9.7 KiB
ReStructuredText
163 lines
9.7 KiB
ReStructuredText
Downhill Simplex Method
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=======================
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.. highlight:: cpp
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optim::DownhillSolver
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---------------------------------
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.. ocv:class:: optim::DownhillSolver
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This class is used to perform the non-linear non-constrained *minimization* of a function, defined on an *n*-dimensional Euclidean space,
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using the **Nelder-Mead method**, also known as **downhill simplex method**. The basic idea about the method can be obtained from
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(`http://en.wikipedia.org/wiki/Nelder-Mead\_method <http://en.wikipedia.org/wiki/Nelder-Mead_method>`_). It should be noted, that
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this method, although deterministic, is rather a heuristic and therefore may converge to a local minima, not necessary a global one.
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It is iterative optimization technique, which at each step uses an information about the values of a function evaluated only at
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*n+1* points, arranged as a *simplex* in *n*-dimensional space (hence the second name of the method). At each step new point is
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chosen to evaluate function at, obtained value is compared with previous ones and based on this information simplex changes it's shape
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, slowly moving to the local minimum. Thus this method is using *only* function values to make decision, on contrary to, say, Nonlinear
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Conjugate Gradient method (which is also implemented in ``optim``).
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Algorithm stops when the number of function evaluations done exceeds ``termcrit.maxCount``, when the function values at the
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vertices of simplex are within ``termcrit.epsilon`` range or simplex becomes so small that it
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can enclosed in a box with ``termcrit.epsilon`` sides, whatever comes first, for some defined by user
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positive integer ``termcrit.maxCount`` and positive non-integer ``termcrit.epsilon``.
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::
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class CV_EXPORTS Solver : public Algorithm
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{
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public:
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class CV_EXPORTS Function
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{
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public:
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virtual ~Function() {}
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virtual double calc(const double* x) const = 0;
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virtual void getGradient(const double* /*x*/,double* /*grad*/) {}
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};
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virtual Ptr<Function> getFunction() const = 0;
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virtual void setFunction(const Ptr<Function>& f) = 0;
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virtual TermCriteria getTermCriteria() const = 0;
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virtual void setTermCriteria(const TermCriteria& termcrit) = 0;
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// x contain the initial point before the call and the minima position (if algorithm converged) after. x is assumed to be (something that
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// after getMat() will return) row-vector or column-vector. *It's size and should
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// be consisted with previous dimensionality data given, if any (otherwise, it determines dimensionality)*
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virtual double minimize(InputOutputArray x) = 0;
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};
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class CV_EXPORTS DownhillSolver : public Solver
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{
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public:
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//! returns row-vector, even if the column-vector was given
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virtual void getInitStep(OutputArray step) const=0;
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//!This should be called at least once before the first call to minimize() and step is assumed to be (something that
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//! after getMat() will return) row-vector or column-vector. *It's dimensionality determines the dimensionality of a problem.*
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virtual void setInitStep(InputArray step)=0;
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};
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It should be noted, that ``optim::DownhillSolver`` is a derivative of the abstract interface ``optim::Solver``, which in
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turn is derived from the ``Algorithm`` interface and is used to encapsulate the functionality, common to all non-linear optimization
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algorithms in the ``optim`` module.
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optim::DownhillSolver::getFunction
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--------------------------------------------
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Getter for the optimized function. The optimized function is represented by ``Solver::Function`` interface, which requires
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derivatives to implement the sole method ``calc(double*)`` to evaluate the function.
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.. ocv:function:: Ptr<Solver::Function> optim::DownhillSolver::getFunction()
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:return: Smart-pointer to an object that implements ``Solver::Function`` interface - it represents the function that is being optimized. It can be empty, if no function was given so far.
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optim::DownhillSolver::setFunction
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-----------------------------------------------
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Setter for the optimized function. *It should be called at least once before the call to* ``DownhillSolver::minimize()``, as
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default value is not usable.
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.. ocv:function:: void optim::DownhillSolver::setFunction(const Ptr<Solver::Function>& f)
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:param f: The new function to optimize.
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optim::DownhillSolver::getTermCriteria
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----------------------------------------------------
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Getter for the previously set terminal criteria for this algorithm.
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.. ocv:function:: TermCriteria optim::DownhillSolver::getTermCriteria()
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:return: Deep copy of the terminal criteria used at the moment.
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optim::DownhillSolver::setTermCriteria
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------------------------------------------
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Set terminal criteria for downhill simplex method. Two things should be noted. First, this method *is not necessary* to be called
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before the first call to ``DownhillSolver::minimize()``, as the default value is sensible. Second, the method will raise an error
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if ``termcrit.type!=(TermCriteria::MAX_ITER+TermCriteria::EPS)``, ``termcrit.epsilon<=0`` or ``termcrit.maxCount<=0``. That is,
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both ``epsilon`` and ``maxCount`` should be set to positive values (non-integer and integer respectively) and they represent
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tolerance and maximal number of function evaluations that is allowed.
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Algorithm stops when the number of function evaluations done exceeds ``termcrit.maxCount``, when the function values at the
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vertices of simplex are within ``termcrit.epsilon`` range or simplex becomes so small that it
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can enclosed in a box with ``termcrit.epsilon`` sides, whatever comes first.
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.. ocv:function:: void optim::DownhillSolver::setTermCriteria(const TermCriteria& termcrit)
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:param termcrit: Terminal criteria to be used, represented as ``TermCriteria`` structure (defined elsewhere in openCV). Mind you, that it should meet ``(termcrit.type==(TermCriteria::MAX_ITER+TermCriteria::EPS) && termcrit.epsilon>0 && termcrit.maxCount>0)``, otherwise the error will be raised.
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optim::DownhillSolver::getInitStep
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-----------------------------------
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Returns the initial step that will be used in downhill simplex algorithm. See the description
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of corresponding setter (follows next) for the meaning of this parameter.
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.. ocv:function:: void optim::getInitStep(OutputArray step)
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:param step: Initial step that will be used in algorithm. Note, that although corresponding setter accepts column-vectors as well as row-vectors, this method will return a row-vector.
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optim::DownhillSolver::setInitStep
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----------------------------------
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Sets the initial step that will be used in downhill simplex algorithm. Step, together with initial point (givin in ``DownhillSolver::minimize``)
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are two *n*-dimensional vectors that are used to determine the shape of initial simplex. Roughly said, initial point determines the position
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of a simplex (it will become simplex's centroid), while step determines the spread (size in each dimension) of a simplex. To be more precise,
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if :math:`s,x_0\in\mathbb{R}^n` are the initial step and initial point respectively, the vertices of a simplex will be: :math:`v_0:=x_0-\frac{1}{2}
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s` and :math:`v_i:=x_0+s_i` for :math:`i=1,2,\dots,n` where :math:`s_i` denotes projections of the initial step of *n*-th coordinate (the result
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of projection is treated to be vector given by :math:`s_i:=e_i\cdot\left<e_i\cdot s\right>`, where :math:`e_i` form canonical basis)
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.. ocv:function:: void optim::setInitStep(InputArray step)
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:param step: Initial step that will be used in algorithm. Roughly said, it determines the spread (size in each dimension) of an initial simplex.
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optim::DownhillSolver::minimize
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-----------------------------------
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The main method of the ``DownhillSolver``. It actually runs the algorithm and performs the minimization. The sole input parameter determines the
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centroid of the starting simplex (roughly, it tells where to start), all the others (terminal criteria, initial step, function to be minimized)
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are supposed to be set via the setters before the call to this method or the default values (not always sensible) will be used.
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.. ocv:function:: double optim::DownhillSolver::minimize(InputOutputArray x)
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:param x: The initial point, that will become a centroid of an initial simplex. After the algorithm will terminate, it will be setted to the point where the algorithm stops, the point of possible minimum.
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:return: The value of a function at the point found.
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optim::createDownhillSolver
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------------------------------------
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This function returns the reference to the ready-to-use ``DownhillSolver`` object. All the parameters are optional, so this procedure can be called
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even without parameters at all. In this case, the default values will be used. As default value for terminal criteria are the only sensible ones,
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``DownhillSolver::setFunction()`` and ``DownhillSolver::setInitStep()`` should be called upon the obtained object, if the respective parameters
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were not given to ``createDownhillSolver()``. Otherwise, the two ways (give parameters to ``createDownhillSolver()`` or miss them out and call the
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``DownhillSolver::setFunction()`` and ``DownhillSolver::setInitStep()``) are absolutely equivalent (and will drop the same errors in the same way,
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should invalid input be detected).
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.. ocv:function:: Ptr<optim::DownhillSolver> optim::createDownhillSolver(const Ptr<Solver::Function>& f,InputArray initStep, TermCriteria termcrit)
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:param f: Pointer to the function that will be minimized, similarly to the one you submit via ``DownhillSolver::setFunction``.
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:param step: Initial step, that will be used to construct the initial simplex, similarly to the one you submit via ``DownhillSolver::setInitStep``.
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:param termcrit: Terminal criteria to the algorithm, similarly to the one you submit via ``DownhillSolver::setTermCriteria``.
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