a95650111f
In particular, the following things are done: *) Consistent tabulation of 4 spaces is ensured *) New function dprintf() is introduced, so now printing of the debug information can be turned on/off via the ALEX_DEBUG macro *) Removed solveLP_aux namespace *) All auxiliary functions are declared as static *) The return codes of solveLP() are encapsulated in enum.
49 lines
2.3 KiB
ReStructuredText
49 lines
2.3 KiB
ReStructuredText
Linear Programming
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==================
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.. highlight:: cpp
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optim::solveLP
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--------------------
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Solve given (non-integer) linear programming problem using the Simplex Algorithm (Simplex Method).
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What we mean here by "linear programming problem" (or LP problem, for short) can be
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formulated as:
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.. math::
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\mbox{Maximize } c\cdot x\\
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\mbox{Subject to:}\\
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Ax\leq b\\
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x\geq 0
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Where :math:`c` is fixed *1*-by-*n* row-vector, :math:`A` is fixed *m*-by-*n* matrix, :math:`b` is fixed *m*-by-*1* column vector and
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:math:`x` is an arbitrary *n*-by-*1* column vector, which satisfies the constraints.
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Simplex algorithm is one of many algorithms that are designed to handle this sort of problems efficiently. Although it is not optimal in theoretical
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sense (there exist algorithms that can solve any problem written as above in polynomial type, while simplex method degenerates to exponential time
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for some special cases), it is well-studied, easy to implement and is shown to work well for real-life purposes.
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The particular implementation is taken almost verbatim from **Introduction to Algorithms, third edition**
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by T. H. Cormen, C. E. Leiserson, R. L. Rivest and Clifford Stein. In particular, the Bland's rule
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(`http://en.wikipedia.org/wiki/Bland%27s\_rule <http://en.wikipedia.org/wiki/Bland%27s_rule>`_) is used to prevent cycling.
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.. ocv:function:: int optim::solveLP(const Mat& Func, const Mat& Constr, Mat& z)
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:param Func: This row-vector corresponds to :math:`c` in the LP problem formulation (see above).
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:param Constr: *m*-by-*n\+1* matrix, whose rightmost column corresponds to :math:`b` in formulation above and the remaining to :math:`A`.
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:param z: The solution will be returned here as a row-vector - it corresponds to (transposed) :math:`c` in the formulation above.
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:return: One of the return codes:
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::
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//!the return codes for solveLP() function
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enum
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{
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SOLVELP_UNBOUNDED = -2, //problem is unbounded (target function can achieve arbitrary high values)
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SOLVELP_UNFEASIBLE = -1, //problem is unfeasible (there are no points that satisfy all the constraints imposed)
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SOLVELP_SINGLE = 0, //there is only one maximum for target function
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SOLVELP_MULTI = 1 //there are multiple maxima for target function - the arbitrary one is returned
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};
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