Solve given (non-integer) linear programming problem using the Simplex Algorithm

[z, res] = cv.solveLP(Func, Constr)
[...] = cv.solveLP(..., 'OptionName', optionValue, ...)

Input

• Func This row-vector corresponds to c in the LP problem formulation (see below). It should contain 32- or 64-bit floating-point numbers. As a convenience, column-vector may be also submitted, in the latter case it is understood to correspond to c'.
• Constr m-by-n+1 matrix, whose rightmost column corresponds to b in formulation above and the remaining to A. It should contain 32 or 64-bit floating-point numbers.

Output

• z The solution will be returned here as a column-vector, it corresponds to x in the formulation above. It will contain 64-bit floating-point numbers.
• res Return code. One of:
  • Unbounded problem is unbounded (target function can achieve arbitrary high values)
  • Unfeasible problem is unfeasible (there are no points that satisfy all the constraints imposed)
  • Single there is only one maximum for target function
  • Multi there are multiple maxima for target function, the arbitrary one is returned

What we mean here by "linear programming problem" (or LP problem, for short) can be formulated as:

Maximize c*x
  subject to  A*x <= b
              x >= 0

Where c is fixed 1-by-n row-vector, A is fixed m-by-n matrix, b is fixed m-by-1 column vector and x is an arbitrary n-by-1 column vector, which satisfies the constraints.

Simplex algorithm is one of many algorithms that are designed to handle this sort of problems efficiently. Although it is not optimal in theoretical sense (there exist algorithms that can solve any problem written as above in polynomial type, while simplex method degenerates to exponential time for some special cases), it is well-studied, easy to implement and is shown to work well for real-life purposes.

The particular implementation is taken almost verbatim from:

Introduction to Algorithms, 3rd edition by T. H. Cormen, C. E. Leiserson, R. L. Rivest and Clifford Stein.

In particular, the Bland's rule is used to prevent cycling.