# Optimization Algorithms

## Overview

The algorithms in this section minimize or maximize function value within specified constraints or without any constraints. More…

// enums

enum cv::SolveLPResult;

// classes

class cv::DownhillSolver;
class cv::MinProblemSolver;

// global functions

int
cv::solveLP(
const Mat& Func,
const Mat& Constr,
Mat& z
);


## Detailed Documentation

The algorithms in this section minimize or maximize function value within specified constraints or without any constraints.

### Global Functions

int
cv::solveLP(
const Mat& Func,
const Mat& Constr,
Mat& z
)


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

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

$\begin{split}\mbox{Maximize } c\cdot x\\ \mbox{Subject to:}\\ Ax\leq b\\ x\geq 0\end{split}$

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 time, 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, third edition by T. H. Cormen, C. E. Leiserson, R. L. Rivest and Clifford Stein. In particular, the Bland’s rule http://en.wikipedia.org/wiki/Bland%27s_rule is used to prevent cycling.

Parameters:

 Func This row-vector corresponds to $$c$$ in the LP problem formulation (see above). 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 $$c^T$$. Constr m -by- n+1 matrix, whose rightmost column corresponds to $$b$$ in formulation above and the remaining to $$A$$. It should containt 32- or 64-bit floating point numbers. z The solution will be returned here as a column-vector - it corresponds to $$c$$ in the formulation above. It will contain 64-bit floating point numbers.

Returns:

One of cv::SolveLPResult