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0ptX is a package for solving and optimizing mixed, integer linear systems
consisting of an arbitrary number of equations, inequations and target functions.
It implements a lattice-based approach for this task, which outperforms
commonly used branch-and-bound algorithms and currently represents the
most efficient way to challenge such systems in a timely manner.

The specification and target(s) of the problem:

The target functions are represented by
a matrix A_{t} ∈ ℝ^{t×n} and a vector s_{t}
∈ {-∞, ∞}^{t} ⋃ ℝ^{t}, where the rows of A_{t}
contain the linear target functions and the entries of s_{t} the desired target values
of an optimal solution.

Further A_{g} is a matrix ∈ ℝ^{g×n}
and b_{g} a vector ∈ ℝ^{g}.
They represent the linear equation system
a solution must fulfill.

The inequation system is denoted by the matrix A_{u} ∈ ℝ^{u×n},
and vectors l_{u} and r_{u}, whose entries define intervals
[l_{u}(j), r_{u}(j)] where the evaluated inequation values
of an optimal solution have to range.

Two additional vectors l_{x} and r_{x} contain the interval bounds,
which have to be fulfilled by the components of a vector x,
in order for being considered as a solution - a third vector d_{x} contains the maximum number
of floating point digits.

Let x be a vector ∈ ℝ^{n}
and set y = A_{t}·x for i = 1,2,..,t.

According to the i-th entry of s_{t}, minimize (-∞) or maximize (∞)
the i-th entry of y. If the i-th entry of s_{t} contains a real number, its distance to the
corresponding entry of y should be minimized.

If the fp-digits of x fulfill:

and x solves the side conditions:

and (d) its target vector y is valid with regards to s_{t}
and a user-provided slack tolerance T, such that:

then x is an optimal solution.

For a short video demonstration click here.

Further A

The inequation system is denoted by the matrix A

Two additional vectors l

The formulation of the optimization problem:

Let x be a vector ∈ ℝ

According to the i-th entry of s

If the fp-digits of x fulfill:

x(i) <= d_{x}(i), for i = 1,2,..,n

and x solves the side conditions:

(a) A_{g}·x = b_{g},

(b) l_{u} <= A_{u}·x <= r_{u},

(c) l_{x} <= x <= r_{x}

(b) l

(c) l

and (d) its target vector y is valid with regards to s

∑ | | y(i) - s_{t}(i) | <= T |

_{i = 1,2,..,t} | |

_{y(i) ∈ ℝ} |

then x is an optimal solution.

For a short video demonstration click here.

See the manual for further details.

To verify that 0ptX fits your business' needs, you can download a free,
trial version of 0ptX.

Caution: If the size of the input problems exceeds a certain limit, the solutions found by the trial version are (intentionally) displayed in scrambled form. In order to unscramble them, you must acquire the licensed version.

Caution: If the size of the input problems exceeds a certain limit, the solutions found by the trial version are (intentionally) displayed in scrambled form. In order to unscramble them, you must acquire the licensed version.