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Linear Programming
Linear Programming

In the following section, Set Theory, Topology and Nonstandard Analysis are presupposed. Simplex and centre method can solve a LP mostly in cubic and quadratic runtime respectively.

Diameter theorem for polytopes: The diameter of an \(n\)-dimensional polytope defined by \(m\) constraints for \(m, n \in {}^{\omega}\mathbb{N}_{\ge 2}\) is at most \(2(m + n – 3)\).

Proof: At most \(\acute{m}\) hyperplanes can be assembled into an incomplete cycle of dimension 2 and there are at most \(n – 2\) alternatives sidewards in the remaining dimensions. Overcoming every minimal distance requires at most two edges and yields the factor 2.\(\square\)

Remark: Dropping the finite limitation allows the analogous transfer of the theorem to polyhedra.

Definition: Let \({|| \cdot ||}_{1}\) the 1-norm. If \(x \in X \subseteq {}^{\omega}\mathbb{R}^{\omega}\) passes to the next step, it is followed by \({}^{*}\) where \(\Delta x := {x}^{*} – x\). The unit vector of \(x \in X^{*}\) is \({_1}{x} := x/||x||\). Let \(x, y \in X \) in \((x, y, …)^T\) be row vectors. If a method requires computation time in seconds and memory in bits of \(\mathcal{O}({\omega}^{\mathcal{O}(1)})\), it is polynomial. If one of both quantities is \(\mathcal{O}({e}^{|\mathcal{O}(\omega)|})\), the method is exponential. Let the eigenproduct (yet: determinant) of a square matrix be the product of its eigenvalues.\(\triangle\)

Theorem: The simplex method1Dantzig, George B.: Lineare Programmierung und Erweiterungen; 1. Aufl.; 1966; Springer; Berlin. is exponential in the worst case.

Proof and algorithm: Let the LP max \(\{{c}^{T}x : c \in {}^{\omega}\mathbb{R}^{n}, x \in P\}\) have the feasible domain \(P := \{x \in {}^{\omega}\mathbb{R}^{n} : Ax \le b, b \in {}^{\omega}\mathbb{R}^{m}, A \in {}^{\omega}\mathbb{R}^{m \times n}, m, n \in {}^{\omega}\mathbb{N}^{*}\}\). Its dual gives \({x}^{*} \in {}^{\omega}\mathbb{R}_{\ge 0}^{m}\). A (big) \(x^+ \in {}^{\omega}\mathbb{R}_{\geq0}\) achieves \(x^* \in {}^{\omega}\mathbb{R}_{\geq0}^n\) for \(x := x^* – {\underline{x}}_n^+\). Solving max \(\{-h \in {}^{\omega}\mathbb{R}_{\le 0} : Ax – b \le \underline{h}_n\}\) yields \(x \in P_{\ge 0}\) if \(b \ge 0\) does not hold. Let \(|\text{min } \{{b}_{1}, …, {b}_{m}\}|\) be the initial value for \(h\) and 0 its target value. Start with \(x := 0\). Only one pivot step causes that \({b}^{*} \ge 0\). Let \(i, j, k \in {}^{\omega}\mathbb{N}^{*}\) and let \({a}_{i}^{T}\) the \(i\)-th row vector of \(A\). If \({c}_{j} \le 0\) for all \(j\), the LP is solved.

If for some \({c}_{j} > 0\) also \({a}_{ij} \le 0\) for all \(i\), the LP is positively unbounded. If for some \({c}_{j} = 0\) also \({a}_{ij} \le 0\) for all \(i\), drop \({c}_{j}\) and \({A}_{.j}\) as well as \({b}_{i}\) and \({a}_{i}\), but only when \({a}_{ij} < 0\) holds. The inequality \({a}_{ij}{x}_{j} \ge 0 > {b}_{i}\) for all \(j\) has no solution, too. Dividing all \({a}_{i}^{T}x \le {b}_{i}\) by \(||{a}_{i}||\) and all \({c}_{j}\) and \({a}_{ij}\) by the minimum of \(|{a}_{ij}|\) such that \({a}_{ij} \ne 0\) for each \(j\) maybe repeated and reversed later. Constraints with \({a}_{i} \le 0\) may always be removed.

For each \({c}_{j} > 0\) and non-base variable \({x}_{j}\) select \({b_k}/{a}_{kj} :=\) min \(\{{b_i}/{a}_{ij} : {a}_{ij} > 0\}\) to obtain a next \({x}^{*} \in P_{\ge 0}\) where \({x}_{j}^{*} = {x}_{j} + {b}_{k}/{a}_{kj}\). To select the steepest edge, pick the pivot \({a}_{kj}\) corresponding to \({x}_{j}\) that maximises \({c}^{T}{_1}{\Delta x}\) or \({c}_{j}^{2}/\grave{v}_{j}^{2}\) with \(v_j := ||{A}_{.j}||\) in the \(k\)-th constraint. Multiple maxima allow to use the rule of best pivot value max\({}_{k,j} {c}_{j}{b}_{k}/{a}_{kj}\) or (slower) the smallest angle min \({{_{(1)}}\underline{1}^{T}_n}{_1}{c}^{*}\).

If \(c^Tx^* = c^Tx\) is repeated, perturb, which means relax, the constraints with \({b}_{i} = 0\) by \(d \in {}^{\omega}\mathbb{R}_{>0}\). For \({b}_{i} := ||{a}_{i}||\), drop \(d\) in the tableau. If another multiple vertex is encountered for once, simply add \(||{a}_{i}||\) to every \({b}_{i}\). Leaving it, after which the relaxation is reverted, may require to solve an LP with \(c > 0\) and \(b = 0\).

This task can also finally arise, if further solutions of the LPs are to be determined for at least two \(c_j = 0\). This avoids having to compute several minimal relaxations that differ strongly but are not selected according to optimality points of view, as with the lexicographical method. Bland’s rule is not optimal, too.

The rectangle rule simply allows to compute \({c}_{j}^{*}, {a}_{ij}^{*}\) and \({b}_{i}^{*}\)2cf. Vanderbei, Robert J.: Linear Programming; 3rd Ed.; 2008; Springer; New York, p. 63.. Despite the diameter theorem for polytopes, no pivoting rule can prevent the simplex method from being exponential as the case may be: An exponential “drift” for e. g. Klee-Minty or Jeroslow polytopes may always provide an unfavourable edge. The result follows in accordance with the state of research.\(\square\)

Remarks: Numbers of length \(\mathcal{O}({\omega})\) can only be processed in \(\mathcal{O}(\vartheta)\) as is generally known. The following fast algorithm could be extremely important despite various numerical difficulties. It was developed in 35 years and answers (e.g. with the help of Gomory cuts) Hilbert’s tenth problem positively. The simplex method, on the other hand, allows simple and precise calculations with rational numbers if the initial problems are appropriately formulated and smaller (for suitably approximated \(||{a}_{i}||\)).

Theorem: The centre method solves every solvable LP in \(\mathcal{O}({\vartheta}^3)\).

Proof and algorithm: Let \(z := m + n\) and \(d \in [0, 1]\) the density of \(A\). First, normalise and scale \({b}^{T}y – {c}^{T}x \le 0, Ax \le b\) as well as \({A}^{T}y \ge c\). Let \(P_r := \{(x, y)^T \in {}^{\omega}\mathbb{R}_{\ge 0}^{z} : {b}^{T}y – {c}^{T}x \le r \in [0, \check{r}], Ax – b \le \underline{r}_m, c – {A}^{T}y \le \underline{r}_n\}\) have the radius \(\check{r} := s|\min \; \{b_1, …, b_m, -c_1, …, -c_n\}|\) and the scaling factor \(s \in [1, 2]\). It follows \(\underline{0}_{z} \in \partial P_{\check{r}}\). By the strong duality theorem3loc. cit., p. 60 – 65., the LP min \(\{ r \in [0, \check{r}] : (x, y)^T \in P_r\}\) solves the LPs max \(\{{c}^{T}x : c \in {}^{\omega}\mathbb{R}^{n}, x \in {P}_{\ge 0}\}\) and min \(\{{b}^{T}y : y \in {}^{\omega}\mathbb{R}_{\ge 0}^{m}, {A}^{T}y \ge c\}\).

Its solution is the geometric centre \(g\) of the polytope \(P_0\). For \(p_k^* := (\text{min}\,p_k + \text{max}\,p_k)/2\) and \(k = 1, …, \grave{z}\) approximate \(g\) by \(p_0 := (x_0, y_0, r_0)^T\) until \(||\Delta p||_1\) is sufficiently small. The solution \(t^o(x^o, y^o, r^o)^T\) of the two-dimensional LP min \(\{ r \in [0, \check{r}] : t \in {}^{\omega}\mathbb{R}_{> 0}, t(x_0, y_0)^T \in P_r\}\) approximates \(g\) better and achieves \(r \le \hat{2}\check{r}\). Repeat this for \(t^o(x^o, y^o)^T\) until \(g \in P_0\) is computed in \(\mathcal{O}({}_2\check{r}^2dmn)\) if it exists.

Solving all two-dimensional LPs \(\text{min}_k r_k\) by bisection methods for \(r_k \in {}^{\omega}\mathbb{R}_{\ge 0}\) and \(k = 1, …, z\) in \(\mathcal{O}({\vartheta}^2)\) each time determines \(q \in {}^{\omega}\mathbb{R}^k\) where \(q_k := \Delta p_k \Delta r_k/r\) and \(r := \text{min}_k \Delta r_k\). Let simplified \(|\Delta p_1| = … = |\Delta p_{z}|\). Here min \(r_{\grave{z}}\) for \(p^* := p + wq\) and \(w \in {}^{\omega}\mathbb{R}_{\ge 0}\) may be solved, too. If \(\text{min}_k \Delta r_k r = 0\) follows, stop computing, otherwise repeat until min \(r = 0\) or min \(r > 0\) is sure.\(\square\)

Remarks: Simplex method and face algorithm4Pan, Ping-Qi: Linear Programming Computation; 1st Ed.; 2014; Springer; New York., p. 580 f. may solve the LP faster for small \(m\) and \(n\). The current stock of constraints or variables can easily be changed because the centre method does not durably transform constraints and is faster than all known (worst-case) LP-algorithms in \(\mathcal{O}(\omega^{37/18}\vartheta)\). Details are first published when no misuse for non-transparent or bad decisions must be feared.

Remarks: The centre method makes use of the fact that the calculated geometric centre and the centre of the sphere coincide in \(P_0\). The former is secured by rotating the coordinate axes (in pairs) like \(c\) with the aid of a simple rotation matrix in a comparable running time. If necessary, the same small modulus relaxes constraints temporarily.

Remarks: If the centre method is optimised for distributed computing in \({}^{\nu}\mathbb{R}^{\nu}\), its runtime only amounts to \(\mathcal{O}(1)\). Integer solutions \(x_j\) satisfying the side conditions \(x_j – \lfloor x_j \rfloor \le r\) prove it successful for (mixed) integer problems and (non-) convex (Pareto) optimisation (according to nature5Vasuki, A: Nature-Inspired Optimization Algorithms; 1st Ed.; 2020; CRC Press; Boca Raton.. The transfer to complex numbers is easy. All of this holds also for following results.

Conclusion: The LP max \(\{{||x – {x}^{o}||}_{1} : {c}^{T}x = {c}^{T}{x}^{o}, Ax \le b, x – {x}^{o} \in {[-1, 1]}^{n}, x \in {}^{\omega}\mathbb{R}_{\ge 0}^{n}\}\) can determine for the first solution \({x}^{o}\) a second one in \(\mathcal{O}({\vartheta}^3)\) if any, where \({y}^{o}\) may be treated analogously.\(\square\)

Conclusion: The LP max \(\{\nu|\lambda_j| + ||x_j||_1: Ax_j = \lambda_j x_j, |\lambda_j| \in [0, r_j], x_j \in {[-1, 1]}^{n*}\}\) can determine for every \(j = 1, …, n\) the eigenvalue \(\lambda_j \in {}^{\omega}\mathbb{R}\) and the eigenvector \(x_j \in {}^{\omega}\mathbb{R}^{n}\) of the matrix \(A \in {}^{\omega}\mathbb{R}^{n \times n}\) in \(\mathcal{O}({\vartheta}^3).\square\)

Conclusion: The LP min \(\{r \in [0, s \, \text{max } \{|{b}_{1}|, …, |{b}_{m}|\}] : \pm(Ax – b) \le \underline{r}_m\}\) can determine an \(x \in {}^{\omega}\mathbb{R}^{n}\) of every solvable linear system (LS) \(Ax = b\) in \(\mathcal{O}({\vartheta}^3)\). The LPs max \(\{{x}_{j} : Ax = 0\}\) yield all solutions to the LS. The matrix \(A\) is regular if and only if the LP max \(\{{||x||}_{1} : Ax = 0\} = 0.\square\)

Conclusion: Let \({\alpha }_{j} := {A}_{.j}^{-1}\) for \(j = 1, …, n\) concerning the matrix \({A}^{-1} \in {}^{\omega}\mathbb{R}^{n \times n}\) and let \({\delta}_{ij}\) the Kronecker delta. A regular \(A\) has an eigenproduct \(\ne 0\) and allows every LS \({A \alpha }_{j} = {({\delta}_{1j}, …, {\delta}_{nj})}^{T}\) to be solved in \(\mathcal{O}({\vartheta}^3)\) as well as matrix multiplication to be executed in \(\mathcal{O}({\vartheta}^2)\) by parallelising.\(\square\)

Corollary:The centre method and two-dimensional bisection or Newton’s methods may solve every solvable convex programme min \(\{{f}_{1}(x) : x \in {}^{\omega}\mathbb{R}^{n}, {({f}_{2}(x), …, {f}_{m}(x))}^{T} \le 0\}\) where the \({f}_{i} \in {}^{\omega}\mathbb{R}\) are convex functions for \(i = 1, …, m\) in polynomial runtime, if the number of operands \({x}_{j}\) of the \({f}_{i}\) is \(\le {\omega}^{\nu-3}\) and if an \(x\) exists so that \({f}_{i}(x) < 0\) for all \(i > 1\)6cf. Bertsekas, Dimitri P.: Nonlinear Programming; 3rd Ed.; 2016; Athena Scientific; Belmont., p. 589 ff..\(\square\)

Corollary: The LP max \(\{x : y_{\acute{m}} – xy_m = b_{\acute{m}}, y_n = y_0 = 0, x \le x_0 \in {}^{\omega}\mathbb{R}\}\) can determine for \(m = 1, …, n\) by Horner scheme an \(x \in {}^{\omega}\mathbb{R}\) solving \(n\)-polynomial \(b_{\acute{n}}x^{\acute{n}} + … + b_1x + b_0 = 0\) in \(\mathcal{O}(\hat{\omega}{\vartheta}^3)\). Aborting TR can analogously solve every continuous inequation system with convex solution set.\(\square\)

Please, input linear programme (separators spaces, last row objective function, first column right-hand sides as in the example):
Result (not the number of the correct decimal places matters, but the time complexity and the number of steps of the simplex method):

© 2008-2021 by Boris Haase

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References

References
1 Dantzig, George B.: Lineare Programmierung und Erweiterungen; 1. Aufl.; 1966; Springer; Berlin.
2 cf. Vanderbei, Robert J.: Linear Programming; 3rd Ed.; 2008; Springer; New York, p. 63.
3 loc. cit., p. 60 – 65.
4 Pan, Ping-Qi: Linear Programming Computation; 1st Ed.; 2014; Springer; New York., p. 580 f.
5 Vasuki, A: Nature-Inspired Optimization Algorithms; 1st Ed.; 2020; CRC Press; Boca Raton.
6 cf. Bertsekas, Dimitri P.: Nonlinear Programming; 3rd Ed.; 2016; Athena Scientific; Belmont., p. 589 ff.