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Linear Programming

Linear Programming

In the following section, we solve linear programmes (LPs). The diameter theorem for polytopes is proven. The well-known simplex method is described, together with the strongly polynomial intex method (inter-/extrapolation) as fast solution of Smale's 9th problem. We may generalise the intex method to (non-) convex programmes (with vector-valued objective function). It is shown that (mixed) integer LPs are polynomial.

Diameter theorem for polytopes: The diameter of an n-dimensional polytope defined by m constraints with m, n ∈ c≥2 is at most 2(m + n - 3).

Proof: We can assemble at most m - 1 hyperplanes into an incomplete cycle (of dimension 2) and have to consider n - 2 alternatives sidewards (in the remaining dimensions). Since we can pass each section with a maximum of two edges, the factor is 2. This theorem can be extended to polyhedra analogously by dropping the requirement of finiteness.⃞

Theorem: The simplex method is not strongly polynomial.

Proof and algorithm: Let P := {x ∈ cn : Ax ≤ b, b ∈ cm, A ∈ cm×n, m, n ∈ cℕ*} be the feasible domain of the LP max {dTx : d ∈ cn, x ∈ P}. By taking the dual or setting x := x+ - x- with x+, x- ≥ 0, we obtain x ≥ 0. We first solve max {-z : Ax - b ≤ (z, , z)Tcm, z ≥ 0} to obtain a feasible x when b ≥ 0 does not hold. Initial and target value are z := |min {b1, ..., bm}| resp. z = 0 and we begin with x := 0 as in the first case. Pivoting if necessary, we may assume that b ≥ 0.

Let i, j, k ∈ cℕ* and let aiT the i-th row vector of A. If dj ≤ 0 for all j, the LP is solved. If for some dj > 0 (dj = 0), aij ≤ 0 for all i, the LP is positively unbounded (for now, we may drop dj and A.j as well as bi and ai, but only when aij < 0 holds). Otherwise, we divide all aiTx ≤ bi by ||ai|| and all dj and aij by the minimum of |aij| such that aij ≠ 0 for each j to improve the runtime performance, if possible. This will be reversed later. If necessary, renormalise by ||ai||.

We can reduce equal constraints to one of them and remove in each step such with ai ≤ 0, since they are redundant. The second case is analogous. If in both cases bi = 0 and ai ≥ 0 for some i, then the LP has maximum 0 and solution x = 0 if b ≥ 0, otherwise it has no solutions. In each step, for each dj > 0 and non-base variable xj, we select the minimum ratio bk/akj for aij > 0.

The variables with * are considered in the next step. The next potential vertex is given by xj* = xj + bk/akj for feasible x*. To select the steepest edge, select the pivot akj corresponding to xj that maximises dT(x* - x)/||x* - x|| resp. dj2/(1 + ||A.j||2) in the k-th constraint. If there are multiple maxima, select max djbk/akj according to the rule of best pivot value or alternatively (less well) the smallest angle min Σ dj*/||d*||.

If we cannot directly maximise the objective function, we relax (perturb) the constraints with bi = 0 by the same, minimal modulus. These do not need to be written into the tableau: We simply set bi = ||ai||. If another multiple vertex is encountered, despite this being unlikely, simply increase the earlier bi by ||ai||.

The cost of eliminating a multiple vertex, after which we revert the relaxation, corresponds to an LP with d > 0 and b = 0. This may be collated with the end of the process when checking whether the LP admits any other solutions. Along the chosen path, the objective function increases (effectively) strictly monotonically. We can then simply calculate dj*, aij* and bi* using the rectangle rule (cf. [775], p. 63).

In the worst-case scenario, the simplex method is not strongly polynomial despite the diameter theorem for polytopes under any given set of pivoting rules, since an exponential "drift" can be constructed with Klee-Minty or Jeroslow polytopes, or others, creating a large deviation from the shortest path by forcing the selection of the least favourable edge. This is consistent with existing proofs. The result follows.⃞

Theorem: The intex method converges linearly and solves every solvable LP in O(mn).

Proof and algorithm: We compute the LP min {h ∈ c≥0 : bTy - dTx ≤ h, 0 ≤ x ∈ cn, 0 ≤ y ∈ cm, Ax - b ≤ (h, , h)Tcm, d - ATy ≤ (h, , h)Tcn} via the (dual) programme min {bTy : 0 ≤ y ∈ cm, ATy ≥ d} for the (primal) programme max {dTx : d ∈ cn, x ∈ P≥0}. First, we normscalise bTy - dTx ≤ 0, Ax ≤ b and ATy ≥ d. We compute all scalar products dTai resp. -dj and sort them. Let v := (xT, yT)Tcm+n. We start with v := 0 and goal h = 0.

We compute x and analogously y from some equations aiTx = bi resp. xj = 0 with the largest corresponding dTai resp. -dj. Initial value for the height h is the minimum needed for the inequalities to be valid. If x and y are optimal, we stop the procedure. Otherwise, we successively interpolate (a few times) all vk* := (max vk + min vk)/2 and extrapolate (vT, h)T through (v*T, h*)T in O(m + n) into the boundary of the polytope if this is beneficial.

When we cannot leave v, we relax all constraints except v ≥ 0 a bit and undo that at the end of one iteration step. If we achieve only slight progress in the interpolation of v* or assume that it is not central, we interpolate again along the progress axis extrapolated on both sides. Since each iteration step in O(mn) processes circa half of h, the claim follows by the strong duality theorem ([775], p. 60 - 65).⃞

Remarks: Simplex method and face algorithm ([916], p. 580 f.) may solve the LP faster for small m and n. We can easily change the current inventory of restrictions or variables, because the intex method is one of the non-transforming methods and currently the fastest known (worst-case) LP-solving algorithm. We simply have to adjust h if necessary, because we can initialise additional variables with 0.

Corollary: Every solvable linear system (LS) Ax = b for x ∈ cn can be solved in O(mn) if we write it as LP min {h ∈ c≥0 : ±(Ax - b) ≤ (h, , h)Tcm}.⃞

Corollary: The LP min {h ∈ c≥0 : ±(Ax - λx) ≤ (h, , h)Tcn} can determine an eigenvector x ∈ cn \ {0} of the matrix A ∈ cn×n for the eigenvalue λ ∈ cℝ in O(n2) if it is solvable.⃞

Corollary: Let αj the j-th column vector of the matrix A-1cn×n and let δij the Kronecker delta. Then every LS Aαj = (δ1j, ..., δnj)T to determine the inverse A-1 of the regular matrix A can be solved for j = 1, ..., n in O(n2). Whether A is regular can be also determined in O(n2).⃞

Remarks: The three corollaries can be easily transferred to complex ones. The intex method is numerically very stable, since the initial data are only a little altered, and can be paralleled at least as well as the simplex method. It may also be applied to (mixed) integer problems and branch-and-bound, in particular for nonconvex optimisation. It can easily be extended to convex programmes (with a vector-valued objective function f1):

Corollary: Every solvable convex programme min {f1(x) : x ∈ cn, (f2(x), , fm(x))T ≤ 0} where the ficℝ are convex functions is strongly polynomial and may be solved by the intex method and two-dimensional bisection or Newton's methods, which handle the fi, in O(p), where p ∈ cℕ* denotes the number of operands xj of the fi, assuming that an x exists so that fi(x) < 0 for all i > 1 (see [939], p. 589 ff.).⃞

Further thoughts: Gomory or Lenstra cuts can find an integer solution of the original problem in polynomial time if we additionally assume that A, b, and d are integers wlog and that m and n are fixed. By dualising, a full-dimensional LP may be obtained as described before within the intex method. This shows that the problem of (mixed) integer linear programming is not NP-complete:

Theorem: (Mixed) integer LPs may be solved in polynomial time.⃞

code of simplex method

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