if an optimal solution is degenerate then

If an optimal solution is degenerate then. (a) There are alternative optimal assist one in moving from an initial feasible solution to the optimal solution.

If min{cT x

then we are not at an optimal solution. 6. If the dual has multiple optimal solutions then every primal optimal basic feasible solution is degenerate. 7 ...

in an optimal simplex tableau with columns corresponding to degenerate optimal dual basic variables. A primal optimal solution is unique if and only if.

• If a sequence of pivots starting from some basic feasible solution ends up at the exact same basic feasible solution then we refer to this as “cycling 

In a transportation problem if a basic feasible solution with m origins and n destinations has less than m +n -1 positive Xij i.e. occupied cells

is not degenerate. ii) For a given optimal solution x*if one optimal basis is degenerate for a network then all optimal bases are degenerate. Hi) For a 

If there is a tie between two slack (or surplus) variables then selection can be made arbitrarily. Again

If F (x0λ0) is degenerate

If an optimal solution is degenerate then. (a) There are alternative optimal solution. (b) The solution is infeasible. (c) The solution is use to the decis 

(a) alternate optimal solution (b) degenerate optimal solution. (c) no feasible solution. 48. If a variable Xj is unrestricted in sign in a primal LPP then 

Definition 3. A basic feasible solution is degenerate if there are more than n tight constraints. We say that a linear programming problem is degenerate if 

But actually we can say something stronger than this. Lemma 1 Given a primal feasible solution x and a dual feasible solution y

We know that a basic feasible solution is degenerate if one of the basic tableau then that tableau was degenerate by definition since one of the basic ...

If the shadow price for this resource is greater than the actual unit cost solution is degenerate there may then be multiple dual optimal solutions

An LP is degenerate if in a basic feasible solution one of the basic variables takes on a zero value. Degeneracy is a problem in practice

The set of primal optimal solutions is bounded if and only if there exists a degenerate then by Theorem 2 (Theorem 1) the dual (primal) optimal solution ...

feasible solution if at least one of the basic variable is zero and at any iteration of the simplex method more than one variable is eligible to.

and dual degenerate if more than one primal solution is optimal. We now introduce a standard approach called lexicographic perturbation

solution of two different sets of equality constraints then this is called degeneracy This will turn out to be important for the simplex algorithm It wasn’ t that I was misinforming you There just wasn’t a better way of describing the situation during that lecture From Lecture 3

1 If there is no optimal solution then the problem is either infeasible or un-bounded 2 If a feasible solution exists then a basic feasible solution exists 3 If an optimal solution exists then a basic optimal solution exists

An LP is degenerate if in a basic feasible solution one of the basic variables takes on a zero value Degeneracy is a problem in practice because it makes the simplex algorithm slower Original LP maximize x1 subject to x1 ?x2 x2 +x3 x2 x3 ? 8 (1) (2) ? 0 (3) x1 x2 ? 0 (4) Standard form =s1 = s2 =

A Degenerate LP De?nition: An LP is degenerate if in a basic feasible solution one of the basic variables takes on a zero value Degeneracy is a problem in practice because it makes the simplex algorithm slower Original LP maximize x 1 + x 2 + x 3 (1) subject to x 1 + x 2 ? 8 (2) ?x 2 + x 3 ? 0 (3) x 1x 2 ? 0 (4) Standard form

Answer 3 Given a basic feasible solution x and associated basis B if y = AT B) 1c B is dual feasible (ATy c) then x must be optimal Call such an y a verifying y" Finally this seems like an answer such that we can actually carry out a reasonably short computation and determine if x is optimal The real question then is what do we do if x

Theorem 1 2 Let x be a primal feasible solution and let u be a dual feasible solution such that complementary slackness holds between x and u Then x and u are primal optimal and dual optimal respectively Proof The rst form of complementary slackness is equivalent to saying that uT(Ax b) = 0 which we can rewrite as uTAx = uTb The second