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When the primal problem has a degenerate optimal solution the dual has?
Slide 16 • If dual has a nondegenerate optimal solution, the primal problem has a unique optimal solution. It is possible, however, that dual has a degenerate solution and the dual has a unique optimal solution.
Is the optimal solution degenerate?
Since all coefficients of variables in the objective function are negative, we now have the optimal solution, (x1,x2,x3,s1,s2) = (0,8,8,0,0) with objective value 16. In a degenerate LP, it is also possible that even in the final solution, some of the basic variables will be zero.
Is the degenerate optimal solution in the primal unique?
So we do have a situation with a degenerate optimal solution in the primal but a unique dual optimal. However, if the degenerate optimal solution is unique, then there must be multiple optimal solutions in the dual. The following table is from Sierksma’s Linear and Integer Programming: Theory and Practice, Volume 1, page 144.
How to prove optimal solution to dual is not unique?
Prove optimal solution to dual is not unique if optimal solution to the primal is degenerate and unique. pages 141-145, proof of Theorem 4.5. However the exact statement is that existance of a degenerate and (additionally) unique solution of the primal implies multiple solutions to the dual.
How to solve the dual and primal problem?
First they had these primal and dual problems: Then they did the canonical form of the primal problem and dual problem (for the dual I have some dificulty with the last 2 equations, need some explainig why they are exactly that):// Never mind I figured it out
When does a primal LP have a unique optimal solution?
Proof 1: 1) Consider a minimization LP in standard form.If there exits a nondegenerate optimal bfs for this LP,then the dual LP will have a unique optimial solution. Proof 2: 2) consider a minimization LP in standard form.If the primal LP has a unique optimal solution,then the dual LP will have a non-degenerate optimal solution.