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Optimization/Simplex Method, proof of unique solution

  1. Sep 21, 2012 #1
    This was on my homework this week. I already turned it in but the professor hasn't posted the solutions yet so I'm genuinely curious what the answer is.

    1. The problem statement, all variables and given/known data
    The LP:

    The problem:

    2. Relevant equations

    c^{T}d = z_{N}^{T}d_{N}

    sorry for the non-use of latex! I tried to use the ## ## and [itex] environments but couldn't get it to work!

    3. The attempt at a solution

    Not sure how standard the notation is since we don't use a textbook, just the professor's notes. Anyway, I'm taking my first analysis course this semester so I'm getting decent with proofs. He gives us a big hint in the problem statement.

    So suppose there exists an i in the index set N such that ##z_{Ni} = 0##. Does this mean that we just let all the remaining components of z_N be strictly positive? I assumed we want to show that if z_N >= 0, we no longer have a unique solution so we require z_N > 0. Optimality occurs when z_N >= 0, but I interpreted the problem as wanting us to show that we need a stronger condition for uniqueness

    From this, I said that d_N must necessarily be the zero vector since we use the convention that:

    d_Nj = 1, if there exists a j0 such that z_Nj0 < 0
    d_Nj = 0, otherwise

    Since we clearly have that z_N is non-negative, d_N must be zero. We have the relation

    (c^T)d = (z_N)^T*d_N (since d_N = 0)

    Then this means that we can find a feasible direction d such that (c^T)d = 0, ie we can step down to x' = x + td. The objective function value at x' is:

    (c^T)(x' + td) = (c^T)x' + t*(c^T)d = (c^T)x' + 0

    which is a contradiction since we know x is a unique solution but we find that x' is also a solution.

    My proof-writing is sketchy but I think I understood the main jist. The problem is I don't get how the hint that we should assume that a single component of z_N is zero helps us. How is this any different than if I assumed for contradiction that z_N was strictly positive for all i? Because for strictly positive z_N, we'd still have d_N = 0.
    Last edited: Sep 21, 2012
  2. jcsd
  3. Sep 21, 2012 #2

    Ray Vickson

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    The hint did not say zNi= 0 for a single i; it said zNi = 0 for some i, and that could mean one or two or three or ... .

    As to LaTeX: if you use "[t e x] your stuff [/t e x]" (but remove the spaces inside [ and ]) you will get a displayed result, like this:
    [tex](c^T)(x' + td) = (c^T)x' + t*(c^T)d = (c^T)x' + 0[/tex] If, instead, you say "[i t e x] your stuff [/ i t e x]" (again, no spaces) you will get it in-line, like this: [itex](c^T)(x' + td) = (c^T)x' + t*(c^T)d = (c^T)x' + 0[/itex]

    Last edited: Sep 21, 2012
  4. Sep 21, 2012 #3
    Yes, but I'm not entirely seeing how that initial supposition will help. Is it okay to suppose there exists an [itex]i_{0} \in N[/itex] such that [itex]z_{Ni_{0}} = 0[/itex] and that for all [itex]i \ne i_{0}[/itex] [itex]z_{Ni} > 0[/itex]? Like I said, I thought the point of the proof was to show that [itex]z_{N} \ge 0[/itex] is not a strong enough condition to guarantee a unique solution.

    For what it's worth, the hint gives so much a way that I understand the direction I should be going but not how to make that initial leap.

    [[thanks for the tex tip btw. I have no idea why it wasn't working the first time.]]
  5. Sep 21, 2012 #4

    Ray Vickson

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    OK. Let me change the notation a bit, to make it easier to type. By re-labelling the variables if necessary, assume that x1, ..., xm are basic and xm+1,..., xn are non-basic in the optimal solution. However, let me re-name the non-basic variables as u1, ..., ur, where r = n-m. Thus, u = xN. The final form of the constraint equations are [tex] x_B = v - T u,[/tex] where
    [tex] v = B^{-1}b, \text{ and } T = B^{-1}N.[/tex] We have
    [tex] Z = \sum c_j x_j = V + \sum_j z_j u_j, \\
    \text{where } V = c_B v = c_B B^{-1} b, \text{ and } z = c_N - c_B B^{-1}N
    = c_N - c_B T.[/tex] (For simplicity I take cB and cN to be row vectors.) In detail, we have:
    [tex] x_1 = v_1 - t_{11} u_1 - \cdots - t_{1 r} u_r\\
    \vdots \\
    x_m = v_m - t_{m1} u_1 - \cdots - t_{m r} u_r\\
    Z = V + z_1 u_1 + \cdots + z_r u_r
    We are assuming two things: (i) this solution is non-degenerate, so that vi > 0 for i = 1,2, ...,m; and (ii) it is optimal.

    Therefore, we need all zj ≥ 0 (because if some zp < 0 we can increase the variable up a bit and make the value of Z smaller (that is, we can reduce V). The non-degeneracy assumption is crucial here; without it (at a degenerate solution) we might have an optimal solution but a non-optimal basis.

    If some zp = 0, we can increase up a bit (i.e., by a truly positive amount) but with no change in Z, because the variable up does not actually enter into the expression for Z in this basis. So, if some component of z is zero we can find an optimal solution that is different from the one we started with, and that would contradict the uniqueness.

    Note: in applied modelling, we often use this type of argument to tell whether or not a solution is unique: if all the "reduced costs" are non-zero, the solution is unique, but if some reduced costs vanish there are alternative optima (at least in the non-degenerate case).

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