### LP opt sol int proof

parent ed10936c
 ... ... @@ -45,7 +45,8 @@ If we allowed swapping between neighboring pixels this would indeed be tricky. If instead we restrict ourselves to only swapping between non-neighboring pixels however, then the score function changes only linearly and we can use linear programming to find the optimal swap. to find the optimal swap. Let the following linear program be defined as the swap LP. Concretely, we setup the following linear program. Let $P$ be a set of $n$ non-neighboring pixels. Every pixel ... ... @@ -77,16 +78,14 @@ there: & & c_{i, j} = \text{Score change for planting } 1 \text{ of crop } j \text{ in pixel } i \end{align*} Then we are trying to maximize the function: \begin{displaymath} \text{maximize } c^T x \end{displaymath} Then we are trying to maximize the function $c^T x$ subject to the constraints that: \begin{enumerate} \item We don't allow less than $0$ or more than $1$ of each crop to be planted in a pixel \item Every Pixel must contain a total amount of crops equaling $1$ \item The amount of each crop we are redistributing must stay the same \item The amount of each crop in $x$ before the swap and in $x^*$ after the swap must be the same \end{enumerate} or formally, a solution $x^*$ must satisfy: \begin{align*} ... ... @@ -97,22 +96,32 @@ or formally, a solution $x^*$ must satisfy: In theory an optimal solution to this linear program could result in fractional amounts of crop being planted in a pixel. However, the following theorem planted in a pixel. However, the following theorems demonstrates that there is always an optimal integer solution. First we will show the specific case of swapping $k$ different crops among $k$ non-neighboring pixels, then we will generalize the statement to $m$ different crops amount $k$ different pixels. Let $\F$ denote all valid integer solutions to the LP above, or formally: \begin{displaymath} \F := \{f \ | \ f \in \Z^{n \cdot C} \land f \text{ denotes a valid field}\} \end{displaymath} \begin{theorem} The linear program for finding the optimal swap between non neighboring pixels (described above) always has an optimal integer solution. \label{kByKOptSwap} If $x \in \F$, then the solution $x^*$ to the swap LP for $k$ pixels swapping $k$ different crops is also in $\F$. \end{theorem} \begin{proof} The essence of our proof is that we first show that when we remap $k$ crops between $k$ pixels there must exist an integer solution, because then we are in a Birkhoff polytope. Then we will generalize the $k$ crop to $k$ pixels remapping to a $m$ crop to $k$ pixels remapping. there must exist an optimal integer solution, because then we are in a Birkhoff polytope. Consider the case where we select $k$ non-neighboring pixels that all contain a different crop. Then we ask ... ... @@ -141,13 +150,76 @@ demonstrates that there is always an optimal integer solution. solution will be at a vertex. Because there is an optimal solution at a vertex we know to be integer, we know there must be an optimal integer solution. \end{proof} Next we use this theorem to prove that the LP for swapping any number of crops types among non-neighboring pixels must result in an integer solution. \begin{theorem} If $x \in \F$, then the solution $x^*$ to the swap LP is also in $\F$. \end{theorem} \begin{proof} We will formulate a swap LP' that must have an optimal integer solution $x^* \in \F$ and demonstrate that $x^*$ can also be translated into an optimal integer solution for our swap LP. We want to use theorem \ref{kByKOptSwap} to show that the swap LP' we are going to construct has an optimal integer solution. Therefore we need to have the same number of crops as pixels. Consider again a set $P$ of non-neighboring pixels. For any two pixels $p_1, p_2 \in P$ that contain the same crop $c$ in $x$ we introduce a new crop $c'$ that behaves exactly the same with all other crops as $c$ does and replace one of the occurrences of $c$ with $c'$. We iterate this procedure until there are no more of the same two crops. By theorem \ref{kByKOptSwap} it now follows, as we have $k$ distinct crops among $k$ pixels, that there must exist an optimal integer solution $x^* \in F$ for the swap LP'. We now show that 1) this solution can be translated into an integer solution for the swap LP and 2) that this solution is optimal. The first part is trivial. Simply replace the `cloned' crops with their originals and we obtain a valid solution for the swap LP. Next we show that an optimal solution $x^*$ for the swap LP must be $\leq$ an optimal solution for the swap LP'. For this simply iterate through all duplicate crops in $x^*$ and replace them with cloned versions. Now again we have the exact same score, as the cloned crop version behave the same as the originals, but have a valid solution for the swap LP'. Obviously the optimal solution for the swap LP' is bounded from below by this solution $x^*$ for the swap LP. What when the number of crop types is smaller then the number of involved pixels? Even then the linear program must have an optimal integer solution. Finally we show that an optimal solution $x^*$ for the swap LP' must be $\leq$ an optimal solution for the swap LP. Again, we replace all of the cloned crop versions with their original crop. The score must remain the same. Thus we have bounded the optimal solution for the swap LP from below with the score of $x^*$ for the solution to the swap LP'. TODO: finish proof As we have shown both that the optimal solution for swap LP $\leq$ the optimal solution for swap LP' and the optimal solution for swap LP' $\leq$ the optimal solution for swap LP it must follow that the optimal score for both linear programs is the same. Therefore, using the swap LP' we can generate an optimal integer solution for the swap LP. \end{proof} \section{Method} ... ...
 ... ... @@ -4,6 +4,7 @@ %% Special characters for number sets, e.g. real or complex numbers. \newcommand{\C}{\mathbb{C}} \newcommand{\Cps}{\mathcal{C}} \newcommand{\F}{\mathcal{F}} \newcommand{\K}{\mathbb{K}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} ... ...
No preview for this file type
 ... ... @@ -4,5 +4,7 @@ \contentsline {proof}{{Proof}{2}{}}{17}{proof.2}% \contentsline {theorem}{{Theorem}{7.{2}}{}}{18}{theorem.7.2.2}% \contentsline {proof}{{Proof}{3}{}}{19}{proof.3}% \contentsline {theorem}{{Theorem}{8.{1}}{}}{22}{theorem.8.1.1}% \contentsline {theorem}{{Theorem}{8.{1}}{}}{23}{theorem.8.1.1}% \contentsline {proof}{{Proof}{4}{}}{23}{proof.4}% \contentsline {theorem}{{Theorem}{8.{2}}{}}{23}{theorem.8.1.2}% \contentsline {proof}{{Proof}{5}{}}{23}{proof.5}%
 ... ... @@ -20,7 +20,7 @@ \contentsline {subsection}{\numberline {7.2.2}The advanced method}{18}{subsection.7.2.2}% \contentsline {chapter}{\chapternumberline {8}The Linear Programming Method}{21}{chapter.8}% \contentsline {section}{\numberline {8.1}Problem Setup}{21}{section.8.1}% \contentsline {section}{\numberline {8.2}Method}{23}{section.8.2}% \contentsline {section}{\numberline {8.2}Method}{24}{section.8.2}% \contentsline {chapter}{\chapternumberline {9}Gradient Descent}{25}{chapter.9}% \contentsline {chapter}{\chapternumberline {10}Conclusion}{27}{chapter.10}% \contentsline {appendix}{\chapternumberline {A}Calculations Appendix}{29}{appendix.A}% ... ...
Supports Markdown
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!