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jiedxu · May 19, 2020 · 21b15ef · 21b15ef
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diff --git a/docs/LP-dual.Rmd b/docs/LP-dual.Rmd
@@ -25,6 +25,12 @@ knitr::include_graphics("images/LP-2.png")
 > 1. The parameters for a (functional) constraint in either problem are the coefficients of a variable in the other problem.  
 > 2. The coefficients in the objective function of either problem are the right-hand sides for the other problem.
 
+### SOB Rules
+
+```{r, out.width='90%', echo=F, fig.align='center', fig.cap='Primal-Dual Table '}
+knitr::include_graphics("images/LP-3.png")
+```
+
 ### Primal-Dual Relationships
 
 ```{definition, name='Complementary optimal solutions property'}
@@ -44,3 +50,11 @@ The following are the only possible relationships between the primal and dual pr
 2. If one problem has feasible solutions and an unbounded objective function (and so no optimal solution), then the other problem has no feasible solutions.  
 3. If one problem has no feasible solutions, then the other problem has either no feasible solutions or an unbounded objective function.
 ```
+
+### Complementary Slackness
+
+Assume that $x$ is a feasible solution to the primal problem and $y$ is a feasible solution to the dual problem. Then, a necessary and sufficient condition for $x$ and $y$ being optimal are:
+$$ \begin{align}
+& \sum_{j=1}^{n} a_{i j} x_{j}=b_{i} \text { or } y_{i}=0 \quad \forall i=1 \ldots m \\
+& \sum_{i=1}^{m} a_{i j} y_{i}=c_{j} \text { or } x_{j}=0 \quad \forall j=1 \ldots n
+\end{align} $$
diff --git a/docs/NLP-KKT.Rmd b/docs/NLP-KKT.Rmd
@@ -112,9 +112,3 @@ So I would allocate 56.133 hours in total to produce 22 and half bottles of IPA,
 The results obtained from modern optimisation algorithms can be validated using the duality gap and KKT conditions.
 
 > For complicated problems, it may be difficult, if not essentially impossible, to derive an optimal solution directly from the KKT conditions. Nevertheless, these conditions still provide valuable clues as to the identity of an optimal solution, and they also permit us to check whether a proposed solution may be optimal. [@hillier2012introduction]
-
-### Lagrangian Duality
-
-> There also are many valuable indirect applications of the KKT conditions. One of these applications arises in the duality theory that has been developed for nonlinear programming to parallel the duality theory for linear programming. [@hillier2012introduction]
-
-The basic idea in Lagrangian duality is to take the constraints in (5.1) into account by augmenting the objective function with a weighted sum of the constraint functions.
diff --git a/docs/NLP-convex.Rmd b/docs/NLP-convex.Rmd
@@ -61,3 +61,24 @@ A convex set is a collection of points such that, for each pair of points in the
 > In mathematics, Slater's condition (or Slater condition) is a sufficient condition for strong duality to hold for a convex optimisation problem.
 
 > If a convex optimization problem with differentiable objective and constraint functions satisfies Slater’s condition, then the KKT conditions provide necessary and sufficient conditions for optimality. [@boyd2004convex]
+
+### Lagrangian Duality
+
+> There also are many valuable indirect applications of the KKT conditions. One of these applications arises in the duality theory that has been developed for nonlinear programming to parallel the duality theory for linear programming. [@hillier2012introduction]
+
+The basic idea in Lagrangian duality is to take the constraints in (5.1) into account by augmenting the objective function with a weighted sum of the constraint functions.
+
+For example:
+$$ \begin{align}
+\max_{\mathbf{x}} \quad & f(\mathbf{x})\\
+\text{s.t.} \quad & \boldsymbol{g}(\boldsymbol{x}) \leq \mathbf{0} \\
+& \boldsymbol{h}(\boldsymbol{x}) = \mathbf{0}
+\end{align} $$
+
+The dual problem is
+$$ \begin{align}
+\min_{\boldsymbol{u}, \boldsymbol{v}} \quad & \theta(\boldsymbol{u}, \boldsymbol{v}) \\
+\text{s.t.} \quad & \boldsymbol{u} \geq 0
+\end{align} $$
+where
+$$ \theta(\boldsymbol{u}, \boldsymbol{v}) = \max _{\boldsymbol{x}} \{f(\boldsymbol{x}) + \boldsymbol{u} \boldsymbol{g}(\boldsymbol{x}) + \boldsymbol{v} \boldsymbol{h}(\boldsymbol{x}) \} $$
diff --git a/docs/images/LP-3.png b/docs/images/LP-3.png
diff --git a/examples/nonlinear/pre/Dyn-1.png → examples/nonlinear/Dyn-1.png b/examples/nonlinear/pre/Dyn-1.png → examples/nonlinear/Dyn-1.png
diff --git a/examples/nonlinear/Re-ExamInstructions2020.docx b/examples/nonlinear/Re-ExamInstructions2020.docx
diff --git a/...inear/pre/assign/DualSolutionExample.xlsx → ...nonlinear/assign/DualSolutionExample.xlsx b/...inear/pre/assign/DualSolutionExample.xlsx → ...nonlinear/assign/DualSolutionExample.xlsx
diff --git a/...ples/nonlinear/pre/assign/KKT_Example.pdf → examples/nonlinear/assign/KKT_Example.pdf b/...ples/nonlinear/pre/assign/KKT_Example.pdf → examples/nonlinear/assign/KKT_Example.pdf
diff --git a/examples/nonlinear/assign/backups/dtu02435a1_1.tex b/examples/nonlinear/assign/backups/dtu02435a1_1.tex
diff --git a/examples/nonlinear/assign/dynamic.pdf b/examples/nonlinear/assign/dynamic.pdf
diff --git a/examples/nonlinear/assign/images/1.JPG b/examples/nonlinear/assign/images/1.JPG
diff --git a/examples/nonlinear/assign/jabbrv-ltwa-all.ldf b/examples/nonlinear/assign/jabbrv-ltwa-all.ldf
diff --git a/examples/nonlinear/assign/jabbrv-ltwa-en.ldf b/examples/nonlinear/assign/jabbrv-ltwa-en.ldf