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Notebook[{
Cell["Lab 4:  Where's the Solution?", "Title",
  Background->RGBColor[0, 0, 1]],

Cell["Math 421\tFall 2004", "Subtitle"],

Cell[CellGroupData[{

Cell["Introduction", "Section"],

Cell["\<\
Today, we'll look at a wee bit more constrained multivariable optimization...\
\
\>", "Text",
  TextAlignment->Left,
  TextJustification->0],

Cell["...It's not a religion, it's just a technique...", "Commentary"]
}, Closed]],

Cell[CellGroupData[{

Cell["My Little Problem...", "Section"],

Cell[TextData[{
  "Let's say we want to solve the following constrained multivariable problem \
with a two inequality constraints.  There are several steps that we go \
through, so let's see if we can understand ",
  StyleBox["why",
    FontWeight->"Bold"],
  " as we work on the logic below.  Suppose we start with something like: \n\t\
minimize ",
  Cell[BoxData[
      \(TraditionalForm\`f(x, y) = \((x - 1)\)\^2 + y\^2 + 1\)]],
  "\n\tsubject to:  ",
  Cell[BoxData[
      \(TraditionalForm\`g(x, y) = x - \(1\/4\) y\^2 \[GreaterEqual] 2\)]],
  "\n\t\t     ",
  Cell[BoxData[
      \(TraditionalForm\`h(x, y) = 2  x - y\^2 \[GreaterEqual] 3\)]],
  "\nWhat can we do with it all?  Let's graph the contour diagram of the \
function ",
  StyleBox["f",
    FontSlant->"Italic"],
  ", and graph the ",
  StyleBox["equality",
    FontWeight->"Bold"],
  " form of the constraints:\n",
  Cell[BoxData[
      \(TraditionalForm\`g(x, y) = \(x - \(1\/4\) y\^2 = 2\)\)]],
  " or  ",
  Cell[BoxData[
      \(TraditionalForm\`g(x, y) = \(x - \(1\/4\) y\^2 - 2 = 0\)\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`h(x, y) = \(2  x - y\^2 = 3\)\)]],
  " or  ",
  Cell[BoxData[
      \(TraditionalForm\`h(x, y) = \(2  x - y\^2 - 3 = 0\)\)]]
}], "Text"],

Cell[BoxData[{
    \(Clear[f, g, h, x, y]\), "\[IndentingNewLine]", 
    \(Off[General::"\<spell1\>"]\), "\[IndentingNewLine]", 
    \(Off[General::"\<spell\>"]\), "\[IndentingNewLine]", 
    \(\(f[x_, y_] = \((x - 1)\)^2 + y^2 + 1;\)\), "\[IndentingNewLine]", 
    \(\(g[x_, y_] = x - y^2/4 - 2;\)\), "\[IndentingNewLine]", 
    \(\(h[x_, y_] = 2  x - y^2 - 3;\)\)}], "Input"],

Cell[TextData[{
  "The function CPLabel basically overlays an array of f[x,y]values on top of \
the standard ",
  StyleBox["Mathematica ",
    FontSlant->"Italic"],
  "ContourPlot command..."
}], "Text"],

Cell[BoxData[
    \(?? ContourPlot\)], "Input"],

Cell["Here we declare it...", "Text"],

Cell[BoxData[
    \(CPLabel[f_, xlimit_, \ ylimit_, \ xclabl_, \ yclabl_, \ opts___] := \n\t
      CompoundExpression[\ 
        Clear[p1c, p1t], \n\t\tp1c = 
          ContourPlot[f, xlimit, ylimit, opts, ContourShading -> False, 
            DisplayFunction -> Identity], \ 
        p1t = Table[Text[N[f, 2], {xlimit[\([1]\)], ylimit[\([1]\)]}], 
            xclabl, yclabl], \n\t\tShow[{p1c, Graphics[p1t]}, \ 
          DisplayFunction -> $DisplayFunction]]\)], "Input"],

Cell[TextData[{
  "... and here we use it...\n\t\tThe first argument is the function\n\t\tThe \
second argument set is the range of ",
  StyleBox["x",
    FontSlant->"Italic"],
  " values\n\t\tThe third argument set is the range of ",
  StyleBox["y",
    FontSlant->"Italic"],
  " values\n\t\tThe fourth argument set is the min, max, and step size for \
function values as you move across the ",
  StyleBox["x",
    FontSlant->"Italic"],
  "-axis\n\t\tThe fifth argument set is the min, max, and step size for \
function values as you move up and down the ",
  StyleBox["y",
    FontSlant->"Italic"],
  "-axis\n\t\t(the other arguments I've used below are just options on a \
standard ContourPlot command... "
}], "Text"],

Cell[BoxData[
    \(\(fplot = 
        CPLabel[f[x, y], {x, 0, 10}, {y, \(-5\), 5}, {x, 0, 10, 
            2}, {y, \(-5\), 5, 2}, Axes -> True, 
          AxesLabel -> {"\<x\>", "\<y\>"}];\)\)], "Input"],

Cell["\<\
We'll need to do some implicit plots... so load the package... \
\>", "Text"],

Cell[BoxData[
    \(Needs["\<Graphics`ImplicitPlot`\>"]\)], "Input"],

Cell["... remember how to use it?", "Text"],

Cell[BoxData[
    \(\(?ImplicitPlot\)\)], "Input"],

Cell["Here is constraint g:", "Text"],

Cell[BoxData[
    \(\(gplot = 
        ImplicitPlot[g[x, y] \[Equal] 0, {x, 0, 10}, {y, \(-5\), 5}, 
          PlotStyle \[Rule] RGBColor[1, 0, 0]];\)\)], "Input"],

Cell["Here is constraint h:", "Text"],

Cell[BoxData[
    \(\(hplot = 
        ImplicitPlot[h[x, y] \[Equal] 0, {x, 0, 10}, {y, \(-5\), 5}, 
          PlotStyle \[Rule] RGBColor[0, 0, 1]];\)\)], "Input"],

Cell["Put it all together:", "Text"],

Cell[BoxData[
    \(\(Show[fplot, gplot, hplot];\)\)], "Input"],

Cell[TextData[StyleBox["Question 1:\tWhere is the feasible region?",
  FontWeight->"Bold"]], "Text",
  Background->RGBColor[0, 1, 1]],

Cell[TextData[{
  "Let's see if there is a critical point in the feasible region!\nJust to be \
different, I won't declare the ",
  StyleBox["grad",
    FontWeight->"Bold"],
  " functions we've been using (feel free to use them if you want...)"
}], "Text"],

Cell[BoxData[
    \(sol = 
      Solve[{\[PartialD]\_x f[x, y] \[Equal] 
            0, \[PartialD]\_y\ f[x, y] \[Equal] 0}, {x, y}]\)], "Input"],

Cell[BoxData[
    \(soln1 = {sol[\([1, 1, 2]\)], sol[\([1, 2, 2]\)]}\)], "Input"],

Cell[TextData[{
  "Using ",
  StyleBox["ListPlot",
    FontWeight->"Bold"],
  ", ",
  "I can plot this point on the graphics I've created above..."
}], "Text"],

Cell[BoxData[
    \(\(?ListPlot\)\)], "Input"],

Cell[BoxData[
    \(\(solnplot = 
        ListPlot[{soln1}, PlotStyle \[Rule] PointSize[0.02]];\)\)], "Input"],

Cell["... and then I can display it all!", "Text"],

Cell[BoxData[
    \(\(Show[fplot, gplot, hplot, solnplot];\)\)], "Input"],

Cell[TextData[StyleBox["Question 2:\tSo... is the critical point in the \
feasible region?",
  FontWeight->"Bold"]], "Text",
  Background->RGBColor[0, 1, 1]],

Cell[TextData[StyleBox["Question 3:\tIf not, what constraint(s) is/are \
violated?",
  FontWeight->"Bold"]], "Text",
  Background->RGBColor[0, 1, 1]],

Cell[TextData[StyleBox["Question 4:\tCan we ever have an optimal solution \
(remember, we are minimizing!) that is strictly in the interior of the \
feasible region?",
  FontWeight->"Bold"]], "Text",
  Background->RGBColor[0, 1, 1]],

Cell["What to do next?", "Subsubsection"],

Cell["Let's form the LaGrangian equation:", "Text"],

Cell[BoxData[
    \(Leqn = f[x, y] - \[Lambda]1\ g[x, y] - \[Lambda]2\ h[x, y]\)], "Input"],

Cell["\<\
... and see if we have any critical points when both constraints are \
binding...\
\>", "Text"],

Cell[BoxData[
    \(soln = 
      Solve[{\[PartialD]\_x Leqn \[Equal] 0, \[PartialD]\_y\ Leqn \[Equal] 0, 
            g[x, y] \[Equal] 0, h[x, y] \[Equal] 0}, {x, 
            y, \[Lambda]1, \[Lambda]2}] // N\)], "Input"],

Cell[TextData[{
  "Looks like we've got 2 of them!  Let's strip off the ",
  StyleBox["x",
    FontSlant->"Italic"],
  " and ",
  StyleBox["y",
    FontSlant->"Italic"],
  " values:"
}], "Text"],

Cell[BoxData[{
    \(bothsoln1 = {soln[\([1, 3, 2]\)], 
        soln[\([1, 4, 2]\)]}\), "\[IndentingNewLine]", 
    \(bothsoln2 = {soln[\([2, 3, 2]\)], soln[\([2, 4, 2]\)]}\)}], "Input"],

Cell["What is the function value for each critical point?", "Text"],

Cell[BoxData[
    \(f[x, y] /. {soln[\([1, 3]\)], soln[\([1, 4]\)]}\)], "Input"],

Cell["...and ", "Text"],

Cell[BoxData[
    \(f[x, y] /. {soln[\([2, 3]\)], soln[\([2, 4]\)]}\)], "Input"],

Cell["Let's plot them!", "Text"],

Cell[BoxData[
    \(\(solnplot = 
        ListPlot[{soln1, bothsoln1, bothsoln2}, 
          PlotStyle \[Rule] PointSize[0.02]];\)\)], "Input"],

Cell["... and look at it all together again!", "Text"],

Cell[BoxData[
    \(\(Show[fplot, gplot, hplot, solnplot];\)\)], "Input"],

Cell[TextData[StyleBox["Question 5:\tCan we do better?   ",
  FontWeight->"Bold"]], "Text",
  Background->RGBColor[0, 1, 1]],

Cell["Probably so, or the notebook would end here, eh?", "Commentary",
  FontColor->RGBColor[0, 0, 1]],

Cell[TextData[{
  "I'd guess that maybe we should be looking at the red constraint... which \
one is it?\nahhh... ",
  StyleBox["g",
    FontSlant->"Italic"],
  " ... ok... then let's make it binding, and we'll ignore constraint ",
  StyleBox["h:",
    FontSlant->"Italic"]
}], "Text"],

Cell[BoxData[
    \(gsoln = 
      Solve[{\[PartialD]\_x Leqn \[Equal] 0, \[PartialD]\_y\ Leqn \[Equal] 0, 
            g[x, y] \[Equal] 0, \[Lambda]2 == 0}, {x, 
            y, \[Lambda]1, \[Lambda]2}] // N\)], "Input"],

Cell[TextData[{
  "Yowsah!  imaginary solutions... I don't imagine we'll want to do much with \
them...\nLet's strip off the ",
  StyleBox["x",
    FontSlant->"Italic"],
  " and ",
  StyleBox["y",
    FontSlant->"Italic"],
  " values on the ",
  StyleBox["real",
    FontWeight->"Bold"],
  " one:"
}], "Text"],

Cell[BoxData[
    \(gsoln1 = {gsoln[\([3, 2, 2]\)], gsoln[\([3, 4, 2]\)]}\)], "Input"],

Cell["What is the function value at this critical point?", "Text"],

Cell[BoxData[
    \(f[x, y] /. {gsoln[\([3, 2]\)], gsoln[\([3, 4]\)]}\)], "Input"],

Cell["Let's plot it!", "Text"],

Cell[BoxData[
    \(\(solnplot = 
        ListPlot[{soln1, bothsoln1, bothsoln2, gsoln1}, 
          PlotStyle \[Rule] PointSize[0.02]];\)\)], "Input"],

Cell["... and look at it all together again!", "Text"],

Cell[BoxData[
    \(\(Show[fplot, gplot, hplot, solnplot];\)\)], "Input"],

Cell[TextData[StyleBox["Question 6:\tCan we do any better?  Why or why not?",
  FontWeight->"Bold"]], "Text",
  Background->RGBColor[0, 1, 1]],

Cell[TextData[StyleBox["Question 7:\tOh yeah... one last thing... what are \
the values of the LaGrange multipliers at the optimal solution?",
  FontWeight->"Bold"]], "Text",
  Background->RGBColor[0, 1, 1]],

Cell[TextData[{
  StyleBox["Question 8:\tInterpret their meaning with respect to the \
constraints they go with (\[Lambda]1 goes with ",
    FontWeight->"Bold"],
  StyleBox["g",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox[" and \[Lambda]2 goes with ",
    FontWeight->"Bold"],
  StyleBox["h",
    FontWeight->"Bold",
    FontSlant->"Italic"],
  StyleBox[")... which constraint(s) is/are binding at the optimal solution?  \
Are the associated values of the LaGrange multipliers positive, negative, or \
0?  Why? ",
    FontWeight->"Bold"]
}], "Text",
  Background->RGBColor[0, 1, 1]]
}, Closed]],

Cell[CellGroupData[{

Cell["Inside the Shadow Cabinet...", "Section"],

Cell[TextData[{
  "OK... now  you've walked through a problem... so let's try one on your \
own!\nLet's minimize the function ",
  StyleBox["f",
    FontSlant->"Italic"],
  " defined below, along with the following constraints:\n\tsubject to:  ",
  Cell[BoxData[
      \(TraditionalForm\`g(x, y) = x \[LessEqual] 3\)]],
  "\n\t\t     ",
  Cell[BoxData[
      \(TraditionalForm\`h(x, y) = y \[LessEqual] 3\)]]
}], "Text"],

Cell[BoxData[{
    \(Clear[f, g, h, x, y]\), "\[IndentingNewLine]", 
    \(\(f[x_, y_] = 
        y \((x - 2)\)^3 - \((x\ y - 3)\)^3 \((x + 4)\)^2 + 
          5;\)\), "\[IndentingNewLine]", 
    \(\(g[x_, y_] = x - 3;\)\), "\[IndentingNewLine]", 
    \(\(h[x_, y_] = y - 3;\)\)}], "Input"],

Cell[BoxData[
    \(\(fplot = 
        CPLabel[f[x, y], {x, \(-4\), 4}, {y, \(-4\), 4}, {x, \(-4\), 4, 
            2}, {y, \(-4\), 4, 2}, Axes -> True, 
          AxesLabel -> {"\<x\>", "\<y\>"}, Contours \[Rule] 50];\)\)], "Input"],

Cell["Go ahead!", "Text"],

Cell[BoxData[
    \(\ \)], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["The Other Side...", "Section"],

Cell[TextData[{
  "So we've played with some \"easy\" functions.. let's step into the beyond. \
 What if you have a function like this:\n\tmaximize ",
  Cell[BoxData[
      \(TraditionalForm\`f(x, 
          y) = \(sin(x\^2)\) \(cos(y)\) - \(cos(x)\) \(sin(y\^2)\)\)]],
  "\n\tsubject to:  ",
  Cell[BoxData[
      \(TraditionalForm\`g(x, y) = 
        x\^2 + \(1\/4\) y\^2 \[GreaterEqual] 2\)]],
  "\n\t\t     ",
  Cell[BoxData[
      \(TraditionalForm\`h(x, y) = x + y \[LessEqual] 6\)]],
  "\nSee what you can do with this one... first, maybe try graphing \
everything to look at it... here's a start:"
}], "Text"],

Cell[BoxData[{
    \(Clear[f, g, h, x, y]\), "\[IndentingNewLine]", 
    \(\(f[x_, y_] = 
        Sin[x^2] Cos[y] - Cos[x] Sin[y^2];\)\), "\[IndentingNewLine]", 
    \(\(g[x_, y_] = x^2 + y^2/4 - 2;\)\), "\[IndentingNewLine]", 
    \(\(h[x_, y_] = x + y - 6;\)\)}], "Input"],

Cell[BoxData[
    \(\(fplot = 
        CPLabel[f[x, y], {x, \(-Pi\), Pi}, {y, \(-Pi\), Pi}, {x, \(-Pi\), Pi, 
            Pi/3}, {y, \(-Pi\), Pi, Pi/3}, Axes -> True, 
          AxesLabel -> {"\<x\>", "\<y\>"}];\)\)], "Input"],

Cell["You do the rest... ", "Text"],

Cell[BoxData[
    \(\ \)], "Input"],

Cell["\<\
Maybe we need something else to solve problems like these, eh?\
\>", "Commentary",
  FontColor->RGBColor[0, 0, 1]]
}, Closed]]
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