(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 415284, 7724]*) (*NotebookOutlinePosition[ 416054, 7750]*) (* CellTagsIndexPosition[ 416010, 7746]*) (*WindowFrame->Normal*) Notebook[{ Cell["Lab 5: Constrained Optimization", "Title", Background->RGBColor[0, 0, 1]], Cell[TextData[{ "Math 233\tFall 2001\t", StyleBox["Chapter 11, Section 8 and beyond", FontFamily->"Arial", FontSize->16, FontWeight->"Bold"], " " }], "Subtitle", TextAlignment->Left, TextJustification->0], Cell[CellGroupData[{ Cell[TextData[StyleBox["Instructions!", FontColor->RGBColor[1, 0, 0]]], "Section"], Cell[TextData[{ "Work in ", StyleBox["groups of 2 or 3", FontWeight->"Bold"], " today, going through the entire notebook. If you have questions, flag me \ down. Write up your answers (complete sentences, please) for a ", StyleBox["group", FontWeight->"Bold"], " hand in on Tuesday of next week (", StyleBox["30 October)", FontWeight->"Bold"], ". \n\nAs a separate hand-in, you should ", StyleBox["each", FontWeight->"Bold"], " start on homework problem 34 from section 11.8. This problems will also \ be due on ", StyleBox["Tuesday (30 October)", FontWeight->"Bold"], "." }], "Subsubtitle"] }, Closed]], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "OBJECTIVE: Today, you will learn:\n\t1. how to maximize or minimize a \ function subject to a single equality constraint using ", StyleBox["Mathematica,", FontSlant->"Italic"], "\n\t2. how to maximize or minimize a function subject to a set of \ equality constraint using ", StyleBox["Mathematica,", FontSlant->"Italic"], "\n", StyleBox["You probably don't have time to complete the following in the lab \ today, but finish it on your own, since it will be helpful for the project \ (and since it has many questions for your lab hand-in!):", FontWeight->"Bold"], "\n\t3. how to maximize or minimize a function subject to a set of \ inequality constraints using ", StyleBox["Mathematica", FontSlant->"Italic"], " \n\t" }], "Text"], Cell[CellGroupData[{ Cell["Technology Guidelines", "Subsection", CellDingbat->"\[LightBulb]"], Cell[TextData[{ StyleBox["NOTE: If you have just finished a module, restart ", CellFrame->True, Background->None], StyleBox["Mathematica", CellFrame->True, FontSlant->"Italic", Background->None], StyleBox[" before executing a new module.\nTO OPEN CELLS, put your cursor \ on the right cell bracket and double click.", CellFrame->True, Background->None], "\nTO STOP AN EXECUTION\n\tSelect the ", StyleBox["Kernel", FontSlant->"Italic"], " pull-down menu and click on ", StyleBox["Abort Evaluation.\n", FontSlant->"Italic"], "ORDER OF EXECUTION\n\tExecute cells in the order given. Do not skip any \ Input cells within a given notebook.\nSAVING NOTEBOOKS\n\tYou can save \ anytime to any directory you choose, and it is wise to save often.\n\t\ However, before you do your final save, delete all your output by selecting \ the \n\t ", StyleBox["Delete All Output", FontSlant->"Italic"], " selection under the ", StyleBox["Kernel", FontSlant->"Italic"], " pull-down menu.\nEXPERIENCING MAJOR PROBLEMS\n\tSave if appropriate, and \ then shut down ", StyleBox["Mathematica", FontSlant->"Italic"], " and start it up again." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Part I: LaGrange on a Skateboard", "Section"], Cell["Acknowledgement:\tIdea for Part I from a lab by Marie Vanisko", \ "Commentary"], Cell["\<\ Today,we'll look at how to help LaGrange figure out his height when \ skateboarding a figure-8 path in a bowl.\ \>", "Text"], Cell[CellGroupData[{ Cell["The Figure-8 Path and the Bowl", "Subsubsection"], Cell[TextData[{ "The following ", StyleBox["Mathematica", FontSlant->"Italic"], " commands are to help you visualize the problem - don't worry about adding \ them to your toolbox of cool ", StyleBox["Mathematica", FontSlant->"Italic"], " factiods. I've set them up as ", StyleBox["initialization cells", FontWeight->"Bold"], " in this notebook. That means that when you go to execute a cell for the \ first time, you will be prompted to see if you want to execute all the \ initialization cells. Answer yes, and then all the heavy duty ", StyleBox["Mathematica", FontSlant->"Italic"], " ", "that you need will be at your fingertips... " }], "Text"], Cell[BoxData[{ \(<< Graphics`Arrow`\[IndentingNewLine]\), "\[IndentingNewLine]", \(Off[General::spell]\ \), "\n", \(Off[General::spell1]\[IndentingNewLine]\), "\[IndentingNewLine]", \(Clear[x, y, t, f]\), "\n", \(\(x1[t_] = 16 \( Sin[t]\^2\) Cos[t];\)\), "\n", \(\(y1[t_] = 16 Sin[t]\^3;\)\), "\n", \(\(pp8 = ParametricPlot[{x1[t], y1[t]}, {t, 0, 2 \[Pi]}, AspectRatio -> Automatic, PlotStyle -> RGBColor[1, 0, 0], AxesLabel -> {"\", "\"}, Epilog \[Rule] {Arrow[{x1[1], y1[1]}, {x1[1.1], y1[1.1]}, HeadScaling \[Rule] Absolute], Arrow[{x1[2.5], y1[2.5]}, {x1[2.6], y1[2.6]}, HeadScaling \[Rule] Absolute], Arrow[{x1[4], y1[4]}, {x1[4.1], y1[4.1]}, HeadScaling \[Rule] Absolute], Arrow[{x1[5.5], y1[5.5]}, {x1[5.6], y1[5.6]}, HeadScaling \[Rule] Absolute]}, \ DisplayFunction \[Rule] Identity];\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(Clear[x, y, t]\), "\n", \(\(ramp[x_, y_] = .05 x\^2;\)\), "\n", \(\(p8ramp = ParametricPlot3D[{x1[t], y1[t], ramp[x1[t], y1[t]], RGBColor[1, 0, 0]}, {t, 0, 2 \[Pi]}, AxesLabel -> {x, y, z}, DisplayFunction -> Identity, PlotPoints \[Rule] 40, ViewPoint -> {\(-0.009\), \ \(-2.516\), \ 2.262}];\)\), "\n", \(\(pramp = Plot3D[ramp[x, y] - 0.1, {x, \(-10\), 10}, {y, \(-20\), 20}, DisplayFunction -> Identity];\)\), "\n", \(\(ctramp = ContourPlot[ramp[x, y], {x, \(-10\), 10}, {y, \(-20\), 20}, ContourShading -> False, DisplayFunction -> Identity];\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(Clear[x, y, t]\), "\[IndentingNewLine]", \(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]]\[IndentingNewLine]\), "\ \[IndentingNewLine]", \(<< Graphics`ImplicitPlot`\)}], "Input", CellOpen->False, InitializationCell->True], Cell["\<\ Go ahead and execute the cell below to unleash the hidden power... The first graph shows the figure-8 path that LaGrange is skateboarding on... \ (in the x-y plane) The second graph shows the figure-8 path superimposed on the actual bowl he \ is skateboarding on... The last graph shows the figure-8 path on top of the contour diagram of the \ bowl he is skating on...\ \>", "Text"], Cell[BoxData[{ \(\(Show[pp8, DisplayFunction -> $DisplayFunction];\)\), "\[IndentingNewLine]", \(\(Show[p8ramp, pramp, DisplayFunction -> $DisplayFunction];\)\), "\n", \(\(Show[pp8, ctramp, DisplayFunction -> $DisplayFunction];\)\)}], "Input"], Cell[TextData[StyleBox["Question 1:\tFrom the geometry of the problem, \ indicate where you will reach the high and low points along the figure-8 \ inside the bowl.", FontWeight->"Bold"]], "Text"], Cell[TextData[{ StyleBox["(Type your answer here)", FontSlant->"Italic"], "\n" }], "Commentary", Background->RGBColor[0, 1, 1]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Finding the Maximum and Minimum Heights: LaGrange Multipliers for LaGrange \ on Skateboard\ \>", "Subsubsection"], Cell["\<\ The commnands in this section are more directly useful to you for homework, \ projects, etc... so pay attention as we go through them.\ \>", "Text", FontColor->RGBColor[1, 0, 1]], Cell["First, let's define the Height function of the bowl:", "Text"], Cell[BoxData[ \(\(height[x_, y_] = .05 x\^2;\)\)], "Input"], Cell["\<\ Now... here is the description of the path that Lagrange is following:\ \>", "Text"], Cell[BoxData[ \(\(eight[x_, y_] = \((x\^2 + y\^2)\)\^3 - \((4 y)\)\^4;\)\)], "Input"], Cell["Now... we'll define the LaGrangian function:", "Text"], Cell[BoxData[{ \(\(Clear[x, y, \[Lambda]];\)\), "\[IndentingNewLine]", \(Lagrange[x_, y_, \[Lambda]_] = height[x, y] - \[Lambda]*eight[x, y]\)}], "Input"], Cell["And Presto: Let's find where the gradient is 0:", "Text"], Cell[BoxData[ \(solution = Solve[{D[Lagrange[x, y, \[Lambda]], x] \[Equal] 0, D[Lagrange[x, y, \[Lambda]], y] \[Equal] 0, D[Lagrange[x, y, \[Lambda]], \[Lambda]] \[Equal] 0}, {x, y, \[Lambda]}]\)], "Input"], Cell[TextData[{ "Kind of hard to digest all of that isn't it. Well, we can use the ", StyleBox["Table", FontWeight->"Bold"], " command to make a nice looking table of data:" }], "Text"], Cell[BoxData[ \(\(?Table\)\)], "Input"], Cell[TextData[{ "I'm going to use some pretty funky commands to grab the pieces of the ", StyleBox["list ", FontWeight->"Bold"], "solution that contains all of the solutions. Basically, I'm grabbing the \ x and y coordinates of the solutions and plugging them into the ", StyleBox["height", FontWeight->"Bold"], " function to figure out how high the skateboarder is. " }], "Text"], Cell[BoxData[{ \(\(results = Table[{solution[\([i, 3, 2]\)], solution[\([i, 2, 2]\)], height[solution[\([i, 3, 2]\)], solution[\([i, 2, 2]\)]]}, {i, 4, Length[solution]}];\)\), "\n", \(display = PrependTo[results, {"\", "\", "\"}] // TableForm\)}], "Input"], Cell[TextData[{ "I did ", StyleBox["not", FontWeight->"Bold"], " include the critical point (0,0) in the above table (basically, it was \ too hard to isolate since there was no \[Lambda] component). Make sure that \ you consider the origin as you answer the following question:" }], "Text"], Cell["\<\ Question 2:\tUsing the results above, identify the points on the contour and \ surface plots below that represent a maximum or minimum. \ \>", "Text", FontWeight->"Bold"], Cell[BoxData[{ \(\(Show[pp8, ctramp, DisplayFunction -> $DisplayFunction];\)\), "\n", \(\(Show[p8ramp, pramp, DisplayFunction -> $DisplayFunction];\)\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Part II: What!? Two Constraints!", "Section"], Cell[CellGroupData[{ Cell["How do we handle 2 constraints?", "Subsubsection"], Cell["\<\ Here is a multivariable problem with two constraints... let's look through it \ in a little more detail... For starters, we want to minimize this function:\ \>", "Text"], Cell[BoxData[{ \(\(Clear[T, f, x, y, g1, g2, \[Lambda]1, \[Lambda]2];\)\), "\[IndentingNewLine]", \(\(T[x_, y_] = \((x - 14)\)\^2 + \((y - 11)\)\^2;\)\)}], "Input"], Cell[TextData[{ "... and we have these two constraints:\n\t\t", StyleBox["g1: (x-11)^2 + (y-13)^2 = 7^2\n\t\tg2: x + y = 19", FontSlant->"Italic"] }], "Text"], Cell[BoxData[{ \(\(g1[x_, y_] = \((x - 11)\)\^2 + \((y - 13)\)\^2 - 7\^2;\)\), "\[IndentingNewLine]", \(\(g2[x_, y_] = x + y - 19;\)\)}], "Input"], Cell[TextData[{ StyleBox["Question 3:", FontWeight->"Bold"], "\t", StyleBox["SInce both of these constraints are equality (as opposed to \ inequality), what do we know about any critical values of T[x,y]? ", FontWeight->"Bold"] }], "Text", FontWeight->"Plain"], Cell[TextData[{ StyleBox["(Type your answer here)", FontSlant->"Italic"], "\n" }], "Commentary", Background->RGBColor[0, 1, 1]], Cell["\<\ Question 4:\tWhen we just have a single constraint, the critical values are \ at points where the gradient of the optimization function and the gradient of \ the constraint function are co-linear (or parallel to each other). What do \ you think is the case when we have more than 1 constraint?\ \>", "Text", FontWeight->"Bold"], Cell[TextData[{ StyleBox["(Type your answer here)", FontSlant->"Italic"], "\n" }], "Commentary", Background->RGBColor[0, 1, 1]] }, Closed]], Cell[CellGroupData[{ Cell["Test it out!:", "Subsubsection"], Cell[TextData[{ "OK... let's try out your idea. First, create the Lagrangian function (but \ this time, we have ", StyleBox["2", FontWeight->"Bold"], " \[Lambda]'s, call them ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Lambda]\_1\)\(\ \)\)\)]], "and ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_2\)]], "." }], "Text"], Cell[BoxData[ \(Leqn[x_, y_, \[Lambda]1_, \[Lambda]2_] = T[x, y] - \[Lambda]1\ g1[x, y] - \[Lambda]2\ g2[x, y]\)], "Input"], Cell["and now go ahead and find where the gradient is 0:", "Text"], Cell[BoxData[ \(bothsoln = Solve[{D[Leqn[x, y, \[Lambda]1, \[Lambda]2], x] \[Equal] 0, D[Leqn[x, y, \[Lambda]1, \[Lambda]2], y] \[Equal] 0, D[Leqn[x, y, \[Lambda]1, \[Lambda]2], \[Lambda]1] \[Equal] 0, D[Leqn[x, y, \[Lambda]1, \[Lambda]2], \[Lambda]2] \[Equal] 0}, {x, y, \[Lambda]1, \[Lambda]2}] // N\)], "Input"], Cell["Let's look at the info in a table again:", "Text"], Cell[BoxData[{ \(\(results = Table[{bothsoln[\([i, 3, 2]\)], bothsoln[\([i, 4, 2]\)]}, {i, 1, Length[bothsoln]}];\)\), "\n", \(display = PrependTo[results, {"\", "\"}] // TableForm\)}], "Input"], Cell["\<\ Here is a graph of the contour diagram of the temperature function, then a \ second one with the two constraints added.\ \>", "Text"], Cell["\<\ Question 5:\tUsing the results above, identify the points on the contour and \ surface plots below that represent a maximum or minimum (and label them as \ maximums or minimums). \ \>", "Text", FontWeight->"Bold"], Cell[BoxData[{ \(\(fplot = CPLabel[T[x, y], {x, 0, 20}, {y, 0, 20}, {x, 0, 20, 4}, {y, 0, 20, 4}, AxesLabel -> {"\", "\"}];\)\), "\[IndentingNewLine]", \ \(\(g1plot = ImplicitPlot[g1[x, y] \[Equal] 0, {x, 0, 20}, {y, 0, 20}, PlotStyle \[Rule] RGBColor[1, 0, 0], DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(g2plot = ImplicitPlot[g2[x, y] \[Equal] 0, {x, 0, 20}, {y, 0, 20}, PlotStyle \[Rule] RGBColor[0, 0, 1], DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(Show[fplot, g1plot, g2plot, DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Part III: What if those constraints have INEQUALITIES?!?!", "Section"], Cell["\<\ Now, we'll see what surprises the Shadow Cabinet has for us today when we \ open it to find constraints with INEQUALITIES!\ \>", "Text"], Cell[CellGroupData[{ Cell["Graphically...", "Subsubsection"], Cell[TextData[{ "Here is our favorite multivariable problem, but this time I've changed the \ equality constraints to ", StyleBox["inequalities", FontWeight->"Bold"], "! How much more difficult does this little twist make life?\nAgain, here \ is the function we want to minimize:" }], "Text"], Cell[BoxData[{ \(\(Clear[T, f, x, y, g1, g2, \[Lambda]1, \[Lambda]2];\)\), "\[IndentingNewLine]", \(\(T[x_, y_] = \((x - 14)\)\^2 + \((y - 11)\)\^2;\)\)}], "Input"], Cell[TextData[{ "... and we have these two constraints:\n\t\t", StyleBox["g1: (x-11)^2 + (y-13)^2 \[LessEqual] 7^2\n\t\tg2: x + y \ \[LessEqual] 19", FontSlant->"Italic"] }], "Text"], Cell[BoxData[{ \(\(g1[x_, y_] = \((x - 11)\)\^2 + \((y - 13)\)\^2 - 7\^2;\)\), "\[IndentingNewLine]", \(\(g2[x_, y_] = x + y - 19;\)\)}], "Input"], Cell["\<\ We could get a visual of this problem by looking at the contour diagram of \ the function and plotting the constraints on top of it. \ \>", "Text"], Cell[BoxData[ \(\(Show[fplot, g1plot, g2plot];\)\)], "Input"], Cell[TextData[{ StyleBox["Question 6:", FontWeight->"Bold"], "\t", StyleBox["Shade in the feasible region on the contour diagram above. The \ feasible region is the area defined by the intersection of the constraints. \ Remember, we have plotted the equality form of the constraints... ", FontWeight->"Bold"] }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Attempt number 1...", "Subsubsection"], Cell["\<\ In an ideal world, we might find a solution to the unconstrained version of \ the problem that solves the constrained version (well, it is usually worth \ checking out at least!) Go ahead and solve for the points where the derivative of T[x,y] is 0:\ \>", "Text"], Cell[BoxData[""], "Input"], Cell[TextData[{ StyleBox["Question 7:", FontWeight->"Bold"], "\t", StyleBox["What is the critical point you found? ", FontWeight->"Bold"] }], "Text"], Cell[TextData[{ StyleBox["(Type your answer here)", FontSlant->"Italic"], "\n" }], "Commentary", Background->RGBColor[0, 1, 1]], Cell["\<\ Here is the big picture again... it might help for question 8...\ \>", "Text"], Cell[BoxData[ \(\(Show[fplot, g1plot, g2plot];\)\)], "Input"], Cell[TextData[{ StyleBox["Question 8:", FontWeight->"Bold"], "\t", StyleBox["Is the solution to the unconstrained problem in the feasible \ region ", FontWeight->"Bold"] }], "Text"], Cell[TextData[{ StyleBox["(Type your answer here)", FontSlant->"Italic"], "\n" }], "Commentary", Background->RGBColor[0, 1, 1]] }, Closed]], Cell[CellGroupData[{ Cell["Attempt number 2...", "Subsubsection"], Cell["\<\ Well... you should end up with something like this from attempt number 1...\ \>", "Text"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 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