(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' 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). 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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[ 273919, 5549]*) (*NotebookOutlinePosition[ 274689, 5575]*) (* CellTagsIndexPosition[ 274645, 5571]*) (*WindowFrame->Normal*) Notebook[{ Cell["Lab 4: Optimization", "Title", Background->RGBColor[0, 0, 1]], Cell[TextData[{ "Math 233\tFall 2004\t", StyleBox["Chapter 15", 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. Your lab report will be due on ", StyleBox["Monday (18 October)", FontWeight->"Bold"], ".\n\n", StyleBox["Group Report:", FontWeight->"Bold"], "\n", StyleBox["The heart of the report should be a summary of your observations \ - you may cut and paste from ", FontWeight->"Bold"], StyleBox["Mathematica", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" to help clarify your explanations. For this lab, you should work \ through all parts of this lab and anwer all questions posed. Focus on the \ mathematics used and be able to explain the mathematics in each Part. Your \ written report ", FontWeight->"Bold"], StyleBox["must", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[" include plots, formula results, and your own summary of what you \ see. Your report should provide a coherent path through your lab, not just \ be a set of bullets. ", FontWeight->"Bold"], " ", StyleBox["Use your experiences in the ", FontWeight->"Bold"], StyleBox["You Try It", FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox[" section (along with the questions you answered from the previous \ section) to write a write a summary of your observations. \n\nBottom Line: \ I'm really looking for your experiences in the lab written from a journal \ sort of fashion. What worked as expected, what didn't, what are you found to \ be most surprising, was it worth your time or not. Reflect on what you have \ seen in each section. \n\n", FontWeight->"Bold"], "The easiest way to create a group report is to open up a Word document \ while doing the lab. When you come to a question, you can highlight the text \ and copy and paste it into the Word document. Your group can then type your \ answers into the Word document. If you want to use a graph in your report to \ help clarify an idea or to answer a question, you can click on the graph (so \ that it is surrounded by a box) and then copy and paste it into the Word \ document. Once you've finished the You Try It section, go back to your \ report and write a summary of your experiences - this summary can either go \ in front of or behind the questions you've already answered. \n" }], "Subsubtitle"] }, Closed]], Cell[CellGroupData[{ Cell["Background", "Section"], Cell[TextData[{ "Multivariable optimization is very much like single-variable optimization, \ but with a little more work involved! Recall that for functions of a single \ variable, we set the derivative equal to zero to find ", StyleBox["critical points", FontSlant->"Italic"], ". Maxima and minima occur at places where the derivative is zero (or \ fails to exist).\nThe idea of setting the derivative to zero in a \ multivariable setting means taking the partials with respect to both x and y \ and setting them to zero ", StyleBox["simultaneously", FontWeight->"Bold"], ", so that we end up with a system of equations that must be solved. When \ life is good, these are linear equations, but that isn't always the case.\n\ Let's get started and see what we need to do... " }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Part I: Finding Critical Points", "Section"], Cell[TextData[{ "Recall that critical points are those (x, y) pairs that simultaneously set \ both partial derivatives to 0. Let's walk through the ", StyleBox["Mathematica", FontSlant->"Italic"], " steps we need in order to find critical points.\nFirst, let's execute a \ command to get rid of some of those annoying red warning messages that \ frequently pop up during our wonderous journey through ", StyleBox["Mathematica", FontSlant->"Italic"], " land." }], "Text"], Cell[BoxData[{ \(Off[General::spell]\ \), "\n", \(Off[General::spell1]\)}], "Input"], Cell["Here is a simple function to work with:", "Text"], Cell[BoxData[ \(f[x_, y_] = 5\ x\^2\ + \ 1\ x\ y\ + \ 5\ y\^2\)], "Input"], Cell[TextData[{ "Now, we'll solve for where the derivatives are 0. There are a couple of \ ways to do this, so first we'll check out the ", StyleBox["Mathematica", FontSlant->"Italic"], " ", StyleBox["Solve", FontWeight->"Bold"], " command" }], "Text"], Cell[BoxData[ \(\(?Solve\)\)], "Input"], Cell[TextData[{ "We want to solve a system of equations rather than a single equation, so \ the form of the command is: ", StyleBox["Solve[{equation1, equation2}, {x,y}] ", FontWeight->"Bold"], "- where we enclose our set of equations in {curly brackets} and our set of \ variables we are solving for in {curly brackets}", ". Note: a trend is developing here... whenever you have a set of stuff \ in ", StyleBox["Mathematica", FontSlant->"Italic"], ", you typically use curly brackets to group them together." }], "Text", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(Solve[{\[PartialD]\_x\ f[x, y] == 0, \[PartialD]\_y\ f[x, y] == 0}, {x, y}]\)], "Input"], Cell[TextData[{ "Another way to find critical points (which will work better in many \ instances) is the ", StyleBox["FindRoot ", FontWeight->"Bold"], "command. This uses ", StyleBox["numerical", FontWeight->"Bold"], " techniques to ", StyleBox["approximate", FontWeight->"Bold"], " where the function is 0 using \"seeds\" (or guesses) that you provide. \ Here is the syntax: ", StyleBox["FindRoot[{", FontWeight->"Bold"], "eqn1,eqn2", StyleBox["}, {x, xseed}, {y, yseed}]", FontWeight->"Bold"] }], "Text"], Cell[BoxData[ \(\(?FindRoot\)\)], "Input"], Cell["Let's see if we can find any roots near (1, 1.5)", "Text"], Cell[CellGroupData[{ Cell[BoxData[ StyleBox[\(FindRoot[{\[PartialD]\_x\ f[x, y] == 0, \[PartialD]\_y\ f[x, y] == 0}, \ {{x, \ 1}, \ {y, \ 1.5}}]\), FontWeight->"Bold"]], "Input"], Cell["So the only critical point is (0,0). ", "Text"] }, Open ]], Cell[TextData[StyleBox["Cool... now we have 2 ways to do something! If one \ doesn't work; try the other!", CellFrame->True, FontWeight->"Bold", FontColor->RGBColor[1, 0, 0], Background->GrayLevel[0.833326]]], "GrayBox", CellFrame->True, TextAlignment->Center, TextJustification->0, Background->GrayLevel[0.833326]] }, Closed]], Cell[CellGroupData[{ Cell["You Try It: Part I", "Section"], Cell[TextData[{ "Find all the critical points for the function g[x,y] = ", Cell[BoxData[ \(TraditionalForm\`x\^2\/2 + 3 y\^3 + 9 y\^2 - 3 x\ y + 9 y - 9 x\)]], ". \nBe careful when typing xy that you don't create a new variable called \ 'xy' instead of the product (x)(y)..." }], "Text"], Cell[BoxData[{ RowBox[{\(Clear[g, x, y]\), " "}], "\n", RowBox[{ StyleBox[\(g[x_, y_]\), FontColor->RGBColor[1, 0, 0]], "="}]}], "Input"], Cell["Question 1:\tWhat are the critical points?", "Text", FontWeight->"Bold", Background->RGBColor[0, 1, 1]] }, Closed]], Cell[CellGroupData[{ Cell["Part II: Analyzing Critical Points", "Section"], Cell[CellGroupData[{ Cell["Graphical", "Subsubsection"], Cell["\<\ We can look at the graph of the function, and at the level curves to figure \ out what is happening:\ \>", "Text"], Cell[BoxData[{ \(f[x_, y_] = 5\ x\^2\ + \ 1\ x\ y\ + \ 5\ y\^2\), "\[IndentingNewLine]", \(\(Plot3D[f[x, y], {x, \(-5\), 5}, {y, \(-5\), 5}, PlotPoints \[Rule] 20, AxesLabel \[Rule] {"\", "\", "\"}, BoxRatios \[Rule] {1, 1, 1}];\)\)}], "Input"], Cell[TextData[{ "I think it is safe to say that the critical point at (0,0) is a minimum. \ In fact, it is a ", StyleBox["global minimum", FontWeight->"Bold"], ", meaning that there are no other minima having a smaller function value." }], "Text"], Cell[TextData[{ "Another interesting way to look at the function is to plot the ", StyleBox["contours", FontWeight->"Bold"], " and the ", StyleBox["gradient", FontWeight->"Bold"], ". You already know the ", StyleBox["ContourPlot", FontWeight->"Bold"], " command, but you don't know the ", StyleBox["PlotGradientField", FontWeight->"Bold"], " command (which needs the package ", StyleBox["Graphics'PlotField", FontWeight->"Bold"], " loaded before it can be used)." }], "Text"], Cell[BoxData[ \(\(?PlotGradientField\)\)], "Input"], Cell[TextData[{ "Go ahead and execute this cell... you will only need to execute it one \ time this ", StyleBox["Mathematica", FontSlant->"Italic"], " session. If you try to execute a ", StyleBox["PlotGradientField", FontWeight->"Bold"], " command before loading the ", StyleBox["PlotField", FontWeight->"Bold"], " graphics package, then ", StyleBox["Mathematica", FontSlant->"Italic"], " won't understand what you are trying to do and will think that you are \ defining a new function for it... So... if you can't get ", StyleBox["PlotGradientField", FontWeight->"Bold"], " to work, you should choose ", StyleBox["Quit Kernel -> Local", FontWeight->"Bold"], " under the ", StyleBox["Kernel", FontWeight->"Bold"], " pull down menu (and start over again)." }], "Text"], Cell[BoxData[ \(Needs["\"]\)], "Input"], Cell["\<\ Now we are ready to create a contour plot and overlay the gradient plot on \ top of it!\ \>", "Text"], Cell[BoxData[ \(\(fconplot = ContourPlot[f[x, y], {x, \(-5\), 5}, {y, \(-5\), 5}, PlotPoints \[Rule] 20, AxesLabel \[Rule] {"\", "\"}, ContourShading \[Rule] False];\)\)], "Input"], Cell[BoxData[ \(\(fgradplot = PlotGradientField[ f[x, y], {x, \(-5\), 5}, {y, \(-5\), 5}];\)\)], "Input"], Cell[BoxData[ \(\(Show[fconplot, fgradplot];\)\)], "Input"], Cell[TextData[StyleBox["Question 2:\tBy looking at this combined graph, how \ can you tell that the critical point at (0,0) is a minimum? ", FontWeight->"Bold"]], "Text", Background->RGBColor[0, 1, 1]] }, Closed]], Cell[CellGroupData[{ Cell["Analytical", "Subsubsection"], Cell[TextData[{ "First, we'll solve things as we did in the book - using the ", StyleBox["Discriminant", FontWeight->"Bold"], ": D=", Cell[BoxData[ \(TraditionalForm\`\(\(f\_xx\)(x\_0, y\_0)\) \(\(f\_yy\)(x\_0, y\_0)\) - \((\(f\_xy\)(x\_0, y\_0))\)\^2\)]] }], "Text"], Cell[TextData[{ "Then, if we are testing the critical point ", Cell[BoxData[ \(TraditionalForm\`\((x\_0, y\_0)\)\)]], ", we have the following rules:\n\tLocal ", StyleBox["minimum", FontWeight->"Bold"], " if D>0 ", StyleBox["and", FontWeight->"Bold"], " ", Cell[BoxData[ \(TraditionalForm\`\(f\_xx\)(x\_0, y\_0) > 0\)]], "\n\tLocal ", StyleBox["maximum", FontWeight->"Bold"], " if D>0 ", StyleBox["and ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`\(f\_xx\)(x\_0, y\_0) < 0\)]], "\n\t", StyleBox["Saddle Point", FontWeight->"Bold"], " if D<0\n\t", StyleBox["Inconclusive test", FontWeight->"Bold"], " if D=0 - you could have a local maximum, local minimum, or a saddle \ point\nLet's test it out on our function (we'll find all the individual \ pieces, then put them together):" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(fxx = \[PartialD]\_\(x, x\)f[x, y]\)], "Input"], Cell[BoxData[ \(fyy = \[PartialD]\_\(y, y\)f[x, y]\)], "Input"], Cell[BoxData[ \(fxy = \[PartialD]\_\(x, y\)f[x, y]\)], "Input"] }, Open ]], Cell[BoxData[ \(discrim = \((fxx)\) \((fyy)\) - \((fxy)\)\^2\)], "Input"], Cell[TextData[{ "Note that we don't need to actually evaluate the discriminant at the \ critical point (0, 0); since the discriminant is a constant (and so is the \ same at all points). But for ", StyleBox["future reference", FontWeight->"Bold"], "... here is how we would do this:" }], "Text"], Cell[BoxData[ \(discrim /. {x \[Rule] 0, y \[Rule] 0}\)], "Input"], Cell[TextData[StyleBox["Question 3:\tBased upon the discriminant or second \ derivative test, what do we have?", FontWeight->"Bold"]], "Text", Background->RGBColor[0, 1, 1]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["You Try It: Part II", "Section"], Cell[CellGroupData[{ Cell["Graphical:", "Subsubsection"], Cell[TextData[{ "Go ahead and classify both critical points from ", StyleBox["You Try It: Part I", FontWeight->"Bold"], " ", "by graphing (try both the ", StyleBox["Plot3D ", FontWeight->"Bold"], "and ", StyleBox["PlotGradientField", FontWeight->"Bold"], " combined with ", StyleBox["ContourPlot", FontWeight->"Bold"], " approaches)... make sure to use plot ranges for x and y that encompass \ all critical points..." }], "Text"], Cell[BoxData[""], "Input"], Cell[TextData[StyleBox["Question 4:\tHow did each critical point classify?", FontWeight->"Bold"]], "Text", Background->RGBColor[0, 1, 1]] }, Closed]], Cell[CellGroupData[{ Cell["Analytical:", "Subsubsection"], Cell[TextData[{ "Here, things work out a little differently than before, since the \ discriminant won't be a constant (it will change depending upon the values of \ your critical point ", Cell[BoxData[ \(TraditionalForm\`\((x\_0, y\_0)\)\)]], ", so you must evaluate it at each critical point." }], "Text"], Cell[TextData[StyleBox["Question 5:\tWhat is the discriminant for g[x,y]? ", FontWeight->"Bold"]], "Text", Background->RGBColor[0, 1, 1]], Cell[TextData[StyleBox["Question 6:\tHow did each critical point classify? \ Do your answers agree with your graphical approach?", FontWeight->"Bold"]], "Text", Background->RGBColor[0, 1, 1]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Part III: Maxima and Minima for functions of 3 or more variables\ \>", "Section"], Cell[TextData[{ "In the case of single variable functions, the critical points occur where \ the tangent line to the graph of the function is horizontal. In the case of \ a function of two variables, the critical points occur where the tangent \ plane to the graph of the function is horizontal. This means that ", StyleBox["both", FontWeight->"Bold"], " partial derivatives must be zero at the critical point. If we extend the \ idea, then we say that the critical points occur where the tangent hyperplane \ to the graph of the function is horizontal. Let's use the function ", Cell[BoxData[ \(TraditionalForm\`f(x\_1, x\_2, x\_3) = 5 x\_1\%2 + 2 x\_2\%2 + x\_3\%4 - 32 x\_3 + 6 \( x\_1\) x\_2 + 5 x\_2\)]] }], "Text"], Cell["So, we take the following steps to find minima and maxima:\t", "Text"], Cell["\<\ This cool command will let us look at matrices in a nice format... \ \>", "Text", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \($PrePrint = If[MatrixQ[#], MatrixForm[#], #] &\)], "Input"], Cell[TextData[{ "\t\t1) find where the partials are 0 (", StyleBox["NOTE:", FontWeight->"Bold"], " ", StyleBox["they must all be 0 at the same point !", FontSlant->"Italic"], ")" }], "Text"], Cell[BoxData[{ \(\(f[x_, y_, z_] = 5 x\^2 + 2 y\^2 + z\^4 - 32 z + 6 x\ y + 5 y;\)\), "\n", \(\(\[PartialD]\_x\ f[x, y, z];\)\), "\[IndentingNewLine]", \(\(\[PartialD]\_y\ f[x, y, z];\)\), "\[IndentingNewLine]", \(\(\[PartialD]\_z\ f[x, y, z];\)\), "\[IndentingNewLine]", \(Solve[{\[PartialD]\_x\ f[x, y, z] == 0, \[PartialD]\_y\ f[x, y, z] == 0, \[PartialD]\_z\ f[x, y, z] \[Equal] 0}, {x, y, z}]\)}], "Input"], Cell[TextData[{ StyleBox["Note: ", FontWeight->"Bold"], " ", StyleBox["Mathematica", FontSlant->"Italic"], " will give us the complex roots of these equations also!" }], "Text", FontColor->RGBColor[0, 0, 1]], Cell[TextData[StyleBox["Question 7:\tWhat is the only critical point \ (real-valued)? ", FontWeight->"Bold"]], "Text", Background->RGBColor[0, 1, 1]], Cell[TextData[{ "\t\n\t2) Classify the critical point. We should form the ", StyleBox["Hessian", FontWeight->"Bold"], " (matrix of 2nd partial derivatives) and then determine whether the \ function is ", StyleBox["concave", FontSlant->"Italic"], ", ", StyleBox["convex", FontSlant->"Italic"], ", or neither ", StyleBox["at the critical point", FontWeight->"Bold"], ". " }], "Text"], Cell[BoxData[{ \(\(fxx = \[PartialD]\_\(x, x\)\ f[x, y, z];\)\), "\[IndentingNewLine]", \(\(fyy = \[PartialD]\_\(y, y\)\ f[x, y, z];\)\), "\[IndentingNewLine]", \(\(fzz = \[PartialD]\_\(z, z\)\ f[x, y, z];\)\), "\[IndentingNewLine]", \(\(fxy = \[PartialD]\_\(x, y\)\ f[x, y, z];\)\), "\[IndentingNewLine]", \(\(fxz = \[PartialD]\_\(x, z\)\ f[x, y, z];\)\), "\[IndentingNewLine]", \(\(fyz = \[PartialD]\_\(x, z\)\ f[x, y, z];\)\), "\[IndentingNewLine]", \(hess = {{fxx, fxy, fxz}, {fxy, fyy, fyz}, {fxz, fyz, fzz}}\)}], "Input"], Cell["Now, let's look at the eigenvalues...", "Text"], Cell[BoxData[ \(Eigenvalues[hess]\)], "Input"], Cell[TextData[StyleBox["Question 8:\tWhat can you say about the critical \ point? Is it a local or a global extremum? (hint: think about what it means \ to be either positive definite or negative definite for all function values \ (not just the critical points - it tells us something more about the shape of \ the function.)", FontWeight->"Bold"]], "Text", Background->RGBColor[0, 1, 1]] }, Closed]], Cell[CellGroupData[{ Cell["You Try It: Part III ", "Section"], Cell[TextData[{ StyleBox["Question 9:\t", FontWeight->"Bold"], StyleBox["Find and classify (as well as possible) all of the critical \ points for the function f[x,y,z] = ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`x\^2\/2 + 3 y\^3 + 9 y\^2 - 3 x\ z + 9 y - 9 x\ + \ 2.0 y\ z\ - \ 4\ z\^2\)], FontWeight->"Bold"], StyleBox[". ", FontWeight->"Bold"] }], "Text", Background->RGBColor[0, 1, 1]] }, Closed]], Cell[CellGroupData[{ Cell["Part IV: An Introduction to Constrained Optimization", "Section"], Cell[TextData[{ "In the real world, we often want to solve optimization problems with \ constraints (", StyleBox["constrained optimization", FontWeight->"Bold"], " problems). 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.8902 L .54926 .88228 L .53406 .87767 L .51642 .87131 L .48358 .857 L .46065 .84483 L .45074 .83892 L .4179 .81642 L .41217 .81199 L .38506 .7885 L .37552 .77915 L .35222 .75333 L .34655 .74631 L .32324 .71346 L .31938 .7073 L .30447 .68062 L .28957 .64778 L .28654 .64011 L .27806 .61494 L .26965 .5821 L .26414 .54926 L .26142 .51642 L .26142 .48358 L .26414 .45074 L .26965 .4179 L .27806 .38506 L .28654 .35989 L .28957 .35222 L .30447 .31938 L .31938 .2927 L .32324 .28654 L .34655 .25369 L .35222 .24667 L Mistroke .37552 .22085 L .38506 .2115 L .41217 .18801 L .4179 .18358 L .45074 .16108 L .46065 .15517 L .48358 .143 L .51642 .12869 L .53406 .12233 L .54926 .11772 L .5821 .1098 L .61494 .10477 L .64778 .1025 L .68062 .10295 L .71346 .10613 L .74631 .11211 L .77915 .12103 L .78324 .12233 L .81199 .13307 L .84483 .14859 L .8568 .15517 L .87767 .16805 L .90529 .18801 L .91051 .19221 L .94193 .22085 L .94335 .22231 L .97088 .25369 L .97619 .26052 L Mfstroke .97619 .68752 m .9593 .71346 L .94335 .73439 L .9331 .74631 L .91051 .76935 L .89961 .77915 L .87767 .79666 L .85523 .81199 L .84483 .81828 L .81199 .83534 L .78935 .84483 L .77915 .84847 L .74631 .85815 L .71346 .86461 L .68062 .86805 L .64778 .86853 L .61494 .86609 L .5821 .86065 L .54926 .85207 L .52804 .84483 L .51642 .84012 L .48358 .82444 L .46221 .81199 L .45074 .80443 L .4179 .77916 L .41788 .77915 L .38506 .74706 L .38438 .74631 L .35811 .71346 L .35222 .70494 L .3373 .68062 L .32095 .64778 L .31938 .6442 L .30841 .61494 L .29929 .5821 L .29334 .54926 L .2904 .51642 L .2904 .48358 L .29334 .45074 L .29929 .4179 L .30841 .38506 L .31938 .3558 L .32095 .35222 L .3373 .31938 L .35222 .29506 L .35811 .28654 L .38438 .25369 L .38506 .25294 L .41788 .22085 L .4179 .22084 L Mistroke .45074 .19557 L .46221 .18801 L .48358 .17556 L .51642 .15988 L .52804 .15517 L .54926 .14793 L .5821 .13935 L .61494 .13391 L .64778 .13147 L .68062 .13195 L .71346 .13539 L .74631 .14185 L .77915 .15153 L .78935 .15517 L .81199 .16466 L .84483 .18172 L .85523 .18801 L .87767 .20334 L .89961 .22085 L .91051 .23065 L .9331 .25369 L .94335 .26561 L .9593 .28654 L .97619 .31248 L Mfstroke .97619 .61392 m .9758 .61494 L .96185 .64778 L .94354 .68062 L .94335 .68093 L .91985 .71346 L .91051 .72442 L .88915 .74631 L .87767 .75655 L .84801 .77915 L .84483 .78127 L .81199 .80044 L .78694 .81199 L .77915 .81503 L .74631 .8257 L .71346 .8328 L .68062 .83656 L .64778 .83709 L .61494 .83442 L .5821 .82845 L .54926 .81901 L .53041 .81199 L .51642 .80577 L .48358 .78823 L .46941 .77915 L .45074 .76551 L .42831 .74631 L .4179 .73613 L .3976 .71346 L .38506 .69715 L .3739 .68062 L .35557 .64778 L .35222 .64082 L .34165 .61494 L .3316 .5821 L .32508 .54926 L .32187 .51642 L .32187 .48358 L .32508 .45074 L .3316 .4179 L .34165 .38506 L .35222 .35918 L .35557 .35222 L .3739 .31938 L .38506 .30285 L .3976 .28654 L .4179 .26387 L .42831 .25369 L .45074 .23449 L .46941 .22085 L Mistroke .48358 .21177 L .51642 .19423 L .53041 .18801 L .54926 .18099 L .5821 .17155 L .61494 .16558 L .64778 .16291 L .68062 .16344 L .71346 .1672 L .74631 .1743 L .77915 .18497 L .78694 .18801 L .81199 .19956 L .84483 .21873 L .84801 .22085 L .87767 .24345 L .88915 .25369 L .91051 .27558 L .91985 .28654 L .94335 .31907 L .94354 .31938 L .96185 .35222 L .9758 .38506 L .97619 .38608 L Mfstroke .54926 .21791 m .5821 .20727 L .61494 .20059 L .64778 .1976 L .68062 .1982 L .71346 .20239 L .74631 .21035 L .77566 .22085 L .77915 .22241 L .81199 .23908 L .83456 .25369 L .84483 .26139 L .8732 .28654 L .87767 .29111 L .90151 .31938 L .91051 .33209 L .92279 .35222 L .93864 .38506 L .94335 .39683 L .94994 .4179 L .95724 .45074 L .96083 .48358 L .96083 .51642 L .95724 .54926 L .94994 .5821 L .94335 .60317 L .93864 .61494 L .92279 .64778 L .91051 .66791 L .90151 .68062 L .87767 .70889 L .8732 .71346 L .84483 .73861 L .83456 .74631 L .81199 .76092 L .77915 .77759 L .77566 .77915 L .74631 .78965 L .71346 .79761 L .68062 .8018 L .64778 .8024 L .61494 .79941 L .5821 .79273 L .54926 .78209 L .54182 .77915 L .51642 .76705 L .48358 .74674 L .48297 .74631 L .45074 .71978 L .44426 .71346 L Mistroke .4179 .6832 L .41597 .68062 L .39467 .64778 L .38506 .62926 L .37882 .61494 L .36748 .5821 L .36017 .54926 L .35659 .51642 L .35659 .48358 L .36017 .45074 L .36748 .4179 L .37882 .38506 L .38506 .37074 L .39467 .35222 L .41597 .31938 L .4179 .3168 L .44426 .28654 L .45074 .28022 L .48297 .25369 L .48358 .25326 L .51642 .23295 L .54182 .22085 L .54926 .21791 L Mfstroke .5821 .248 m .61494 .24026 L .64778 .23683 L .68062 .23751 L .71346 .24235 L .74631 .2516 L .75222 .25369 L .77915 .26573 L .81199 .2858 L .81302 .28654 L .84483 .31359 L .85044 .31938 L .87675 .35222 L .87767 .35357 L .89574 .38506 L .90899 .4179 L .91051 .42165 L .91747 .45074 L .92162 .48358 L .92162 .51642 L .91747 .54926 L .91051 .57835 L .90899 .5821 L .89574 .61494 L .87767 .64643 L .87675 .64778 L .85044 .68062 L .84483 .68641 L .81302 .71346 L .81199 .7142 L .77915 .73427 L .75222 .74631 L .74631 .7484 L .71346 .75765 L .68062 .76249 L .64778 .76317 L .61494 .75974 L .5821 .752 L .56486 .74631 L .54926 .73957 L .51642 .72165 L .50436 .71346 L .48358 .69675 L .46698 .68062 L .45074 .66167 L .44069 .64778 L .42174 .61494 L .4179 .607 L .40846 .5821 L .39998 .54926 L Mistroke .39584 .51642 L .39584 .48358 L .39998 .45074 L .40846 .4179 L .4179 .393 L .42174 .38506 L .44069 .35222 L .45074 .33833 L .46698 .31938 L .48358 .30325 L .50436 .28654 L .51642 .27835 L .54926 .26043 L .56486 .25369 L .5821 .248 L Mfstroke .64778 .28305 m .68062 .28389 L .69889 .28654 L .71346 .28978 L .74631 .30117 L .77915 .31922 L .77938 .31938 L .81199 .34602 L .81796 .35222 L .84305 .38506 L .84483 .38782 L .85988 .4179 L .87032 .45074 L .87533 .48358 L .87533 .51642 L .87032 .54926 L .85988 .5821 L .84483 .61218 L .84305 .61494 L .81796 .64778 L .81199 .65398 L .77938 .68062 L .77915 .68078 L .74631 .69883 L .71346 .71022 L .69889 .71346 L .68062 .71611 L .64778 .71695 L .61857 .71346 L .61494 .71275 L .5821 .70329 L .54926 .68764 L .53789 .68062 L .51642 .66419 L .49944 .64778 L .48358 .62865 L .47438 .61494 L .45759 .5821 L .45074 .56446 L .44717 .54926 L .44211 .51642 L .44211 .48358 L .44717 .45074 L .45074 .43554 L .45759 .4179 L .47438 .38506 L .48358 .37135 L .49944 .35222 L .51642 .33581 L .53789 .31938 L Mistroke .54926 .31236 L .5821 .29671 L .61494 .28725 L .61857 .28654 L .64778 .28305 L Mfstroke .61494 .34806 m .64778 .34221 L .68062 .34336 L .71346 .35168 L .71507 .35222 L .74631 .36819 L .76765 .38506 L .77915 .3971 L .79399 .4179 L .809 .45074 L .81199 .46121 L .816 .48358 L .816 .51642 L .81199 .53879 L .809 .54926 L .79399 .5821 L .77915 .6029 L .76765 .61494 L .74631 .63181 L .71507 .64778 L .71346 .64832 L .68062 .65664 L .64778 .65779 L .61494 .65194 L .60011 .64778 L .5821 .63844 L .55013 .61494 L .54926 .61411 L .52348 .5821 L .51642 .57081 L .50844 .54926 L .50139 .51642 L .50139 .48358 L .50844 .45074 L .51642 .42919 L .52348 .4179 L .54926 .38589 L .55013 .38506 L .5821 .36156 L .60011 .35222 L .61494 .34806 L s .64778 .44773 m .67816 .45074 L .68062 .45179 L .70907 .48358 L .70907 .51642 L .68062 .54821 L .67816 .54926 L .64778 .55227 L .63931 .54926 L .61494 .52978 L .60685 .51642 L .60685 .48358 L .61494 .47022 L .63931 .45074 L .64778 .44773 L s [(26.)] .02381 .02381 0 0 Mshowa [(21.)] .02381 .18254 0 0 Mshowa [(18.)] .02381 .34127 0 0 Mshowa [(17.)] .02381 .5 0 0 Mshowa [(18.)] .02381 .65873 0 0 Mshowa [(21.)] .02381 .81746 0 0 Mshowa [(26.)] .02381 .97619 0 0 Mshowa [(19.)] .18254 .02381 0 0 Mshowa [(14.)] .18254 .18254 0 0 Mshowa [(11.)] .18254 .34127 0 0 Mshowa [(10.)] .18254 .5 0 0 Mshowa [(11.)] .18254 .65873 0 0 Mshowa [(14.)] .18254 .81746 0 0 Mshowa [(19.)] .18254 .97619 0 0 Mshowa [(14.)] .34127 .02381 0 0 Mshowa [(9.)] .34127 .18254 0 0 Mshowa [(6.)] .34127 .34127 0 0 Mshowa [(5.)] .34127 .5 0 0 Mshowa [(6.)] .34127 .65873 0 0 Mshowa [(9.)] .34127 .81746 0 0 Mshowa [(14.)] .34127 .97619 0 0 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" }], "Text"], Cell[TextData[{ StyleBox["Question 10:\tSo what is the lowest value of f(x,y) as you trace \ the curve ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`x = \(1\/4\) y\^2\)]], StyleBox["? ", FontWeight->"Bold"] }], "Text", Background->RGBColor[0, 1, 1]], Cell["\<\ This is not a very practical way to solve these problems. \ \>", "Text"], Cell[TextData[StyleBox["Question 11:\tAt the optimal point on the graph \ above, sketch the gradient vectors for both the function f(x,y) and the \ constraint g(x,y). You don't need to do any hard work here, remember the \ relationship between the gradient and level curves!", FontWeight->"Bold"]], "Text", Background->RGBColor[0, 1, 1]], Cell[TextData[StyleBox["Question 12:\tWhat is the relationship between \ \[EmptyDownTriangle]f and \[EmptyDownTriangle]g at the optimal point? Are \ the parallel? Perpendicular? Is there no apparent relationship?", FontWeight->"Bold"]], "Text", Background->RGBColor[0, 1, 1]], Cell[TextData[{ "So the contour of the constraint is tangent to the optimal contour of the \ objective function ", StyleBox["at the optimal point", FontWeight->"Bold"], ". We can write this mathematically as: ", StyleBox["\[EmptyDownTriangle]f = \[Lambda]\[EmptyDownTriangle]g ", "MathCaption"], StyleBox["at the optimal point!", FontWeight->"Bold"], " Where ", StyleBox["\[Lambda]", "MathCaption"], " is a scalar - we will call it the ", StyleBox["LaGrange Multiplier", FontWeight->"Bold"], ".\nThe ", StyleBox["Method of LaGrange", FontWeight->"Bold"], " for solving constrained optimization problems is to \"rewrite\" the \ problem as an unconstrained optimization problem - we will find the critical \ points of the \"new\" function\n\t\t ", StyleBox["\[ScriptCapitalL](x,y,\[Lambda]) = f(x,y) - \[Lambda][g(x,y)]\n", "MathCaption"], "1. Find ", StyleBox["\[ScriptCapitalL](x,y,\[Lambda]) = f(x,y) - \[Lambda][g(x,y)]", "MathCaption"], "\n\t\tFor this problem, we have: \[ScriptCapitalL](x,y,\[Lambda])=", Cell[BoxData[ \(TraditionalForm\`\((x - 1)\)\^2 + y\^2 + 1 - \[Lambda][x - \(1\/4\) y\^2 - 0]\)]], "\n\t\twhich simplifies to ", Cell[BoxData[ \(TraditionalForm\`\((x - 1)\)\^2 + y\^2 + 1 - \[Lambda]x + \(1\/4\) \[Lambda]y\^2\)]], "\n2. Take ", StyleBox["all", FontWeight->"Bold"], " partial derivatives (we have an unconstrained optimization problem, find \ critical points!)\n\t\t", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["\[ScriptCapitalL]", "MathCaption"], "x"], TraditionalForm]]], StyleBox[" = 2(x - 1) - \[Lambda]\n\t\t", "MathCaption"], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["\[ScriptCapitalL]", "MathCaption"], "y"], TraditionalForm]]], StyleBox[" = 2y + ", "MathCaption"], Cell[BoxData[ \(TraditionalForm\`\(1\/2\) y\)]], StyleBox["\[Lambda]\n\t\t", "MathCaption"], Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["\[ScriptCapitalL]", "MathCaption"], StyleBox["\[Lambda]", "MathCaption"]], TraditionalForm]]], " = ", Cell[BoxData[ \(TraditionalForm\`\(-x\) + \(1\/4\) y\^2\)]], "\n\n3. Set all partials to 0 and solve simultaneously:\n\t\t", StyleBox[" 2(x - 1) - \[Lambda] = 0\t(A)\n\t\t2y + ", "MathCaption"], Cell[BoxData[ \(TraditionalForm\`\(1\/2\) y\)]], StyleBox["\[Lambda] = 0\t\t(B)\n\t\t", "MathCaption"], " ", StyleBox["x -", "MathCaption"], Cell[BoxData[ \(TraditionalForm\`\(1\/4\) y\^2\)]], StyleBox[" = 0 \t\t(C)", "MathCaption"], "\n\t\t \t\n\tWe can solve (A) for ", StyleBox["\[Lambda] ", "MathCaption"], "= ", StyleBox["2(x - 1)\t(D)\n\t", "MathCaption"], "We can solve (C) for x=", Cell[BoxData[ \(TraditionalForm\`\(1\/4\) y\^2\)]], StyleBox[" \t(E)\n\t", "MathCaption"], "Then, we can substitute for x in (D) using (E):\t", StyleBox["\[Lambda] ", "MathCaption"], "= ", StyleBox["2(", "MathCaption"], Cell[BoxData[ \(TraditionalForm\`\(1\/4\) y\^2\)]], StyleBox[" - 1)", "MathCaption"], " or ", StyleBox["\[Lambda] ", "MathCaption"], "= ", Cell[BoxData[ \(TraditionalForm\`\(1\/2\) y\^2\)]], StyleBox[" - 2", "MathCaption"], "\n\tAnd finally, substitute this into (B):\t ", StyleBox["2y + ", "MathCaption"], Cell[BoxData[ \(TraditionalForm\`\(1\/2\) y\)]], StyleBox["(", "MathCaption"], Cell[BoxData[ \(TraditionalForm\`\(1\/2\) y\^2\)]], StyleBox[" - 2) = 0\t", "MathCaption"], "\t", StyleBox["\t", "MathCaption"], "\n\tThis simplifies to ", StyleBox["2y + ", "MathCaption"], Cell[BoxData[ \(TraditionalForm\`\(1\/4\) y\^3\)]], StyleBox[" - y = 0 ", "MathCaption"], "or ", StyleBox["y(", "MathCaption"], Cell[BoxData[ \(TraditionalForm\`\(1\/4\) y\^2\)]], StyleBox[" + 1) = 0 ", "MathCaption"], "\n\tSo, ", StyleBox["y = 0 ", "MathCaption"], " or ", StyleBox["0 = ", "MathCaption"], Cell[BoxData[ \(TraditionalForm\`\((\(1\/4\) y\^2 + 1)\)\)]], StyleBox[" ", "MathCaption"], StyleBox["which is not possible!", FontWeight->"Bold"], "\n\tSo when y=0, x=0, from (E).\n\tSo we come up with x=0 and y=0!\n" }], "Text"] }, Closed]] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 692}}, AutoGeneratedPackage->None, WindowToolbars->{"RulerBar", "EditBar"}, CellGrouping->Manual, WindowSize->{1016, 646}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, StyleDefinitions -> "DemoText.nb" ] (******************************************************************* Cached data follows. 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