(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 184428, 5016] NotebookOptionsPosition[ 174578, 4739] NotebookOutlinePosition[ 175194, 4763] CellTagsIndexPosition[ 175151, 4760] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell["\<\ Penalty functions, using Newton's method for several dimensions\ \>", "Subtitle", Evaluatable->False, CellHorizontalScrolling->False, TextAlignment->Center], Cell["Standard setup stuff", "Section"], Cell[BoxData[{ RowBox[{ RowBox[{"Clear", "[", RowBox[{"norm", ",", "x"}], "]"}], ";"}], "\n", RowBox[{ RowBox[{ RowBox[{"norm", "[", "x_", "]"}], "=", SqrtBox[ RowBox[{"x", ".", "x"}]]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"dp", " ", "=", " ", "9"}], ";", " ", RowBox[{ RowBox[{"nn", "[", "x_", "]"}], " ", ":=", " ", RowBox[{"NumberForm", "[", RowBox[{ RowBox[{"N", "[", "x", "]"}], ",", " ", RowBox[{"{", RowBox[{"20", ",", "dp"}], "}"}]}], "]"}]}], ";"}]}], "Input", PageWidth->Infinity, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["Simple quadratic example", "Section"], Cell["\<\ The problem we did in class using Lagrange multipliers: quadratic objective \ function and linear constraints. We know from that analysis that the exact \ solution is (-1,0,1). We use the Courant (quadratic loss) penalty function \ phi.\ \>", "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"Clear", "[", RowBox[{ "f", ",", "g1", ",", " ", "g2", ",", "phi", ",", " ", "rho", ",", " ", "pi", ",", " ", "g", ",", "h", ",", "x1", ",", "x2", ",", "x3", ",", " ", "x"}], "]"}], ";"}], "\n", RowBox[{ RowBox[{ RowBox[{"f", "[", RowBox[{"{", RowBox[{"x1_", ",", "x2_", ",", "x3_"}], "}"}], "]"}], "=", RowBox[{ RowBox[{"x1", "^", "2"}], " ", "+", " ", RowBox[{"4", " ", "x2"}], " ", "-", " ", "x3"}]}], ";"}], "\n", RowBox[{ RowBox[{ RowBox[{"g1", "[", RowBox[{"{", RowBox[{"x1_", ",", "x2_", ",", "x3_"}], "}"}], "]"}], " ", "=", " ", RowBox[{"x1", " ", "-", " ", "x2", " ", "+", " ", "1"}]}], ";"}], "\n", RowBox[{ RowBox[{ RowBox[{"g2", "[", RowBox[{"{", RowBox[{"x1_", ",", "x2_", ",", "x3_"}], "}"}], "]"}], " ", "=", " ", RowBox[{ "x1", " ", "+", " ", "x2", " ", "-", " ", "x3", " ", "+", " ", "2"}]}], ";"}], "\n", RowBox[{ RowBox[{ RowBox[{"phi", "[", "x_", "]"}], " ", "=", " ", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"g1", "[", "x", "]"}], "^", "2"}], " ", "+", " ", RowBox[{ RowBox[{"g2", "[", "x", "]"}], "^", "2"}]}], ")"}], "/", "2"}]}], ";"}]}], "Input", PageWidth->Infinity, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell["\<\ We define our penalised objective function pi, its gradient gpi, and its \ Hessian hpi, every time we change the value of the parameter rho.\ \>", "Text"], Cell[BoxData[ RowBox[{"findpigh", " ", ":=", " ", RowBox[{"Block", "[", RowBox[{ RowBox[{"{", "}"}], ",", "\n", " ", RowBox[{ RowBox[{ RowBox[{"pi", "[", "x_", "]"}], " ", "=", " ", RowBox[{ RowBox[{"f", "[", "x", "]"}], " ", "+", " ", RowBox[{"rho", " ", RowBox[{"phi", "[", "x", "]"}]}]}]}], ";", "\n", " ", RowBox[{ RowBox[{"gpi", "[", RowBox[{"{", RowBox[{"x1_", ",", "x2_", ",", "x3_"}], "}"}], "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"D", "[", " ", RowBox[{ RowBox[{"pi", "[", RowBox[{"{", RowBox[{"x1", ",", "x2", ",", "x3"}], "}"}], "]"}], ",", " ", "x1"}], "]"}], ",", RowBox[{"D", "[", " ", RowBox[{ RowBox[{"pi", "[", RowBox[{"{", RowBox[{"x1", ",", "x2", ",", "x3"}], "}"}], "]"}], ",", " ", "x2"}], "]"}], ",", "\n", "\t\t\t", RowBox[{"D", "[", " ", RowBox[{ RowBox[{"pi", "[", RowBox[{"{", RowBox[{"x1", ",", "x2", ",", "x3"}], "}"}], "]"}], ",", " ", "x3"}], "]"}]}], "}"}]}], ";", "\[IndentingNewLine]", RowBox[{ RowBox[{"hpi", "[", RowBox[{"{", RowBox[{"x1_", ",", "x2_", ",", "x3_"}], "}"}], "]"}], "=", RowBox[{"{", RowBox[{ RowBox[{"D", "[", " ", RowBox[{ RowBox[{"gpi", "[", RowBox[{"{", RowBox[{"x1", ",", "x2", ",", "x3"}], "}"}], "]"}], ",", " ", "x1"}], "]"}], ",", RowBox[{"D", "[", " ", RowBox[{ RowBox[{"gpi", "[", RowBox[{"{", RowBox[{"x1", ",", "x2", ",", "x3"}], "}"}], "]"}], ",", " ", "x2"}], "]"}], ",", "\n", "\t\t\t", RowBox[{"D", "[", " ", RowBox[{ RowBox[{"gpi", "[", RowBox[{"{", RowBox[{"x1", ",", "x2", ",", "x3"}], "}"}], "]"}], ",", " ", "x3"}], "]"}]}], "}"}]}], ";"}]}], "\n", "]"}]}]], "Input"], Cell["\<\ At any point we will want to print the current x, the values of the \ constraints, the gradient gpi and its magnitude, the Hessian hpi and its \ eigenvalues. The eigenvalues show how well- or ill-conditioned our current \ problem is.\ \>", "Text"], Cell[BoxData[ RowBox[{"printxgh", " ", ":=", " ", RowBox[{"Block", "[", RowBox[{ RowBox[{"{", "}"}], ",", "\n", "\t\t", RowBox[{ RowBox[{"Print", "[", RowBox[{ "k1", ",", " ", "\"\<.\>\"", ",", " ", "k2", ",", "\"\< \>\"", ",", RowBox[{ RowBox[{"x", "[", RowBox[{"k1", ",", "k2"}], "]"}], "//", "nn"}], ",", " ", "\"\< g1 = \>\"", ",", " ", RowBox[{"g1", "[", RowBox[{"x", "[", RowBox[{"k1", ",", "k2"}], "]"}], "]"}], ",", " ", "\"\< g2 = \>\"", ",", " ", RowBox[{"g2", "[", RowBox[{"x", "[", RowBox[{"k1", ",", "k2"}], "]"}], "]"}]}], "]"}], ";", "\[IndentingNewLine]", "\t ", RowBox[{"Print", "[", RowBox[{"\"\< gpi \>\"", ",", RowBox[{"gpi", "[", RowBox[{"x", "[", RowBox[{"k1", ",", "k2"}], "]"}], "]"}], ",", "\"\< \>\"", ",", " ", RowBox[{"N", "[", RowBox[{"norm", "[", RowBox[{"gpi", "[", RowBox[{"x", "[", RowBox[{"k1", ",", "k2"}], "]"}], "]"}], "]"}], "]"}]}], "]"}], ";", "\n", "\t\t", RowBox[{"Print", "[", RowBox[{"\"\< hpi \>\"", ",", " ", RowBox[{"N", "[", RowBox[{ RowBox[{"hpi", "[", RowBox[{"x", "[", RowBox[{"k1", ",", "k2"}], "]"}], "]"}], " ", "//", " ", "MatrixForm"}], "]"}], ",", " ", "\"\< ev = \>\"", ",", " ", RowBox[{"Eigenvalues", "[", RowBox[{"N", "[", RowBox[{"hpi", "[", RowBox[{"x", "[", RowBox[{"k1", ",", "k2"}], "]"}], "]"}], "]"}], "]"}]}], " ", "]"}], ";"}]}], "\n", "]"}]}]], "Input"], Cell["\<\ Note: we are going to keep TWO iteration counters: k1 and k2. k1 is the \ iteration number of the Penalty Function Method, and k2 is the iteration \ number of the optimization method we use (Newton's method, here). The x \ values are kept in x[k1,k2]; for convenience we make x[k1] the same thing as \ x[k1, 0].\ \>", "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"Clear", "[", RowBox[{"init", ",", " ", "x0", ",", " ", "rho0"}], "]"}], ";"}], "\n", RowBox[{ RowBox[{ RowBox[{"init", "[", RowBox[{"x0_", ",", " ", "rho0_"}], "]"}], ":=", RowBox[{"Block", "[", RowBox[{ RowBox[{"{", "}"}], ",", "\t\t", "\n", "\t\t", RowBox[{ RowBox[{"Clear", "[", RowBox[{"rho", ",", " ", "x", ",", "k1", ",", " ", "k2"}], "]"}], ";", "\n", "\t\t", RowBox[{ RowBox[{"x", "[", RowBox[{"0", ",", "0"}], "]"}], "=", "x0"}], ";", " ", RowBox[{ RowBox[{"x", "[", "0", "]"}], " ", "=", " ", "x0"}], ";", " ", RowBox[{"k1", "=", "0"}], ";", " ", RowBox[{"k2", " ", "=", " ", "0"}], ";", "\n", "\t\t", RowBox[{"rho", " ", "=", " ", "rho0"}], ";", " ", "findpigh", ";", " ", "printxgh", ";"}]}], "\n", "]"}]}], ";"}]}], "Input"], Cell[BoxData[{ RowBox[{ RowBox[{"Clear", "[", RowBox[{"changerho", ",", " ", "newrho"}], "]"}], ";"}], "\n", 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(Also, the Hessian matrix stays constant for a \ given value of rho.) So for this example, in future we just do one iteration \ of Newton's method at each step.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"changerho", "[", "10", "]"}], ";", " ", "newton", ";"}]], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{ "1", "\[InvisibleSpace]", "\<\".\"\>", "\[InvisibleSpace]", "0", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", TagBox[ RowBox[{"{", RowBox[{ InterpretationBox["\<\"-1.000000000\"\>", -1., AutoDelete->True], ",", InterpretationBox["\<\"-3.000000000\"\>", -3., AutoDelete->True], ",", InterpretationBox["\<\"-1.000000000\"\>", -1., AutoDelete->True]}], "}"}], NumberForm[#, {20, 9}]& ], "\[InvisibleSpace]", "\<\" g1 = \"\>", "\[InvisibleSpace]", "3.`", "\[InvisibleSpace]", "\<\" g2 = \"\>", "\[InvisibleSpace]", RowBox[{"-", "1.`"}]}], SequenceForm[1, ".", 0, " ", NumberForm[{-1., -3., -1.}, {20, 9}], " g1 = ", 3., " g2 = ", -1.], Editable->False]], "Print", CellChangeTimes->{3.4178011620191097`*^9}], Cell[BoxData[ 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