LP: Formulating dual problems
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\nGiven LP problem: optimise $cx$ subject to constraints $Ax \\le/\\ge b,$ formulate the dual problem.
\n{mm}: $\\simplify {{c[0]}x_1+{c[1]}x_2+{c[2]}x_3+{c[3]}x_4} $
\nSubject to:
\n$\\simplify {{a1[0]}x_1+{a1[1]}x_2+{a1[2]}x_3+{a1[3]}x_4} \\le \\var{b[0]}$
\n$\\simplify {{a2[0]}x_1+{a2[1]}x_2+{a2[2]}x_3+{a2[3]}x_4} \\le \\var{b[1]}$
\n$x_1,x_2,x_3,x_4 \\ge 0$
\n\nDual problem (use variables \"y1,y2,...\" for $y_1,y_2,...$, and \">=\" or \"<=\" for $\\ge$ or $\\le$):
\n
{mmd}: [[0]]
Subject to (express constraints in the form $A^T y >= c$):
\n[[1]]$\\ge$[[5]]
\n[[2]]$\\ge$[[6]]
\n[[3]]$\\ge$[[7]]
\n[[4]]$\\ge$[[8]]
\n\n
Assuming that dual variables $y_i \\ge 0, ~ i = 1, ...$
Formulating duality problems
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\n\n{mm}: $\\simplify {{c[0]}x_1+{c[1]}x_2+{c[2]}x_3+{c[3]}x_4} $
\nSubject to:
\n$\\simplify {{a1[0]}x_1+{a1[1]}x_2+{a1[2]}x_3+{a1[3]}x_4} \\le \\var{b[0]}$
\n$\\simplify {{a20}x_1+{a21}x_2+{a22}x_3+{a23}x_4} \\le \\var{b[1]}$
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\n$x_1,x_2,x_3,x_4 \\ge 0$
\n\nDual problem (use variables \"y1,y2,...\" for $y_1,y_2,...$, and \">=\" or \"<=\" for $\\ge$ or $\\le$):
\n{mmd}: [[0]]
\nSubject to (express constraints in the form $A^T y \\ge c$):
\n[[1]]$\\ge$[[5]]
\n[[2]]$\\ge$[[6]]
\n[[3]]$\\ge$[[7]]
\n[[4]]$\\ge$[[8]]
\n\n
Assuming that dual variables $y_i \\le 0, ~ i = 1, ...$
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Formulating duality problems
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\n\n{mm}: $\\simplify {{c[0]}x_1+{c[1]}x_2+{c[2]}x_3+{c[3]}x_4} $
\nSubject to:
\n$\\simplify {{a1[0]}x_1+{a1[1]}x_2+{a1[2]}x_3+{a1[3]}x_4} \\ge \\var{b[0]}$
\n$\\simplify {{a20}x_1+{a21}x_2+{a22}x_3+{a23}x_4} \\le \\var{b[1]}$
\n$\\simplify {{aa1[0]}x_1+{aa1[1]}x_2+{aa1[2]}x_3+{aa1[3]}x_4} \\ge \\var{b[2]}$
\n$x_1,x_2,x_3,x_4 \\ge 0$
\n\nFormulate the dual problem (use variables \"y1,y2,...\" for $y_1,y_2,...$, and \">=\" or \"<=\" for $\\ge$ or $\\le$)
\nassuming that dual variables $y_2 \\ge 0, ~y_1,y_3 \\le 0$:
\n\n{mmd}: [[0]]
\nSubject to (express constraints in the form $A^T y >= c$):
\n[[1]]$\\ge$[[5]]
\n[[2]]$\\ge$[[6]]
\n[[3]]$\\ge$[[7]]
\n[[4]]$\\ge$[[8]]
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Formulating duality problems
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\n\n{mm}: $\\simplify {{c[0]}x_1+{c[1]}x_2+{c[2]}x_3+{c[3]}x_4+{c[4]}x_5} $
\nSubject to:
\n$\\simplify {{a1[0]}x_1+{a1[1]}x_2+{a1[2]}x_3+{a1[3]}x_4+{a1[4]}x_5} \\le \\var{b[0]}$
\n$\\simplify {{a20}x_1+{a21}x_2+{a22}x_3+{a23}x_4+{a24}x_5} \\le \\var{b[1]}$
\n$\\simplify {{aa1[0]}x_1+{aa1[1]}x_2+{aa1[2]}x_3+{aa1[3]}x_4+{aa1[4]}x_5} \\ge \\var{b[2]}$
\n$\\simplify {{aa2[0]}x_1+{aa2[1]}x_2+{aa2[2]}x_3+{aa2[3]}x_4+{aa2[4]}x_5} \\ge \\var{b[3]}$
\n$\\simplify {{aa3[0]}x_1+{aa3[1]}x_2+{aa3[2]}x_3+{aa3[3]}x_4+{aa3[4]}x_5} = \\var{b[4]}$
\n$\\simplify {{aa4[0]}x_1+{aa4[1]}x_2+{aa4[2]}x_3+{aa4[3]}x_4+{aa4[4]}x_5} = \\var{b[5]}$
\n$x_1,x_2,x_3,x_4,x_5 \\ge 0$
\n\nFormulate the dual problem (use variables \"y1,y2,...\" for $y_1,y_2,...$, and \">=\" or \"<=\" for $\\ge$ or $\\le$)
\nassuming that dual variables $y_1,y_2 \\ge 0$ and $y_3,y_4 \\le 0$:
\n\n{mmd}: [[0]]
\nSubject to (express constraints in the form $A^T y \\ge c$):
\n[[1]]$\\ge$[[6]]
\n[[2]]$\\ge$[[7]]
\n[[3]]$\\ge$[[8]]
\n[[4]]$\\ge$[[9]]
\n[[5]]$\\ge$[[10]]
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Formulating duality problems
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\n\n{mm}: $\\simplify {{c[0]}x_1+{c[1]}x_2+{c[2]}x_3+{c[3]}x_4+{c[4]}x_5} $
\nSubject to:
\n$\\simplify {{a1[0]}x_1+{a1[1]}x_2+{a1[2]}x_3+{a1[3]}x_4+{a1[4]}x_5} = \\var{b[0]}$
\n$\\simplify {{a20}x_1+{a21}x_2+{a22}x_3+{a23}x_4+{a24}x_5} \\le \\var{b[1]}$
\n$\\simplify {{aa1[0]}x_1+{aa1[1]}x_2+{aa1[2]}x_3+{aa1[3]}x_4+{aa1[4]}x_5} \\ge \\var{b[2]}$
\n$\\simplify {{aa2[0]}x_1+{aa2[1]}x_2+{aa2[2]}x_3+{aa2[3]}x_4+{aa2[4]}x_5} = \\var{b[3]}$
\n$\\simplify {{aa3[0]}x_1+{aa3[1]}x_2+{aa3[2]}x_3+{aa3[3]}x_4+{aa3[4]}x_5} \\ge \\var{b[4]}$
\n$\\simplify {{aa4[0]}x_1+{aa4[1]}x_2+{aa4[2]}x_3+{aa4[3]}x_4+{aa4[4]}x_5} \\le \\var{b[5]}$
\n$x_1,x_2,x_3,x_4,x_5 \\ge 0$
\n\nFormulate the dual problem (use variables \"y1,y2,...\" for $y_1,y_2,...$, and \">=\" or \"<=\" for $\\ge$ or $\\le$)
\nassuming that dual variables $y_2,y_6 \\le 0$ and $y_3,y_5 \\ge 0$:
\n\n{mmd}: [[0]]
\nSubject to (express constraints in the form $A^T y \\ge c$):
\n[[1]]$\\ge$[[6]]
\n[[2]]$\\ge$[[7]]
\n[[3]]$\\ge$[[8]]
\n[[4]]$\\ge$[[9]]
\n[[5]]$\\ge$[[10]]
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Formulating duality problems
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"eqx", "eq2", "eq3", "eq4", "eq5", "eq6", "eq0les", "eq1les", "eq2les", "eq3les", "eq4les", "eq5les", "eq6les", "eq0gr", "eq1gr", "eq2gr", "eq3gr", "eq4gr", "eq5gr", "eq6gr", "eq0e", "eq1e", "eq2e", "eq3e", "eq4e", "eq5e", "eq6e", "m1", "m2", "m0", "m3", "m4", "m5"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "The following LP problem is given:
\n{mm}: $\\simplify {{c[0]}x_1+{c[1]}x_2+{c[2]}x_3} $
\nSubject to:
\n$\\simplify {{a1[0]}x_1+{a1[1]}x_2+{a1[2]}x_3} ~ \\var{eqx[0]} ~ \\var{b[0]}$
\n$\\simplify {{a20}x_1+{a21}x_2+{a22}x_3} ~ \\var{eqx[1]} ~ \\var{b[1]}$
\n$\\simplify {{aa1[0]}x_1+{aa1[1]}x_2+{aa1[2]}x_3} ~ \\var{eqx[2]} ~ \\var{b[2]}$
\n$\\simplify {{aa2[0]}x_1+{aa2[1]}x_2+{aa2[2]}x_3} ~ \\var{eqx[3]} ~ \\var{b[3]}$
\n$\\simplify {{aa3[0]}x_1+{aa3[1]}x_2+{aa3[2]}x_3} ~ \\var{eqx[4]} ~ \\var{b[4]}$
\n$\\simplify {{aa4[0]}x_1+{aa4[1]}x_2+{aa4[2]}x_3} ~ \\var{eqx[5]} ~ \\var{b[5]}$
\n$x_1,x_2,x_3 \\ge 0.$
\n\nThe dual problem is formulated as follows:
\n{mmd}: $\\simplify {{b[0]}y_1+{b[1]}y_2+{b[2]}y_3+{b[3]}y_4+{b[4]}y_5+{b[5]}y_6 }$
\nSubject to:
\n$\\simplify { {a1[0]}y_1 + {a20}y_2 + {aa1[0]}y_3 + {aa2[0]}y_4 + {aa3[0]}y_5 + {aa4[0]}y_6 >= {c[0]} }$
\n$\\simplify { {a1[1]}y_1 + {a21}y_2 + {aa1[1]}y_3 + {aa2[1]}y_4 + {aa3[1]}y_5 + {aa4[1]}y_6 >= {c[1]}}$
\n$\\simplify { {a1[2]}y_1 + {a22}y_2 + {aa1[2]}y_3 + {aa2[2]}y_4 + {aa3[2]}y_5 + {aa4[2]}y_6 >= {c[2]}}$
\n\nFind the range/sign of each dual variable (put in the boxes \"1\" for \"$\\le 0$\", \"2\" for \"$\\ge 0$\" and \"3\" for \"$any ~ number$\"):
\n\n
($~y_1, ~y_2,~ y_3,~ y_4,~ y_5,~ y_6~$): [[0]]
\n
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Formulating duality problems
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"eqx", "eq2", "eq3", "eq4", "eq5", "eq6", "eq0les", "eq1les", "eq2les", "eq3les", "eq4les", "eq5les", "eq6les", "eq0gr", "eq1gr", "eq2gr", "eq3gr", "eq4gr", "eq5gr", "eq6gr", "eq0e", "eq1e", "eq2e", "eq3e", "eq4e", "eq5e", "eq6e", "m1", "m2", "m0", "m3", "m4", "m5"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "The following LP problem is given:
\n{mm}: $\\simplify {{c[0]}x_1+{c[1]}x_2+{c[2]}x_3+{c[3]}x_4+{c[4]}x_5} $
\nSubject to:
\n$\\simplify {{a1[0]}x_1+{a1[1]}x_2+{a1[2]}x_3+{a1[3]}x_4+{a1[4]}x_5} ~ \\var{eqx[0]} ~ \\var{b[0]}$
\n$\\simplify {{a20}x_1+{a21}x_2+{a22}x_3+{a23}x_4+{a24}x_5} ~ \\var{eqx[1]} ~ \\var{b[1]}$
\n$\\simplify {{aa1[0]}x_1+{aa1[1]}x_2+{aa1[2]}x_3+{aa1[3]}x_4+{aa1[4]}x_5} ~ \\var{eqx[2]} ~ \\var{b[2]}$
\n$\\simplify {{aa2[0]}x_1+{aa2[1]}x_2+{aa2[2]}x_3+{aa2[3]}x_4+{aa2[4]}x_5} ~ \\var{eqx[3]} ~ \\var{b[3]}$
\n$\\simplify {{aa3[0]}x_1+{aa3[1]}x_2+{aa3[2]}x_3+{aa3[3]}x_4+{aa3[4]}x_5} ~ \\var{eqx[4]} ~ \\var{b[4]}$
\n$\\simplify {{aa4[0]}x_1+{aa4[1]}x_2+{aa4[2]}x_3+{aa4[3]}x_4+{aa4[4]}x_5} ~ \\var{eqx[5]} ~ \\var{b[5]}$
\n$x_1,x_2,x_3,x_4,x_5 \\ge 0.$
\n\nThe dual problem is formulated as follows:
\n{mmd}: $\\simplify {{b[0]}y_1+{b[1]}y_2+{b[2]}y_3+{b[3]}y_4+{b[4]}y_5+{b[5]}y_6 }$
\nSubject to:
\n$\\simplify { {a1[0]}y_1 + {a20}y_2 + {aa1[0]}y_3 + {aa2[0]}y_4 + {aa3[0]}y_5 + {aa4[0]}y_6 >= {c[0]} }$
\n$\\simplify { {a1[1]}y_1 + {a21}y_2 + {aa1[1]}y_3 + {aa2[1]}y_4 + {aa3[1]}y_5 + {aa4[1]}y_6 >= {c[1]}}$
\n$\\simplify { {a1[2]}y_1 + {a22}y_2 + {aa1[2]}y_3 + {aa2[2]}y_4 + {aa3[2]}y_5 + {aa4[2]}y_6 >= {c[2]}}$
\n$\\simplify { {a1[3]}y_1 + {a23}y_2 + {aa1[3]}y_3 + {aa2[3]}y_4 + {aa3[3]}y_5 + {aa4[3]}y_6 >= {c[3]}}$
\n$\\simplify { {a1[4]}y_1 + {a24}y_2 + {aa1[4]}y_3 + {aa2[4]}y_4 + {aa3[4]}y_5 + {aa4[4]}y_6 >= {c[4]}}$
\n\nFind the range/sign of each dual variable (put in the boxes \"1\" for \"$\\le 0$\", \"2\" for \"$\\ge 0$\" and \"3\" for \"$any ~ number$\"):
\n\n
($~y_1, ~y_2,~ y_3,~ y_4,~ y_5,~ y_6~$): [[0]]
\n
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