plotPolytope: Plot the polytope (bounded convex set) of a linear... In gMOIP: Tools for 2D and 3D Plots of Single and Multi-Objective Linear/Integer Programming Models

Description

Plot the polytope (bounded convex set) of a linear mathematical program

Usage

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 plotPolytope( A, b, obj = NULL, type = rep("c", ncol(A)), nonneg = rep(TRUE, ncol(A)), crit = "max", faces = type, plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = FALSE, latex = FALSE, labels = NULL, ... )

Arguments

 A The constraint matrix. b Right hand side. obj A vector with objective coefficients. type A character vector of same length as number of variables. If entry k is 'i' variable k must be integer and if 'c' continuous. nonneg A boolean vector of same length as number of variables. If entry k is TRUE then variable k must be non-negative. crit Either max or min (only used if add the iso profit line) faces A character vector of same length as number of variables. If entry k is 'i' variable k must be integer and if 'c' continuous. Useful if e.g. want to show the linear relaxation of an IP. plotFaces If True then plot the faces. plotFeasible If True then plot the feasible points/segments (relevant for IPLP/MILP). plotOptimum Show the optimum corner solution point (if alternative solutions only one is shown) and add the iso profit line. latex If True make latex math labels for TikZ. labels If NULL don't add any labels. If 'n' no labels but show the points. If 'coord' add coordinates to the points. Otherwise number all points from one. ... If 2D, further arguments passed on the the ggplot plotting functions. This must be done as lists. Currently the following arguments are supported: argsFaces: A list of arguments for plotHull2D. argsFeasible: A list of arguments for ggplot2 functions: geom_point: A list of arguments for ggplot2::geom_point. geom_line: A list of arguments for ggplot2::geom_line. argsLabels: A list of arguments for ggplot2 functions: geom_text: A list of arguments for ggplot2::geom_text. argsOptimum: geom_point: A list of arguments for ggplot2::geom_point. geom_abline: A list of arguments for ggplot2::geom_abline. geom_label: A list of arguments for ggplot2::geom_label. argsTheme: A list of arguments for ggplot2::theme. If 3D further arguments passed on the the RGL plotting functions. This must be done as lists. Currently the following arguments are supported: argsAxes3d: A list of arguments for rgl::axes3d. argsPlot3d: A list of arguments for rgl::plot3d to open the RGL window. argsTitle3d: A list of arguments for rgl::title3d. argsFaces: A list of arguments for plotHull3D. argsFeasible: A list of arguments for rgl functions: points3d: A list of arguments for rgl::points3d. segments3d: A list of arguments for rgl::segments3d. triangles3d: A list of arguments for rgl::triangles3d. argsLabels: A list of arguments for rgl functions: points3d: A list of arguments for rgl::points3d. text3d: A list of arguments for rgl::text3d. argsOptimum: A list of arguments for rgl functions: points3d: A list of arguments for rgl::points3d.

Value

If 2D a ggplot2 object. If 3D a RGL window with the 3D plot.

Note

The feasible region defined by the constraints must be bounded (i.e. no extreme rays) otherwise you may see strange results.

Author(s)

Lars Relund lars@relund.dk

Examples

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 #### 2D examples #### # Define the model max/min coeff*x st. Ax<=b, x>=0 A <- matrix(c(-3,2,2,4,9,10), ncol = 2, byrow = TRUE) b <- c(3,27,90) obj <- c(7.75, 10) ## LP model # The polytope with the corner points plotPolytope( A, b, obj, type = rep("c", ncol(A)), crit = "max", faces = rep("c", ncol(A)), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = FALSE, labels = NULL, argsFaces = list(argsGeom_polygon = list(fill = "red")) ) # With optimum and labels: plotPolytope( A, b, obj, type = rep("c", ncol(A)), crit = "max", faces = rep("c", ncol(A)), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = TRUE, labels = "coord", argsOptimum = list(lty="solid") ) # Minimize: plotPolytope( A, b, obj, type = rep("c", ncol(A)), crit = "min", faces = rep("c", ncol(A)), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = TRUE, labels = "n" ) # Note return a ggplot so can e.g. add other labels on e.g. the axes: p <- plotPolytope( A, b, obj, type = rep("c", ncol(A)), crit = "max", faces = rep("c", ncol(A)), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = TRUE, labels = "coord" ) p + ggplot2::xlab("x") + ggplot2::ylab("y") # More examples ## LP-model with no non-negativity constraints A <- matrix(c(-3, 2, 2, 4, 9, 10, 1, -2), ncol = 2, byrow = TRUE) b <- c(3, 27, 90, 2) obj <- c(7.75, 10) plotPolytope( A, b, obj, type = rep("c", ncol(A)), nonneg = rep(FALSE, ncol(A)), crit = "max", faces = rep("c", ncol(A)), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = FALSE, labels = NULL ) ## The package don't plot feasible regions that are unbounded e.g if we drop the 2 and 3 constraint A <- matrix(c(-3,2), ncol = 2, byrow = TRUE) b <- c(3) obj <- c(7.75, 10) # Wrong plot plotPolytope( A, b, obj, type = rep("c", ncol(A)), crit = "max", faces = rep("c", ncol(A)), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = FALSE, labels = NULL ) # One solution is to add a bounding box and check if the bounding box is binding A <- rbind(A, c(1,0), c(0,1)) b <- c(b, 10, 10) plotPolytope( A, b, obj, type = rep("c", ncol(A)), crit = "max", faces = rep("c", ncol(A)), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = FALSE, labels = NULL ) ## ILP model A <- matrix(c(-3,2,2,4,9,10), ncol = 2, byrow = TRUE) b <- c(3,27,90) obj <- c(7.75, 10) # ILP model with LP faces: plotPolytope( A, b, obj, type = rep("i", ncol(A)), crit = "max", faces = rep("c", ncol(A)), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = TRUE, labels = "coord", argsLabels = list(size = 4, color = "blue"), argsFeasible = list(color = "red", size = 3) ) #ILP model with IP faces: plotPolytope( A, b, obj, type = rep("i", ncol(A)), crit = "max", faces = rep("i", ncol(A)), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = TRUE, labels = "coord" ) ## MILP model A <- matrix(c(-3,2,2,4,9,10), ncol = 2, byrow = TRUE) b <- c(3,27,90) obj <- c(7.75, 10) # Second coordinate integer plotPolytope( A, b, obj, type = c("c", "i"), crit = "max", faces = c("c", "i"), plotFaces = FALSE, plotFeasible = TRUE, plotOptimum = TRUE, labels = "coord", argsFeasible = list(color = "red") ) # First coordinate integer and with LP faces: plotPolytope( A, b, obj, type = c("i", "c"), crit = "max", faces = c("c", "c"), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = TRUE, labels = "coord" ) # First coordinate integer and with LP faces: plotPolytope( A, b, obj, type = c("i", "c"), crit = "max", faces = c("i", "c"), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = TRUE, labels = "coord" ) #### 3D examples #### # Ex 1 view <- matrix( c(-0.412063330411911, -0.228006735444069, 0.882166087627411, 0, 0.910147845745087, -0.0574885793030262, 0.410274744033813, 0, -0.042830865830183, 0.97196090221405, 0.231208890676498, 0, 0, 0, 0, 1), nc = 4) loadView(v = view) A <- matrix( c( 3, 2, 5, 2, 1, 1, 1, 1, 3, 5, 2, 4 ), nc = 3, byrow = TRUE) b <- c(55, 26, 30, 57) obj <- c(20, 10, 15) # LP model plotPolytope(A, b, plotOptimum = TRUE, obj = obj, labels = "coord") plotPolytope(A, b, plotOptimum = TRUE, obj = obj, labels = "coord", argsFaces = list(drawLines = FALSE, argsPolygon3d = list(alpha = 0.95)), argsLabels = list(points3d = list(color = "blue"))) # ILP model plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","i"), plotOptimum = TRUE, obj = obj) # MILP model plotPolytope(A, b, faces = c("c","c","c"), type = c("i","c","i"), plotOptimum = TRUE, obj = obj) plotPolytope(A, b, faces = c("c","c","c"), type = c("c","i","i"), plotOptimum = TRUE, obj = obj) plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","c"), plotOptimum = TRUE, obj = obj) plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","c"), plotFaces = FALSE) plotPolytope(A, b, type = c("i","c","c"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE) plotPolytope(A, b, type = c("c","i","c"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE) plotPolytope(A, b, type = c("c","c","i"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE) # Ex 2 view <- matrix( c(-0.812462985515594, -0.029454167932272, 0.582268416881561, 0, 0.579295456409454, -0.153386667370796, 0.800555109977722, 0, 0.0657325685024261, 0.987727105617523, 0.14168381690979, 0, 0, 0, 0, 1), nc = 4) loadView(v = view) A <- matrix( c( 1, 1, 1, 3, 0, 1 ), nc = 3, byrow = TRUE) b <- c(10, 24) obj <- c(20, 10, 15) plotPolytope(A, b, plotOptimum = TRUE, obj = obj, labels = "coord") # ILP model plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","i"), plotOptimum = TRUE, obj = obj) # MILP model plotPolytope(A, b, faces = c("c","c","c"), type = c("i","c","i"), plotOptimum = TRUE, obj = obj) plotPolytope(A, b, faces = c("c","c","c"), type = c("c","i","i"), plotOptimum = TRUE, obj = obj) plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","c"), plotOptimum = TRUE, obj = obj) plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","c"), plotFaces = FALSE) plotPolytope(A, b, type = c("i","c","c"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE) plotPolytope(A, b, type = c("c","i","c"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE) plotPolytope(A, b, type = c("c","c","i"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE) # Ex 3 view <- matrix( c(0.976349174976349, -0.202332556247711, 0.0761845782399178, 0, 0.0903248339891434, 0.701892614364624, 0.706531345844269, 0, -0.196427255868912, -0.682940244674683, 0.703568696975708, 0, 0, 0, 0, 1), nc = 4) loadView(v = view) A <- matrix( c( -1, 1, 0, 1, 4, 0, 2, 1, 0, 3, -4, 0, 0, 0, 4 ), nc = 3, byrow = TRUE) b <- c(5, 45, 27, 24, 10) obj <- c(5, 45, 15) plotPolytope(A, b, plotOptimum = TRUE, obj = obj, labels = "coord") # ILP model plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","i"), plotOptimum = TRUE, obj = obj) # MILP model plotPolytope(A, b, faces = c("c","c","c"), type = c("i","c","i"), plotOptimum = TRUE, obj = obj) plotPolytope(A, b, faces = c("c","c","c"), type = c("c","i","i"), plotOptimum = TRUE, obj = obj) plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","c"), plotOptimum = TRUE, obj = obj) plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","c"), plotFaces = FALSE) plotPolytope(A, b, type = c("i","c","c"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE) plotPolytope(A, b, type = c("c","i","c"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE) plotPolytope(A, b, type = c("c","c","i"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE) # Ex 4 view <- matrix( c(-0.452365815639496, -0.446501553058624, 0.77201122045517, 0, 0.886364221572876, -0.320795893669128, 0.333835482597351, 0, 0.0986008867621422, 0.835299551486969, 0.540881276130676, 0, 0, 0, 0, 1), nc = 4) loadView(v = view) Ab <- matrix( c( 1, 1, 2, 5, 2, -1, 0, 3, -1, 2, 1, 3, 0, -3, 5, 2 # 0, 1, 0, 4, # 1, 0, 0, 4 ), nc = 4, byrow = TRUE) A <- Ab[,1:3] b <- Ab[,4] obj = c(1,1,3) plotPolytope(A, b, plotOptimum = TRUE, obj = obj, labels = "coord") # ILP model plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","i"), plotOptimum = TRUE, obj = obj) # MILP model plotPolytope(A, b, faces = c("c","c","c"), type = c("i","c","i"), plotOptimum = TRUE, obj = obj) plotPolytope(A, b, faces = c("c","c","c"), type = c("c","i","i"), plotOptimum = TRUE, obj = obj) plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","c"), plotOptimum = TRUE, obj = obj) plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","c"), plotFaces = FALSE) plotPolytope(A, b, type = c("i","c","c"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE) plotPolytope(A, b, type = c("c","i","c"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE) plotPolytope(A, b, faces = c("c","c","c"), type = c("c","c","i"), plotOptimum = TRUE, obj = obj)

gMOIP documentation built on Aug. 23, 2021, 5:09 p.m.