1 00:00:04,180 --> 00:00:09,549 welcome everybody to this webinar we're 2 00:00:07,960 --> 00:00:11,680 going to be talking about numerical 3 00:00:09,549 --> 00:00:14,500 issues and we Furr getting into that 4 00:00:11,680 --> 00:00:18,099 well Robbie's been around for twenty ten 5 00:00:14,500 --> 00:00:19,750 years and the teaming Grobe has over 20 6 00:00:18,099 --> 00:00:22,119 years experience in optimization 7 00:00:19,750 --> 00:00:25,000 problems and a numerical issues for sure 8 00:00:22,119 --> 00:00:27,669 and actually we have built this 9 00:00:25,000 --> 00:00:31,210 experience helping our customers now all 10 00:00:27,669 --> 00:00:34,239 of them use Rob in a daily basis and and 11 00:00:31,210 --> 00:00:37,329 the kind of applications that they solve 12 00:00:34,239 --> 00:00:40,510 it is really wide from accounting to 13 00:00:37,329 --> 00:00:43,510 programming to you really name it and 14 00:00:40,510 --> 00:00:46,510 it's from this experience instead this 15 00:00:43,510 --> 00:00:48,370 webinar is drawing on so let's go to the 16 00:00:46,510 --> 00:00:52,120 global guidelines for numerical issues 17 00:00:48,370 --> 00:00:55,539 or you may say why things go wrong and 18 00:00:52,120 --> 00:00:57,730 more importantly how to avoid it so what 19 00:00:55,539 --> 00:01:00,730 are numerical issues that's probably a 20 00:00:57,730 --> 00:01:03,579 hard thing to say we may say that 21 00:01:00,730 --> 00:01:05,490 actually people talk about numerical 22 00:01:03,579 --> 00:01:08,530 issues whenever they encounter 23 00:01:05,490 --> 00:01:11,140 unexpected results from their algorithms 24 00:01:08,530 --> 00:01:14,080 or even they get inconsistent results 25 00:01:11,140 --> 00:01:17,770 for the same problem and the source of 26 00:01:14,080 --> 00:01:20,710 these are numerous however the most 27 00:01:17,770 --> 00:01:24,250 common ones I will share these for these 28 00:01:20,710 --> 00:01:26,470 are or can be more easily hampered the 29 00:01:24,250 --> 00:01:29,710 first one is rounding which I'm going to 30 00:01:26,470 --> 00:01:32,220 be talking about and then there is the 31 00:01:29,710 --> 00:01:34,600 problem of the big gap between 32 00:01:32,220 --> 00:01:36,130 theoretical numbers and what the 33 00:01:34,600 --> 00:01:38,980 computers actually understand and 34 00:01:36,130 --> 00:01:43,780 represent those numbers and coming from 35 00:01:38,980 --> 00:01:46,480 that gap it's also a common thing for us 36 00:01:43,780 --> 00:01:49,270 to forget that these numbers have limits 37 00:01:46,480 --> 00:01:51,970 and those limits impose to what we can 38 00:01:49,270 --> 00:01:53,920 ask for regarded on precision and 39 00:01:51,970 --> 00:01:57,790 finally we're going to be talking out so 40 00:01:53,920 --> 00:01:59,800 about it conditioning so let's get first 41 00:01:57,790 --> 00:02:03,130 on trying to understand where these 42 00:01:59,800 --> 00:02:05,710 issues come from the first and probably 43 00:02:03,130 --> 00:02:08,170 the easiest one is rounding coefficient 44 00:02:05,710 --> 00:02:11,860 probably this is best explained with an 45 00:02:08,170 --> 00:02:14,109 example and this can really be very very 46 00:02:11,860 --> 00:02:16,569 dangerous so this is a very simple 47 00:02:14,109 --> 00:02:17,680 example I assume that you give the 48 00:02:16,569 --> 00:02:20,230 solver 49 00:02:17,680 --> 00:02:21,760 these two constraints and you may look 50 00:02:20,230 --> 00:02:24,219 at them and you say well you know what 51 00:02:21,760 --> 00:02:26,739 this really looked like the same 52 00:02:24,219 --> 00:02:30,299 constraint and the numbers are 53 00:02:26,739 --> 00:02:33,989 well close to be really one-third that 54 00:02:30,299 --> 00:02:38,469 0.333 could be interpreted as one sir 55 00:02:33,989 --> 00:02:40,629 however we always will interpret the 56 00:02:38,469 --> 00:02:43,450 numbers that you gave the solver asked 57 00:02:40,629 --> 00:02:45,430 the true number and if you do that and 58 00:02:43,450 --> 00:02:47,049 you do the linear algebra on do the 59 00:02:45,430 --> 00:02:49,750 substitution what you will realize is 60 00:02:47,049 --> 00:02:53,980 essentially these two constraints are 61 00:02:49,750 --> 00:02:57,310 really implying that X is one and Y is 62 00:02:53,980 --> 00:02:58,810 zero and this comes from assuming that 63 00:02:57,310 --> 00:03:02,919 the numbers you're creating the solver 64 00:02:58,810 --> 00:03:05,769 are what you really want them to me so 65 00:03:02,919 --> 00:03:08,769 if instead of using this kind of 66 00:03:05,769 --> 00:03:12,099 approximate a fair number you have had 67 00:03:08,769 --> 00:03:14,290 full precision let's say these two 68 00:03:12,099 --> 00:03:16,719 equations a very different story will 69 00:03:14,290 --> 00:03:19,599 have a Rison so essentially you will do 70 00:03:16,719 --> 00:03:22,090 the algebra you do the substitutions and 71 00:03:19,599 --> 00:03:24,970 then you end up with these expression 72 00:03:22,090 --> 00:03:29,229 down here which is 2 times 10 to the 73 00:03:24,970 --> 00:03:31,690 minus 16 times X plus 1 plus the same 74 00:03:29,229 --> 00:03:33,639 coefficient and this is really really 75 00:03:31,690 --> 00:03:35,609 really close to b1 and actually probably 76 00:03:33,639 --> 00:03:38,500 will say well this is one and 77 00:03:35,609 --> 00:03:41,049 essentially what we will do is that drop 78 00:03:38,500 --> 00:03:44,439 one of those two constraints and except 79 00:03:41,049 --> 00:03:47,349 anything that satisfies these linear 80 00:03:44,439 --> 00:03:49,900 constraint as feasible solutions whereas 81 00:03:47,349 --> 00:03:52,479 if you rounded as I showed before you 82 00:03:49,900 --> 00:03:56,109 will be telling the solver only to care 83 00:03:52,479 --> 00:03:58,269 for X 1 and y equals 0 which in most 84 00:03:56,109 --> 00:04:00,489 cases is not what you really want it so 85 00:03:58,269 --> 00:04:03,189 this is very easy to avoid and and and 86 00:04:00,489 --> 00:04:05,349 one thing to look out for errors is 87 00:04:03,189 --> 00:04:08,859 whenever you're feeling the numbers in 88 00:04:05,349 --> 00:04:13,299 model be sure to use the full precision 89 00:04:08,859 --> 00:04:15,400 allowed by the solver which is 16 digits 90 00:04:13,299 --> 00:04:18,009 that's the standard precision I'm going 91 00:04:15,400 --> 00:04:21,009 to get into that and actually probably a 92 00:04:18,009 --> 00:04:25,030 good question to ask yourself well why 2 93 00:04:21,009 --> 00:04:27,039 times 10 to the minus 16 what's so 94 00:04:25,030 --> 00:04:29,620 special about that number why this 95 00:04:27,039 --> 00:04:32,219 global use disregard those numbers and 96 00:04:29,620 --> 00:04:34,419 that brings us to the next 97 00:04:32,219 --> 00:04:37,330 that the real numbers are not really 98 00:04:34,419 --> 00:04:40,179 real at least not in the computer or not 99 00:04:37,330 --> 00:04:41,499 the way computers handle these numbers 100 00:04:40,179 --> 00:04:43,419 so that's what I mean 101 00:04:41,499 --> 00:04:45,069 and there is an easy check for that so 102 00:04:43,419 --> 00:04:48,129 you can open up your Python interpreter 103 00:04:45,069 --> 00:04:51,580 and type that expression and you will 104 00:04:48,129 --> 00:04:53,319 see that you will get a true and look at 105 00:04:51,580 --> 00:04:55,119 it and you say now that that cannot be 106 00:04:53,319 --> 00:04:58,449 true but that's what the computer is 107 00:04:55,119 --> 00:05:01,509 saying well you may think this is coming 108 00:04:58,449 --> 00:05:04,180 from this is really tiny number so maybe 109 00:05:01,509 --> 00:05:07,360 it's not so wrong to assume that is that 110 00:05:04,180 --> 00:05:09,249 expression is almost one but surely it 111 00:05:07,360 --> 00:05:12,249 is huge numbers that shouldn't happen 112 00:05:09,249 --> 00:05:14,229 but that's not true either so if you do 113 00:05:12,249 --> 00:05:17,979 these other experiments right now I have 114 00:05:14,229 --> 00:05:21,999 10 to the 16 plus 1 I compare it to 10 115 00:05:17,979 --> 00:05:25,089 to the 16 again Python will say that's 116 00:05:21,999 --> 00:05:28,509 true that's really strange is those are 117 00:05:25,089 --> 00:05:32,110 not really small numbers to make a long 118 00:05:28,509 --> 00:05:34,990 story short this has nothing to do with 119 00:05:32,110 --> 00:05:38,789 Python really this will be truly tested 120 00:05:34,990 --> 00:05:41,379 in Excel in our in C in Java whatever 121 00:05:38,789 --> 00:05:45,219 programming language that you use if you 122 00:05:41,379 --> 00:05:47,769 use the standard provided floating-point 123 00:05:45,219 --> 00:05:51,699 numbers underneath these expressions 124 00:05:47,769 --> 00:05:54,789 will always hold true and it get worse 125 00:05:51,699 --> 00:05:58,089 really so given these limitations I mean 126 00:05:54,789 --> 00:06:00,309 these kind of comparisons then not even 127 00:05:58,089 --> 00:06:02,649 the basic algebra rules that we learn in 128 00:06:00,309 --> 00:06:05,469 school are true anymore so not even 129 00:06:02,649 --> 00:06:08,439 associativity like here so if I add one 130 00:06:05,469 --> 00:06:10,899 plus the tiny quantity and then again 131 00:06:08,439 --> 00:06:12,969 the tiny quantity that's not the same as 132 00:06:10,899 --> 00:06:14,800 first adding these two tiny quantities 133 00:06:12,969 --> 00:06:18,849 and then one this is not people 134 00:06:14,800 --> 00:06:21,729 so really numbers storing the computer 135 00:06:18,849 --> 00:06:23,740 using they'd standard hardware it's not 136 00:06:21,729 --> 00:06:27,430 what we expected or what we learned to 137 00:06:23,740 --> 00:06:31,869 do with numbers in high school and I'm 138 00:06:27,430 --> 00:06:35,589 sure there are ways around it just to 139 00:06:31,869 --> 00:06:39,519 mention some of them gene UMP is one way 140 00:06:35,589 --> 00:06:42,370 around it there are other tools but all 141 00:06:39,519 --> 00:06:44,589 these tools rely on software base 142 00:06:42,370 --> 00:06:44,960 implementations or handing out numbers 143 00:06:44,589 --> 00:06:47,000 and 144 00:06:44,960 --> 00:06:50,270 those can be very slow and I'm going to 145 00:06:47,000 --> 00:06:53,240 be talking a bit about that and actually 146 00:06:50,270 --> 00:06:56,630 you can expect where these limits come 147 00:06:53,240 --> 00:06:58,910 from in Python you can say import system 148 00:06:56,630 --> 00:07:01,430 and then print the floating information 149 00:06:58,910 --> 00:07:06,020 and you'll see there is this quantity 150 00:07:01,430 --> 00:07:09,169 down here epsilon that is 2 on all those 151 00:07:06,020 --> 00:07:11,509 and the numbers times 10 to the minus 16 152 00:07:09,169 --> 00:07:14,500 and what's that epsilon that's the 153 00:07:11,509 --> 00:07:16,759 smallest number that you can add to one 154 00:07:14,500 --> 00:07:19,639 and still distinguish it from one 155 00:07:16,759 --> 00:07:23,120 anything below that and one won't be 156 00:07:19,639 --> 00:07:26,810 able to register in the computer numbers 157 00:07:23,120 --> 00:07:29,419 and we use and as I said there is no way 158 00:07:26,810 --> 00:07:31,070 around this and that's what I mean by 159 00:07:29,419 --> 00:07:35,539 real numbers none being real at least 160 00:07:31,070 --> 00:07:38,000 not in the computer so that really means 161 00:07:35,539 --> 00:07:41,270 that we have to be very careful on how 162 00:07:38,000 --> 00:07:44,030 we do arithmetic cela computer now let 163 00:07:41,270 --> 00:07:47,030 me explain a bit more about this so what 164 00:07:44,030 --> 00:07:50,710 does this mean most peripherals provide 165 00:07:47,030 --> 00:07:52,940 a cheap base implementation of these 166 00:07:50,710 --> 00:07:56,120 floating-point numbers these kind of 167 00:07:52,940 --> 00:07:59,449 approximation of real numbers and this 168 00:07:56,120 --> 00:08:02,300 hardware is really really key to get 169 00:07:59,449 --> 00:08:04,130 good performance I mean this using the 170 00:08:02,300 --> 00:08:06,590 hardware base floating-point numbers 171 00:08:04,130 --> 00:08:09,080 it's 10 100 and sometimes even a 172 00:08:06,590 --> 00:08:11,120 thousand times faster then you use 173 00:08:09,080 --> 00:08:14,419 seeing any of the standard extended 174 00:08:11,120 --> 00:08:17,900 precision around there so if you really 175 00:08:14,419 --> 00:08:20,509 care about performance you should deal 176 00:08:17,900 --> 00:08:24,199 with these numbers under limit and 177 00:08:20,509 --> 00:08:26,719 that's what GU Ruby does we use in 178 00:08:24,199 --> 00:08:30,229 almost all code these double precision 179 00:08:26,719 --> 00:08:32,089 64-bit numbers and this is not too bad 180 00:08:30,229 --> 00:08:35,180 actually I'm going to show you that 181 00:08:32,089 --> 00:08:38,209 although these numbers are limits if use 182 00:08:35,180 --> 00:08:43,279 within reason they do provide meaningful 183 00:08:38,209 --> 00:08:45,850 answers 99.999 percent of the time there 184 00:08:43,279 --> 00:08:50,149 is also this last part which is very key 185 00:08:45,850 --> 00:08:53,720 since these numbers don't really satisfy 186 00:08:50,149 --> 00:08:56,060 so should he be adding up is not really 187 00:08:53,720 --> 00:08:58,640 an exact operation 188 00:08:56,060 --> 00:09:00,440 whenever you're testing if I mean 189 00:08:58,640 --> 00:09:03,170 while their own equation holds you 190 00:09:00,440 --> 00:09:05,780 cannot really test for equality of the 191 00:09:03,170 --> 00:09:07,880 expressions you have to take into 192 00:09:05,780 --> 00:09:11,510 account that there might be some 193 00:09:07,880 --> 00:09:12,530 rounding error going on and what do we 194 00:09:11,510 --> 00:09:15,410 do to handle that 195 00:09:12,530 --> 00:09:19,070 well that's Torrance's and user scale 196 00:09:15,410 --> 00:09:21,230 and this is key to understand so what 197 00:09:19,070 --> 00:09:23,240 will guru be do we always as I said 198 00:09:21,230 --> 00:09:27,260 before when we were talking about these 199 00:09:23,240 --> 00:09:31,100 rounding issues we always trust numbers 200 00:09:27,260 --> 00:09:33,590 that you keep so for example if somehow 201 00:09:31,100 --> 00:09:35,870 in your model you write 1 plus 10 to the 202 00:09:33,590 --> 00:09:41,180 minus 7 times X less than or equal to 1 203 00:09:35,870 --> 00:09:42,740 with X being binary results may detect 204 00:09:41,180 --> 00:09:45,410 this expression because it couldn't be 205 00:09:42,740 --> 00:09:49,150 hidden and decided well that implies 206 00:09:45,410 --> 00:09:52,070 that X must be 0 which is a correct 207 00:09:49,150 --> 00:09:55,760 implication from what the input that you 208 00:09:52,070 --> 00:09:58,820 gave the server is however I have to 209 00:09:55,760 --> 00:10:01,190 make an emphasis that this may happen 210 00:09:58,820 --> 00:10:03,260 and why may happen and it doesn't happen 211 00:10:01,190 --> 00:10:06,140 all the time well because during resolve 212 00:10:03,260 --> 00:10:08,660 or even by cutting planes we have to 213 00:10:06,140 --> 00:10:11,510 limit how much time we invest in trying 214 00:10:08,660 --> 00:10:14,180 to detect this kind of stuff so not all 215 00:10:11,510 --> 00:10:15,800 the time we will be performing an 216 00:10:14,180 --> 00:10:17,990 exhaustive search and all the 217 00:10:15,800 --> 00:10:21,880 implications hidden in the model and 218 00:10:17,990 --> 00:10:27,290 that's why it may be comforting to this 219 00:10:21,880 --> 00:10:30,230 however when you solve the model and you 220 00:10:27,290 --> 00:10:33,320 get a feasible solution you have to 221 00:10:30,230 --> 00:10:35,750 check if that solution is feasible but 222 00:10:33,320 --> 00:10:37,910 as I said before you cannot really check 223 00:10:35,750 --> 00:10:40,940 for equality on the constraint and you 224 00:10:37,910 --> 00:10:43,520 need to introduce the idea of tolerances 225 00:10:40,940 --> 00:10:45,860 so whenever we have a feasible solution 226 00:10:43,520 --> 00:10:48,320 produced by any race they go by solving 227 00:10:45,860 --> 00:10:51,050 a linear system there are at least three 228 00:10:48,320 --> 00:10:53,420 things that we have to check one is 229 00:10:51,050 --> 00:10:57,080 deciding if a given solution is integer 230 00:10:53,420 --> 00:10:59,470 or not and what we do is that we define 231 00:10:57,080 --> 00:11:02,650 this tolerance called the inter-facility 232 00:10:59,470 --> 00:11:08,030 tolerance to decide whether a given 233 00:11:02,650 --> 00:11:09,360 quantity is interior enough the same 234 00:11:08,030 --> 00:11:11,940 applies to 235 00:11:09,360 --> 00:11:14,279 I'm a feasibility whenever we have to 236 00:11:11,940 --> 00:11:17,850 check that an inequality or an equation 237 00:11:14,279 --> 00:11:21,290 actually is satisfied we don't test for 238 00:11:17,850 --> 00:11:25,529 equality here we always allow a little 239 00:11:21,290 --> 00:11:28,680 wiggle room to say and again whenever 240 00:11:25,529 --> 00:11:31,829 you are checking for a certificate of 241 00:11:28,680 --> 00:11:34,500 dual optimality the same happens the 242 00:11:31,829 --> 00:11:36,810 dual space and that is called the 243 00:11:34,500 --> 00:11:38,910 optimality tolerance and those are three 244 00:11:36,810 --> 00:11:44,130 different quantities that actually you 245 00:11:38,910 --> 00:11:48,089 can move a bit so well does this mean 246 00:11:44,130 --> 00:11:51,000 and this is very subtle so let me try to 247 00:11:48,089 --> 00:11:54,750 be very careful with what I say first 248 00:11:51,000 --> 00:11:57,620 the tolerances that I was describing for 249 00:11:54,750 --> 00:12:00,420 there are absolute quantities they won't 250 00:11:57,620 --> 00:12:03,209 change depending on the input numbers 251 00:12:00,420 --> 00:12:06,029 that you give so if you change the scale 252 00:12:03,209 --> 00:12:08,459 of a constrain or if a variable the 253 00:12:06,029 --> 00:12:10,980 meaning in terms of these Torrance's we 254 00:12:08,459 --> 00:12:12,839 change and you have to be careful about 255 00:12:10,980 --> 00:12:16,519 that I'm going to talk a bit more about 256 00:12:12,839 --> 00:12:19,350 it it also means that you may have a 257 00:12:16,519 --> 00:12:23,070 feasible solution that satisfied the set 258 00:12:19,350 --> 00:12:25,380 tolerance that I said before but even 259 00:12:23,070 --> 00:12:28,519 having a solution that violates use 260 00:12:25,380 --> 00:12:31,920 let's say by 10 to the minus 7 a 261 00:12:28,519 --> 00:12:34,230 constrained pre-sold or even a cutting 262 00:12:31,920 --> 00:12:36,720 plane may decide at some intermediate 263 00:12:34,230 --> 00:12:39,930 state on the algorithm to cut it off and 264 00:12:36,720 --> 00:12:43,410 to mark that solution as infeasible and 265 00:12:39,930 --> 00:12:46,110 why that can be troublesome well because 266 00:12:43,410 --> 00:12:49,740 it means that at least for models where 267 00:12:46,110 --> 00:12:53,190 the numerix are challenging the optimal 268 00:12:49,740 --> 00:12:58,140 value may change depending on the part 269 00:12:53,190 --> 00:13:00,180 of the argument age so essentially what 270 00:12:58,140 --> 00:13:01,980 this means is that these tolerances 271 00:13:00,180 --> 00:13:05,310 especially for problems were is hard to 272 00:13:01,980 --> 00:13:06,240 get accurate solutions imply that there 273 00:13:05,310 --> 00:13:09,990 is a gray zone 274 00:13:06,240 --> 00:13:12,000 there is a grey zone of solutions that 275 00:13:09,990 --> 00:13:14,130 may be considered feasible in some 276 00:13:12,000 --> 00:13:17,190 situations but in some other situations 277 00:13:14,130 --> 00:13:19,560 may be considered infeasible where as 278 00:13:17,190 --> 00:13:22,920 whatever is strictly feasible will 279 00:13:19,560 --> 00:13:23,370 remain feasible always and then outside 280 00:13:22,920 --> 00:13:25,200 that 281 00:13:23,370 --> 00:13:28,890 torence region then things will be 282 00:13:25,200 --> 00:13:32,550 considered infeasible so whenever the 283 00:13:28,890 --> 00:13:35,160 problem has a big gray area that's the 284 00:13:32,550 --> 00:13:38,850 case where you're observing numerical 285 00:13:35,160 --> 00:13:41,310 issues with your mother so how bad is 286 00:13:38,850 --> 00:13:43,650 really this because I mean up to now 287 00:13:41,310 --> 00:13:46,230 this is sounding really really bad 288 00:13:43,650 --> 00:13:48,540 however people use optimization very 289 00:13:46,230 --> 00:13:51,060 often and it's not that common for them 290 00:13:48,540 --> 00:13:53,460 to get into issues so how come there 291 00:13:51,060 --> 00:13:55,370 must be something going on in here and 292 00:13:53,460 --> 00:13:59,940 indeed there is 293 00:13:55,370 --> 00:14:02,279 so first the fault tolerance that we use 294 00:13:59,940 --> 00:14:05,130 in grub is 10 to the minus 6 for primal 295 00:14:02,279 --> 00:14:08,430 and dual feasibility and for integral it 296 00:14:05,130 --> 00:14:10,710 is 10 to the minus 5 and let's say that 297 00:14:08,430 --> 00:14:13,140 you defined your model and your right 298 00:14:10,710 --> 00:14:17,300 hand size are in the order of 10 to the 299 00:14:13,140 --> 00:14:20,880 4 10 to the 3 it means that the gray so 300 00:14:17,300 --> 00:14:22,410 the area where you may have some 301 00:14:20,880 --> 00:14:23,940 solution to be treated sometimes 302 00:14:22,410 --> 00:14:26,550 feasible and sometimes is infeasible 303 00:14:23,940 --> 00:14:29,339 it's really tiny is use around 10 to the 304 00:14:26,550 --> 00:14:32,190 minus 9 or 1 in a billion so that's 305 00:14:29,339 --> 00:14:33,000 really really really small and for most 306 00:14:32,190 --> 00:14:35,459 problems 307 00:14:33,000 --> 00:14:38,700 that's just insignificant so that gray 308 00:14:35,459 --> 00:14:40,620 area doesn't play any role and that's 309 00:14:38,700 --> 00:14:44,400 why I said that the scaling is key 310 00:14:40,620 --> 00:14:47,160 because you are the one who choose how 311 00:14:44,400 --> 00:14:49,440 big is going to be the coefficients your 312 00:14:47,160 --> 00:14:52,890 constraint or how big are the ranges in 313 00:14:49,440 --> 00:14:56,160 your variables also this means since the 314 00:14:52,890 --> 00:14:58,770 tolerance as I said before are absolute 315 00:14:56,160 --> 00:15:02,820 and not relative if you have constraints 316 00:14:58,770 --> 00:15:05,730 with very small active ranges the solver 317 00:15:02,820 --> 00:15:07,920 may consider them juice to be an 318 00:15:05,730 --> 00:15:10,790 equation and the changes that you're 319 00:15:07,920 --> 00:15:13,470 expecting to see let's say if the 320 00:15:10,790 --> 00:15:16,950 solutions moving the range of 10 to the 321 00:15:13,470 --> 00:15:21,660 minus 9 and 10 to the minus 10 then that 322 00:15:16,950 --> 00:15:24,810 is just not being seen by the solver so 323 00:15:21,660 --> 00:15:27,090 you should also try to avoid 324 00:15:24,810 --> 00:15:30,630 ranges for constraint are just too small 325 00:15:27,090 --> 00:15:35,340 unless they are really equations 326 00:15:30,630 --> 00:15:38,280 okay so what about variables because 327 00:15:35,340 --> 00:15:40,970 this was talking about constraints so 328 00:15:38,280 --> 00:15:43,800 the same really applies to variables 329 00:15:40,970 --> 00:15:46,110 ranges for variables shouldn't also be 330 00:15:43,800 --> 00:15:48,540 to be for the same reason because the 331 00:15:46,110 --> 00:15:51,390 bounds are again primal constraints 332 00:15:48,540 --> 00:15:54,000 although we treat them differently you 333 00:15:51,390 --> 00:15:56,880 should also be careful not to have wide 334 00:15:54,000 --> 00:15:58,980 range of primal variables and also for 335 00:15:56,880 --> 00:16:01,350 interior variables I mean if you have an 336 00:15:58,980 --> 00:16:03,810 entire variable with really wide ranges 337 00:16:01,350 --> 00:16:05,520 let's say I'm million maybe you should 338 00:16:03,810 --> 00:16:08,250 reconsider treating them as a linear 339 00:16:05,520 --> 00:16:10,440 maybe their use continues especially if 340 00:16:08,250 --> 00:16:13,920 that's kind of the range where they move 341 00:16:10,440 --> 00:16:16,380 I mean just a unit change here in the 342 00:16:13,920 --> 00:16:18,660 order of the millions won't change much 343 00:16:16,380 --> 00:16:21,750 but you will be paying the price of 344 00:16:18,660 --> 00:16:25,730 forcing this over to work harder to find 345 00:16:21,750 --> 00:16:28,230 kind of an intercession for those ranges 346 00:16:25,730 --> 00:16:31,320 what about the objective function so 347 00:16:28,230 --> 00:16:35,370 again the same applies because remember 348 00:16:31,320 --> 00:16:37,440 whenever you are looking at feasible 349 00:16:35,370 --> 00:16:40,230 solutions you hold so how to check the 350 00:16:37,440 --> 00:16:44,700 duality conditions really stay that the 351 00:16:40,230 --> 00:16:48,150 giving help resolve is optimal so what I 352 00:16:44,700 --> 00:16:50,310 recommend is to deal your problem so 353 00:16:48,150 --> 00:16:53,280 that the optimal solution is in the 354 00:16:50,310 --> 00:16:56,070 range of either minus 10 to the 4 or 10 355 00:16:53,280 --> 00:16:58,740 to the 4 otherwise what you're doing is 356 00:16:56,070 --> 00:17:01,350 really pushing the dual feasibility so 357 00:16:58,740 --> 00:17:05,220 let's say you're talking about billions 358 00:17:01,350 --> 00:17:07,589 of dollars a good idea will be to change 359 00:17:05,220 --> 00:17:09,810 the units in your elective function to 360 00:17:07,589 --> 00:17:13,230 talk about thousands or maybe 10 361 00:17:09,810 --> 00:17:16,110 thousands of dollars instead of I know 362 00:17:13,230 --> 00:17:18,089 using iron dollars or cents which will 363 00:17:16,110 --> 00:17:21,630 make the objective function blow up and 364 00:17:18,089 --> 00:17:23,430 make everything really harder also for 365 00:17:21,630 --> 00:17:26,069 the same reasons that you don't want to 366 00:17:23,430 --> 00:17:28,620 have constraints with very small ranges 367 00:17:26,069 --> 00:17:31,980 you should also try to avoid objective 368 00:17:28,620 --> 00:17:34,830 functions that are really tiny so if 369 00:17:31,980 --> 00:17:37,650 your problem does have a meaningful 370 00:17:34,830 --> 00:17:39,750 objective function what I will recommend 371 00:17:37,650 --> 00:17:43,410 is to either make it strictly bigger 372 00:17:39,750 --> 00:17:44,160 than 1 or relieve all of that minus 1 373 00:17:43,410 --> 00:17:47,070 and 374 00:17:44,160 --> 00:17:50,610 is to move away from zero again because 375 00:17:47,070 --> 00:17:53,610 then there is a better handling of the 376 00:17:50,610 --> 00:17:55,260 coefficients on the dual side and the 377 00:17:53,610 --> 00:17:56,910 only exception for this is when your 378 00:17:55,260 --> 00:17:59,460 problem is really a feasibility problem 379 00:17:56,910 --> 00:18:02,190 in which case all the coefficients in 380 00:17:59,460 --> 00:18:05,640 the objective can be zero and that won't 381 00:18:02,190 --> 00:18:08,070 hurt performance at all actually and not 382 00:18:05,640 --> 00:18:11,370 all these is not too hard to achieve 383 00:18:08,070 --> 00:18:13,770 it's really just changing the scaling in 384 00:18:11,370 --> 00:18:16,740 your roses change essentially changing 385 00:18:13,770 --> 00:18:19,430 the units on how you measure the DVDs in 386 00:18:16,740 --> 00:18:22,920 your rows and also in the columns and 387 00:18:19,430 --> 00:18:25,860 also in objective function however as I 388 00:18:22,920 --> 00:18:26,220 said before Ruby will not do these for 389 00:18:25,860 --> 00:18:29,280 you 390 00:18:26,220 --> 00:18:32,880 it's after the user really to define 391 00:18:29,280 --> 00:18:36,300 what's the right range for all these 392 00:18:32,880 --> 00:18:39,650 quantities so how did you get tight 393 00:18:36,300 --> 00:18:42,120 ranges for constraints and variables 394 00:18:39,650 --> 00:18:46,950 probably the first thing that you should 395 00:18:42,120 --> 00:18:50,490 try is to use other information not 396 00:18:46,950 --> 00:18:53,340 available in the model to impose tighter 397 00:18:50,490 --> 00:18:55,800 bounds either in constraints or in 398 00:18:53,340 --> 00:18:59,790 variables so let's say I know I'm trying 399 00:18:55,800 --> 00:19:02,910 to plan providing a service of dispatch 400 00:18:59,790 --> 00:19:07,230 and delivery I'm in principle I could 401 00:19:02,910 --> 00:19:09,750 have any quantity of trucks however I 402 00:19:07,230 --> 00:19:11,910 know because I know the business that I 403 00:19:09,750 --> 00:19:14,160 am only considering three possible 404 00:19:11,910 --> 00:19:15,810 locations to store the tracks and those 405 00:19:14,160 --> 00:19:18,930 locations won't handle more than a 406 00:19:15,810 --> 00:19:20,880 hundred trucks so you can give that 407 00:19:18,930 --> 00:19:24,390 piece of information to the solver and 408 00:19:20,880 --> 00:19:27,180 the server will use that information to 409 00:19:24,390 --> 00:19:30,060 really prune both the search and make 410 00:19:27,180 --> 00:19:32,520 everything much faster and if you think 411 00:19:30,060 --> 00:19:35,610 about it a little bit if you don't give 412 00:19:32,520 --> 00:19:37,800 that information solver the solver has 413 00:19:35,610 --> 00:19:39,690 to consider the possibility of going 414 00:19:37,800 --> 00:19:43,670 beyond although you won't have any use 415 00:19:39,690 --> 00:19:46,620 for that solution if it's relevant so 416 00:19:43,670 --> 00:19:49,410 the first thing and most important is 417 00:19:46,620 --> 00:19:52,290 try to get whatever extra information 418 00:19:49,410 --> 00:19:55,220 you have at hand to get tied bounds on 419 00:19:52,290 --> 00:19:58,820 your variables after that 420 00:19:55,220 --> 00:20:01,040 change units from towns 2,000 tons let's 421 00:19:58,820 --> 00:20:04,000 say or from dollars to thousands of 422 00:20:01,040 --> 00:20:07,880 dollars all the time try to choose the 423 00:20:04,000 --> 00:20:09,860 proper units so that the range of 424 00:20:07,880 --> 00:20:13,130 constraints and variables is not too 425 00:20:09,860 --> 00:20:16,610 large were too large I mean hopefully 426 00:20:13,130 --> 00:20:21,050 not beyond 10 to the 4 and finally there 427 00:20:16,610 --> 00:20:23,900 is a common use case on where you can 428 00:20:21,050 --> 00:20:25,330 get very big and small coefficients in 429 00:20:23,900 --> 00:20:28,310 the objective function and that's what 430 00:20:25,330 --> 00:20:30,200 people do when they are really trying to 431 00:20:28,310 --> 00:20:33,010 do hierarchical multi objective 432 00:20:30,200 --> 00:20:35,690 optimization they will blend all these 433 00:20:33,010 --> 00:20:37,490 different objectives with huge 434 00:20:35,690 --> 00:20:39,140 coefficients for the most important 435 00:20:37,490 --> 00:20:40,700 objectives and then with smaller 436 00:20:39,140 --> 00:20:43,580 coefficients for the less important 437 00:20:40,700 --> 00:20:45,320 objectives so that will naturally two 438 00:20:43,580 --> 00:20:48,100 huge clear pitch in the objective 439 00:20:45,320 --> 00:20:50,210 function however now Grobe supports 440 00:20:48,100 --> 00:20:53,060 internally this multi objective 441 00:20:50,210 --> 00:20:56,330 optimization where each objective now 442 00:20:53,060 --> 00:20:59,660 can have reasonable values reasonable 443 00:20:56,330 --> 00:21:04,880 coefficients and everything believe me 444 00:20:59,660 --> 00:21:09,290 will be much easier to follow so let me 445 00:21:04,880 --> 00:21:12,530 try to stress why scaling is relevant 446 00:21:09,290 --> 00:21:14,900 what I'm going to do is an experiment a 447 00:21:12,530 --> 00:21:17,300 quick experiment actually so what I'm 448 00:21:14,900 --> 00:21:19,820 going to take is like a problem and then 449 00:21:17,300 --> 00:21:21,950 I'm going to randomly rescale the 450 00:21:19,820 --> 00:21:25,070 columns of the problems so that the 451 00:21:21,950 --> 00:21:29,240 optimal value in theory doesn't change 452 00:21:25,070 --> 00:21:32,120 so I'm going to change the objective and 453 00:21:29,240 --> 00:21:35,180 the coefficients here and then decrease 454 00:21:32,120 --> 00:21:37,660 the bounds if they are finite okay I'm 455 00:21:35,180 --> 00:21:40,670 going to do that within our range 456 00:21:37,660 --> 00:21:42,470 coefficient and I'm going to run this 457 00:21:40,670 --> 00:21:43,370 script you'll get this script so you can 458 00:21:42,470 --> 00:21:46,730 look at it later on 459 00:21:43,370 --> 00:21:48,740 this is our common problem and this 460 00:21:46,730 --> 00:21:52,100 means that I'm going to scale the 461 00:21:48,740 --> 00:21:55,190 columns with a factor up to of one that 462 00:21:52,100 --> 00:21:57,260 it could be smaller randomly and then I 463 00:21:55,190 --> 00:21:59,660 let grow be optimized the resulting 464 00:21:57,260 --> 00:22:01,040 model and what you see is right away is 465 00:21:59,660 --> 00:22:03,650 you get this warning I mean the 466 00:22:01,040 --> 00:22:05,870 coefficients are to be actually here you 467 00:22:03,650 --> 00:22:07,460 see here Tina's order of magnitude 468 00:22:05,870 --> 00:22:08,060 difference between the smallest and the 469 00:22:07,460 --> 00:22:09,120 largest 470 00:22:08,060 --> 00:22:13,620 that's hardly nards 471 00:22:09,120 --> 00:22:16,950 but Robbie goes on work a bit and he 472 00:22:13,620 --> 00:22:20,580 finds the solution about a thousand 473 00:22:16,950 --> 00:22:23,520 iterations and the value is 4.49 III 474 00:22:20,580 --> 00:22:27,780 something like that okay it wasn't too 475 00:22:23,520 --> 00:22:31,370 bad but if I now scale the problem with 476 00:22:27,780 --> 00:22:35,460 a bigger coefficient so up to a thousand 477 00:22:31,370 --> 00:22:38,070 well you see now the matrix coefficient 478 00:22:35,460 --> 00:22:41,160 range and it grew now it's 18 orders of 479 00:22:38,070 --> 00:22:44,870 magnitude you see again the warning and 480 00:22:41,160 --> 00:22:49,260 now we are still getting the same 481 00:22:44,870 --> 00:22:52,340 optimal value however and this is key it 482 00:22:49,260 --> 00:22:56,550 took longer in numbers of iterations and 483 00:22:52,340 --> 00:22:59,730 what if I push it even more well I can 484 00:22:56,550 --> 00:23:01,890 push it now to let's say a million 485 00:22:59,730 --> 00:23:04,290 I mean scaling coefficients for up to a 486 00:23:01,890 --> 00:23:08,040 million the matrix range is really crazy 487 00:23:04,290 --> 00:23:12,179 now Robbie will do sting and still is 488 00:23:08,040 --> 00:23:14,130 able to get the optimal value to longer 489 00:23:12,179 --> 00:23:17,390 but now you're getting these extra 490 00:23:14,130 --> 00:23:20,840 warning what is saying is saying that 491 00:23:17,390 --> 00:23:23,850 although he tried really hard to manage 492 00:23:20,840 --> 00:23:26,610 violations when he unscaled 493 00:23:23,850 --> 00:23:29,429 or when he finished the process of doing 494 00:23:26,610 --> 00:23:33,090 depreciation he couldn't find a solution 495 00:23:29,429 --> 00:23:35,790 with a small primer or do violations so 496 00:23:33,090 --> 00:23:38,010 in this case it's really extreme still 497 00:23:35,790 --> 00:23:40,080 got the objective function right took 498 00:23:38,010 --> 00:23:43,559 longer in number of iterations but 499 00:23:40,080 --> 00:23:45,540 couldn't find a nice solution with small 500 00:23:43,559 --> 00:23:48,059 violations and it's no surprise I mean 501 00:23:45,540 --> 00:23:51,900 you are scaling for up to a million what 502 00:23:48,059 --> 00:23:55,770 if I push it even further well if I go 503 00:23:51,900 --> 00:23:57,600 all the way to 100 millions or rescaling 504 00:23:55,770 --> 00:24:00,870 columns now the problems really break 505 00:23:57,600 --> 00:24:04,140 down so Rob is not able to solve them 506 00:24:00,870 --> 00:24:05,480 all the carry to be invisible and this 507 00:24:04,140 --> 00:24:09,559 has to do 508 00:24:05,480 --> 00:24:12,230 that decided Roby we trust the numbers 509 00:24:09,559 --> 00:24:15,080 that you were giving so in theory if the 510 00:24:12,230 --> 00:24:16,850 numbers were pure scaling columns 511 00:24:15,080 --> 00:24:19,549 shouldn't change anything 512 00:24:16,850 --> 00:24:22,490 however since numbers are storing this 513 00:24:19,549 --> 00:24:24,049 finite precision being when you rescale 514 00:24:22,490 --> 00:24:26,720 the problem you're really changing the 515 00:24:24,049 --> 00:24:30,020 definition of it and you are risking 516 00:24:26,720 --> 00:24:32,540 even getting unfeasible solutions and 517 00:24:30,020 --> 00:24:35,390 also please remember I mean as we were 518 00:24:32,540 --> 00:24:37,700 increasing the scaling range essentially 519 00:24:35,390 --> 00:24:39,740 making the problem harder to solve the 520 00:24:37,700 --> 00:24:42,440 first time scrubby was able to solve 521 00:24:39,740 --> 00:24:44,929 economy anyway but but the price that 522 00:24:42,440 --> 00:24:45,470 you pay for that it's extra computing 523 00:24:44,929 --> 00:24:48,710 time 524 00:24:45,470 --> 00:24:52,419 so again rescaled your problem may 525 00:24:48,710 --> 00:24:56,330 actually result in a faster optimization 526 00:24:52,419 --> 00:24:59,299 so what else can you do to handle these 527 00:24:56,330 --> 00:25:02,450 issues so one thing that you can do is 528 00:24:59,299 --> 00:25:04,309 actually do a combination of row and 529 00:25:02,450 --> 00:25:06,200 column scaling to get better 530 00:25:04,309 --> 00:25:07,790 coordination so let me explain this with 531 00:25:06,200 --> 00:25:11,720 an example so let's say that I have this 532 00:25:07,790 --> 00:25:16,370 problem I have 10 to the minus 7 X plus 533 00:25:11,720 --> 00:25:17,059 y and X + 3 Z less than or equal to 10 534 00:25:16,370 --> 00:25:21,320 to the 4 535 00:25:17,059 --> 00:25:24,650 what I can do is replace X with 10 to 536 00:25:21,320 --> 00:25:27,440 the 4 X Prime and then I get this 537 00:25:24,650 --> 00:25:31,280 constraint on top but later on what I 538 00:25:27,440 --> 00:25:34,910 can also do I mean I can simplify this 539 00:25:31,280 --> 00:25:40,400 constraint and instead of having 10 to 540 00:25:34,910 --> 00:25:42,700 the 4 X prime here and then to the 3 Z 541 00:25:40,400 --> 00:25:46,490 and then to before here I have this 542 00:25:42,700 --> 00:25:49,940 simpler expression so by combining I 543 00:25:46,490 --> 00:25:51,860 move from a world where I had 10 order 544 00:25:49,940 --> 00:25:56,200 of magnitudes of difference in the 545 00:25:51,860 --> 00:25:59,690 coefficient matrix to only 4 which is 546 00:25:56,200 --> 00:26:02,390 better but not only that I mean since 547 00:25:59,690 --> 00:26:05,210 the precision and the tolerance is 548 00:26:02,390 --> 00:26:07,490 absolute as I was saying before actually 549 00:26:05,210 --> 00:26:09,590 solving the system with this constraint 550 00:26:07,490 --> 00:26:12,950 with scale it down constraint it's 551 00:26:09,590 --> 00:26:15,169 easier than solving the same problem but 552 00:26:12,950 --> 00:26:18,409 with the scale up constraint and why is 553 00:26:15,169 --> 00:26:19,260 that it's because I will still be asking 554 00:26:18,409 --> 00:26:22,950 for a 10 to the 555 00:26:19,260 --> 00:26:25,260 - seeks tolerance in this case whereas 556 00:26:22,950 --> 00:26:27,240 if I ask for a 10 to the -6 tolerance in 557 00:26:25,260 --> 00:26:29,640 this constraint it will be in a sense 558 00:26:27,240 --> 00:26:33,390 easier for the solver to find such X 559 00:26:29,640 --> 00:26:36,510 Prime and Z don't lie to yourself so a 560 00:26:33,390 --> 00:26:39,030 common guideline that we provide is the 561 00:26:36,510 --> 00:26:41,970 range of coefficients in the rows and 562 00:26:39,030 --> 00:26:44,010 colors should be smaller or not to be so 563 00:26:41,970 --> 00:26:46,320 if you encounter this problem it's not 564 00:26:44,010 --> 00:26:49,080 too hard to say oh where I can hide that 565 00:26:46,320 --> 00:26:51,870 six orders of magnitude just having 566 00:26:49,080 --> 00:26:54,930 these equations now everything looks 567 00:26:51,870 --> 00:26:57,690 nice isn't it now instead of having 1 568 00:26:54,930 --> 00:27:00,420 and 10 to the 6 I only have 1 and 10 569 00:26:57,690 --> 00:27:02,910 however you didn't really change the 570 00:27:00,420 --> 00:27:05,520 problem actually the same kind of 571 00:27:02,910 --> 00:27:08,250 epsilon some feasible solutions still 572 00:27:05,520 --> 00:27:10,710 persist in the model and actually Prasad 573 00:27:08,250 --> 00:27:12,900 will detect this and we'll get back into 574 00:27:10,710 --> 00:27:15,150 this original stuff so you're not 575 00:27:12,900 --> 00:27:18,950 changing the problem by doing this trick 576 00:27:15,150 --> 00:27:25,470 what you couldn't do instead of that is 577 00:27:18,950 --> 00:27:28,350 do this so you can change Y by Y prime 578 00:27:25,470 --> 00:27:30,570 where they are related in this form and 579 00:27:28,350 --> 00:27:33,630 now what you will get is essentially if 580 00:27:30,570 --> 00:27:35,430 you push these three orders of 581 00:27:33,630 --> 00:27:37,950 magnitudes here in this coefficient and 582 00:27:35,430 --> 00:27:41,250 you increase now the range of the 583 00:27:37,950 --> 00:27:46,380 variable but the side gain from these is 584 00:27:41,250 --> 00:27:50,310 that now if Y prime is epsilon violative 585 00:27:46,380 --> 00:27:53,760 what you will get in X is really a very 586 00:27:50,310 --> 00:27:56,370 small possible violation of only 10 to 587 00:27:53,760 --> 00:27:58,620 the minus 3 which is really better so 588 00:27:56,370 --> 00:27:59,580 this is improving the numeric I'm the 589 00:27:58,620 --> 00:28:03,030 problem whereas the previous 590 00:27:59,580 --> 00:28:06,240 substitution is not really helping what 591 00:28:03,030 --> 00:28:08,910 about Big M constraints so in principle 592 00:28:06,240 --> 00:28:10,800 is you rescale the meaning of your 593 00:28:08,910 --> 00:28:13,080 constraint of your variables there 594 00:28:10,800 --> 00:28:14,460 shouldn't be any big amps I mean because 595 00:28:13,080 --> 00:28:16,740 the range of the variables will be 596 00:28:14,460 --> 00:28:20,520 willing 10 to the 4 and then everything 597 00:28:16,740 --> 00:28:23,070 will work fine however for some reason 598 00:28:20,520 --> 00:28:26,910 you cannot really get away from those 599 00:28:23,070 --> 00:28:28,020 big constants like here and you cannot 600 00:28:26,910 --> 00:28:31,110 change that 601 00:28:28,020 --> 00:28:32,610 with a by changing this scaling on the 602 00:28:31,110 --> 00:28:35,700 meaning of X 603 00:28:32,610 --> 00:28:38,880 what you can do that will improve 604 00:28:35,700 --> 00:28:41,970 numerical accuracy is to define these 605 00:28:38,880 --> 00:28:44,730 constraints as SOS constraints or or as 606 00:28:41,970 --> 00:28:47,150 a general implication constraints which 607 00:28:44,730 --> 00:28:50,340 was introduced in Robeson point five 608 00:28:47,150 --> 00:28:53,360 however although you will get more 609 00:28:50,340 --> 00:28:56,610 reliable solutions and actually this 610 00:28:53,360 --> 00:29:00,540 implication of Y being zero then X being 611 00:28:56,610 --> 00:29:02,820 0 will be always satisfied the price 612 00:29:00,540 --> 00:29:06,660 that you will be paying is that extra 613 00:29:02,820 --> 00:29:09,150 computing time so it will fix the 614 00:29:06,660 --> 00:29:11,010 problem of numerical stability but the 615 00:29:09,150 --> 00:29:13,530 price you will be paying is more 616 00:29:11,010 --> 00:29:16,410 computing time so don't forget that 617 00:29:13,530 --> 00:29:19,470 so in summary kind of the quick 618 00:29:16,410 --> 00:29:23,130 takeaways variables shouldn't be in 619 00:29:19,470 --> 00:29:25,440 general an arranged outside minus 10 to 620 00:29:23,130 --> 00:29:28,440 the 4 and 10 to the 4 also the same 621 00:29:25,440 --> 00:29:30,270 applies for the constraints the active 622 00:29:28,440 --> 00:29:32,730 range of the constraints meaning where 623 00:29:30,270 --> 00:29:35,370 the solutions really move and also the 624 00:29:32,730 --> 00:29:37,169 same apply for the good solutions 625 00:29:35,370 --> 00:29:40,830 it could solution shouldn't be too far 626 00:29:37,169 --> 00:29:44,370 away from that range either also you 627 00:29:40,830 --> 00:29:46,919 should not expect or try to compare 628 00:29:44,370 --> 00:29:49,500 quantities that have two different 629 00:29:46,919 --> 00:29:54,000 absolute values I mean if you try to 630 00:29:49,500 --> 00:29:56,490 compare 10 to the minus 10 week one and 631 00:29:54,000 --> 00:29:58,860 those quantities were computed true 632 00:29:56,490 --> 00:30:03,059 linear algebra you're really asking for 633 00:29:58,860 --> 00:30:05,669 trouble avoid those things and actually 634 00:30:03,059 --> 00:30:07,770 if you change these in your current 635 00:30:05,669 --> 00:30:09,900 models and you might get surprised 636 00:30:07,770 --> 00:30:12,090 because it may give you a boost in 637 00:30:09,900 --> 00:30:15,840 performance and and why is that well 638 00:30:12,090 --> 00:30:19,230 because grow B will try really hard to 639 00:30:15,840 --> 00:30:22,320 whenever we find numerical issues to 640 00:30:19,230 --> 00:30:25,950 solve it by ourselves either by going 641 00:30:22,320 --> 00:30:29,370 even to extra precision or by changing 642 00:30:25,950 --> 00:30:31,620 the algorithm internally all bar turning 643 00:30:29,370 --> 00:30:33,900 off some parts of the code that in 644 00:30:31,620 --> 00:30:37,919 general help to solve it quicker and so 645 00:30:33,900 --> 00:30:40,440 on and so forth so really selecting the 646 00:30:37,919 --> 00:30:42,929 right ranges for everything may be 647 00:30:40,440 --> 00:30:45,059 really key to even improve the 648 00:30:42,929 --> 00:30:46,490 performance of your core amount so give 649 00:30:45,059 --> 00:30:51,389 me the try 650 00:30:46,490 --> 00:30:54,029 so how do you check if your given model 651 00:30:51,389 --> 00:30:57,210 has some wreckage which is an important 652 00:30:54,029 --> 00:31:00,240 question how we go about that so 653 00:30:57,210 --> 00:31:03,059 probably the first thing to do is try to 654 00:31:00,240 --> 00:31:05,490 separate whatever you're doing angel 655 00:31:03,059 --> 00:31:07,830 code from whatever solver is doing and 656 00:31:05,490 --> 00:31:11,460 an easy way to do that is to use this 657 00:31:07,830 --> 00:31:13,440 parameter guru up our dump equal one if 658 00:31:11,460 --> 00:31:16,110 you do that just before calling optimize 659 00:31:13,440 --> 00:31:19,860 then optimize will generate two files 660 00:31:16,110 --> 00:31:23,009 one is called guru B dot R EU and the 661 00:31:19,860 --> 00:31:25,649 other is guru B dot PRM the first one is 662 00:31:23,009 --> 00:31:28,559 an anonymized version of your model and 663 00:31:25,649 --> 00:31:31,830 the second one is the set of all the 664 00:31:28,559 --> 00:31:33,629 options non default values that you're 665 00:31:31,830 --> 00:31:35,210 using just when you enter the 666 00:31:33,629 --> 00:31:38,490 optimization column this is key because 667 00:31:35,210 --> 00:31:42,210 essentially those parameters will manage 668 00:31:38,490 --> 00:31:44,879 and affect which algorithms are used and 669 00:31:42,210 --> 00:31:46,470 how they're used so to really be able to 670 00:31:44,879 --> 00:31:49,440 reproduce whatever you're seeing you 671 00:31:46,470 --> 00:31:52,590 need these two files so after getting 672 00:31:49,440 --> 00:31:56,159 those what you can do it let's say try 673 00:31:52,590 --> 00:31:58,799 to read the model back in Python and 674 00:31:56,159 --> 00:32:00,720 inspect so you can do that easily by 675 00:31:58,799 --> 00:32:03,840 printing the stats and that will give 676 00:32:00,720 --> 00:32:06,149 you the matrix range the objective 677 00:32:03,840 --> 00:32:09,120 coefficient range and everything and 678 00:32:06,149 --> 00:32:11,190 that will let you see if the ranges of 679 00:32:09,120 --> 00:32:16,279 these coefficients are within reason or 680 00:32:11,190 --> 00:32:19,350 not also if everything looks fine there 681 00:32:16,279 --> 00:32:21,899 you should read them into the model the 682 00:32:19,350 --> 00:32:23,549 parameters that you were using and after 683 00:32:21,899 --> 00:32:26,220 reading those parameters you can call 684 00:32:23,549 --> 00:32:31,230 optimize and you definitely want to look 685 00:32:26,220 --> 00:32:33,110 at the outer and the grubby outputs are 686 00:32:31,230 --> 00:32:36,450 designed to provide as much information 687 00:32:33,110 --> 00:32:38,340 as possible we will output in different 688 00:32:36,450 --> 00:32:40,470 situations many warning messages the 689 00:32:38,340 --> 00:32:43,500 most common one is these warning that 690 00:32:40,470 --> 00:32:46,740 the matrix provisions looks bad or the 691 00:32:43,500 --> 00:32:49,980 objective coefficients or whatever also 692 00:32:46,740 --> 00:32:52,830 you see this warning mark with Torrance 693 00:32:49,980 --> 00:32:55,320 tied into some given value that's really 694 00:32:52,830 --> 00:32:59,000 signalling that simply is struggling to 695 00:32:55,320 --> 00:33:00,120 find stable solutions and it's usually 696 00:32:59,000 --> 00:33:02,310 indication 697 00:33:00,120 --> 00:33:04,800 model indeed is having to medical issues 698 00:33:02,310 --> 00:33:07,560 for sure if you see one and switching to 699 00:33:04,800 --> 00:33:09,900 quad precision it means that Grobe not 700 00:33:07,560 --> 00:33:13,260 only couldn't manage by changing the 701 00:33:09,900 --> 00:33:15,150 algorithm but had to try on working on 702 00:33:13,260 --> 00:33:16,580 extended floating-point precision it 703 00:33:15,150 --> 00:33:19,380 talked a little bit about that later on 704 00:33:16,580 --> 00:33:21,330 and of course you see numerical error or 705 00:33:19,380 --> 00:33:24,450 numerical trouble encounter if you see 706 00:33:21,330 --> 00:33:27,270 restart crossover that means that now 707 00:33:24,450 --> 00:33:29,160 the barrier cruiser is having issues 708 00:33:27,270 --> 00:33:31,230 with your model and suboptimal 709 00:33:29,160 --> 00:33:33,240 termination is the same warning 710 00:33:31,230 --> 00:33:35,040 variables drop from the basis is use 711 00:33:33,240 --> 00:33:36,890 another warning saying that this simply 712 00:33:35,040 --> 00:33:40,200 is struggling with your problem 713 00:33:36,890 --> 00:33:44,370 landscape primal-dual violation all that 714 00:33:40,200 --> 00:33:46,470 comes again from working in your 715 00:33:44,370 --> 00:33:49,260 original model it means that Grubb is 716 00:33:46,470 --> 00:33:52,500 not able to satisfy the tolerances that 717 00:33:49,260 --> 00:33:55,020 you asked for for this given problem so 718 00:33:52,500 --> 00:33:58,050 all those are signal strong signals that 719 00:33:55,020 --> 00:33:59,870 you should think about looking in detail 720 00:33:58,050 --> 00:34:02,670 to what's going on 721 00:33:59,870 --> 00:34:05,910 also you look into the quality of the 722 00:34:02,670 --> 00:34:09,060 solutions so you can always preen 723 00:34:05,910 --> 00:34:10,560 what's the worst bound being violated 724 00:34:09,060 --> 00:34:12,150 what's the worst constraint being 725 00:34:10,560 --> 00:34:14,100 violated and what's the word integrity 726 00:34:12,150 --> 00:34:17,910 condition being violated so in this 727 00:34:14,100 --> 00:34:20,430 particular example bounds are very okay 728 00:34:17,910 --> 00:34:23,160 ten to the minus eight that's pretty 729 00:34:20,430 --> 00:34:25,350 good however in this problem we are 730 00:34:23,160 --> 00:34:28,140 seeing a constraint violation of 10 to 731 00:34:25,350 --> 00:34:30,390 the minus 4 and actually almost 10 to 732 00:34:28,140 --> 00:34:32,700 the minus 3 that's really bad however 733 00:34:30,390 --> 00:34:36,720 the integral it is I looks fine so these 734 00:34:32,700 --> 00:34:39,030 will be signalling that my solution is 735 00:34:36,720 --> 00:34:41,220 not really conforming and in and that 736 00:34:39,030 --> 00:34:44,090 probably Robin is struggling to find a 737 00:34:41,220 --> 00:34:46,410 solution confirming recurrence 738 00:34:44,090 --> 00:34:49,020 you can also inspect the condition 739 00:34:46,410 --> 00:34:51,540 number and I'm talking about that in a 740 00:34:49,020 --> 00:34:54,930 little bit essentially the condition 741 00:34:51,540 --> 00:34:59,220 number is also an indication actually if 742 00:34:54,930 --> 00:35:01,230 it's really be that the solver may be 743 00:34:59,220 --> 00:35:05,850 encountering numerical issues in your 744 00:35:01,230 --> 00:35:08,790 model other symptoms so whenever you see 745 00:35:05,850 --> 00:35:11,160 that I changed some parameter and now my 746 00:35:08,790 --> 00:35:13,810 solution changes the problem goes from 747 00:35:11,160 --> 00:35:17,400 infeasible to feasible or the optimal 748 00:35:13,810 --> 00:35:20,140 the value changes all these are strong 749 00:35:17,400 --> 00:35:24,480 symptoms showing that your problem has 750 00:35:20,140 --> 00:35:28,480 numerical issues now most of these 751 00:35:24,480 --> 00:35:31,210 problems are usually just America 752 00:35:28,480 --> 00:35:33,430 meaning that they are not truly boxed in 753 00:35:31,210 --> 00:35:35,980 Krabi would have a few but it's really 754 00:35:33,430 --> 00:35:39,370 rare most of the time please believe me 755 00:35:35,980 --> 00:35:42,940 come from numerical problems reading the 756 00:35:39,370 --> 00:35:45,670 model one way to see if that's the case 757 00:35:42,940 --> 00:35:48,280 is for example trying to solve your 758 00:35:45,670 --> 00:35:50,230 model with tightening Torrance's let's 759 00:35:48,280 --> 00:35:53,080 instead of asking 10 to the minus 6 you 760 00:35:50,230 --> 00:35:56,170 may try 10 to the minus 8 it now the 761 00:35:53,080 --> 00:35:58,750 solver becomes stable for the solutions 762 00:35:56,170 --> 00:36:02,200 or with the answer that's a strong 763 00:35:58,750 --> 00:36:05,680 indication that your problem is lacking 764 00:36:02,200 --> 00:36:08,800 in good scaling and beware although 765 00:36:05,680 --> 00:36:11,680 tidal torrents may solve the problem in 766 00:36:08,800 --> 00:36:14,980 terms of stability again you will be 767 00:36:11,680 --> 00:36:16,990 paying by extra numerical computation so 768 00:36:14,980 --> 00:36:21,690 you are paying the price on running time 769 00:36:16,990 --> 00:36:24,550 don't forget that now if you already 770 00:36:21,690 --> 00:36:26,890 improve everything the scales look nice 771 00:36:24,550 --> 00:36:29,140 and you're still getting strange answers 772 00:36:26,890 --> 00:36:34,150 well I honestly paid and call us I mean 773 00:36:29,140 --> 00:36:36,010 that's what we are for so that was about 774 00:36:34,150 --> 00:36:38,500 detecting or assessing it 775 00:36:36,010 --> 00:36:41,050 do you have numerical issues so what can 776 00:36:38,500 --> 00:36:43,600 you do once you see that you have 777 00:36:41,050 --> 00:36:45,790 numerical issues and there are some 778 00:36:43,600 --> 00:36:49,120 parameters within Robbie that help 779 00:36:45,790 --> 00:36:52,270 manage issues but again this will not 780 00:36:49,120 --> 00:36:54,850 solve them it will just try to walk 781 00:36:52,270 --> 00:36:57,670 around them and that's as I said many 782 00:36:54,850 --> 00:37:00,610 times now you will be paying in running 783 00:36:57,670 --> 00:37:04,780 time so let's go with this resolved 784 00:37:00,610 --> 00:37:07,690 resolved can sometimes introduce some 785 00:37:04,780 --> 00:37:11,860 issues one way to look at it is your if 786 00:37:07,690 --> 00:37:15,130 your problem you now load the pre-salt 787 00:37:11,860 --> 00:37:18,730 version and you can print the stats of 788 00:37:15,130 --> 00:37:21,580 this Persol model if that start shows 789 00:37:18,730 --> 00:37:24,670 that now your coefficients are looking 790 00:37:21,580 --> 00:37:26,380 really strange then maybe resolve is the 791 00:37:24,670 --> 00:37:28,020 culprit which is doesn't happen that 792 00:37:26,380 --> 00:37:30,510 often but it may 793 00:37:28,020 --> 00:37:32,610 if that's the case you can try to read 794 00:37:30,510 --> 00:37:34,830 off all together with resolve equals 795 00:37:32,610 --> 00:37:36,930 zero or you can say okay 796 00:37:34,830 --> 00:37:39,000 aggregation shouldn't be used 797 00:37:36,930 --> 00:37:41,310 aggregation essentially what we'll do is 798 00:37:39,000 --> 00:37:43,860 remember that bad example of don't lie 799 00:37:41,310 --> 00:37:46,400 to yourself of hiding ten to the six 800 00:37:43,860 --> 00:37:49,320 with six constraints what aggregate will 801 00:37:46,400 --> 00:37:52,740 essentially reweave those stuff and 802 00:37:49,320 --> 00:37:54,900 we'll show implicit B coefficients back 803 00:37:52,740 --> 00:37:55,860 into the models so you disable it 804 00:37:54,900 --> 00:37:58,590 altogether 805 00:37:55,860 --> 00:38:00,960 that won't show up in the constraint but 806 00:37:58,590 --> 00:38:02,580 remember you still have those big 807 00:38:00,960 --> 00:38:06,420 numbers whenever you're solving will be 808 00:38:02,580 --> 00:38:08,490 anyway so be careful all you can also 809 00:38:06,420 --> 00:38:11,280 disable the aggregation of Finland 810 00:38:08,490 --> 00:38:13,880 essentially Grobe will try to strengthen 811 00:38:11,280 --> 00:38:17,760 constraints and sometimes while doing so 812 00:38:13,880 --> 00:38:20,520 may introduce kind of ugly coefficients 813 00:38:17,760 --> 00:38:22,680 if that's the case then you can try by 814 00:38:20,520 --> 00:38:26,280 disabling this piece and see if that 815 00:38:22,680 --> 00:38:29,040 helps performance also think about using 816 00:38:26,280 --> 00:38:31,620 the right in general barrier is the 817 00:38:29,040 --> 00:38:33,360 fastest in general not always however 818 00:38:31,620 --> 00:38:35,580 unfortunately 819 00:38:33,360 --> 00:38:38,250 barri it's also more sensible to 820 00:38:35,580 --> 00:38:41,370 numerical issues so you should try 821 00:38:38,250 --> 00:38:44,750 either primal dual simplex and bar by 822 00:38:41,370 --> 00:38:48,420 itself and see if that helps to get more 823 00:38:44,750 --> 00:38:50,130 robust solutions for your model so it's 824 00:38:48,420 --> 00:38:53,820 always worth a try 825 00:38:50,130 --> 00:38:55,800 selecting the root node method you can 826 00:38:53,820 --> 00:38:57,620 also try to make the algorithms less 827 00:38:55,800 --> 00:39:02,460 sensitive to medical issues 828 00:38:57,620 --> 00:39:05,130 the easiest way probably is to use the 829 00:39:02,460 --> 00:39:07,590 parameter called numeric focus by 830 00:39:05,130 --> 00:39:09,420 default is set to minus one which means 831 00:39:07,590 --> 00:39:12,240 we Robbie will handle it internally and 832 00:39:09,420 --> 00:39:14,880 decide what to do you can disable it 833 00:39:12,240 --> 00:39:17,040 altogether setting it to zero and you 834 00:39:14,880 --> 00:39:19,590 can increase it all the way to one two 835 00:39:17,040 --> 00:39:21,870 and three what Robbie will do depending 836 00:39:19,590 --> 00:39:23,880 on these levels is to be more 837 00:39:21,870 --> 00:39:26,130 conservative on the Markowitz torrents 838 00:39:23,880 --> 00:39:28,170 which is handling how this impacts 839 00:39:26,130 --> 00:39:30,650 people's first like there he will also 840 00:39:28,170 --> 00:39:33,720 select when to switch to quad precision 841 00:39:30,650 --> 00:39:36,120 it will also limit gamma ray passes 842 00:39:33,720 --> 00:39:37,980 which in some cases especially for an 843 00:39:36,120 --> 00:39:40,680 American challenging problems may 844 00:39:37,980 --> 00:39:41,690 introduce issues he married this the 845 00:39:40,680 --> 00:39:44,930 number of 846 00:39:41,690 --> 00:39:48,049 passes that we do on aggregation and so 847 00:39:44,930 --> 00:39:49,940 on so this is a common thing for people 848 00:39:48,049 --> 00:39:51,859 to try when they're encountering 849 00:39:49,940 --> 00:39:55,039 numerical issues to change the Murray 850 00:39:51,859 --> 00:39:58,130 focus to either one or even three you 851 00:39:55,039 --> 00:40:00,950 can also try to change the way simplex 852 00:39:58,130 --> 00:40:05,059 do its pricing by selecting different 853 00:40:00,950 --> 00:40:07,190 norm adjust values if barrier is having 854 00:40:05,059 --> 00:40:10,280 trouble you may try to use borrow 855 00:40:07,190 --> 00:40:12,260 manginas parameter if also you can try a 856 00:40:10,280 --> 00:40:15,020 different strategy for the second across 857 00:40:12,260 --> 00:40:18,950 over babys and you can also try to 858 00:40:15,020 --> 00:40:20,720 disable cuts altogether sometimes again 859 00:40:18,950 --> 00:40:23,869 for numerically challenging problems 860 00:40:20,720 --> 00:40:27,500 they may introduce more numerical issues 861 00:40:23,869 --> 00:40:30,079 into your problem again you have to 862 00:40:27,500 --> 00:40:32,780 remember all these is not really fixed 863 00:40:30,079 --> 00:40:36,710 for the numerical issues this is a work 864 00:40:32,780 --> 00:40:39,410 around them the real way to fix them is 865 00:40:36,710 --> 00:40:44,839 really going back into the model and 866 00:40:39,410 --> 00:40:46,940 trying to change how you pose it so this 867 00:40:44,839 --> 00:40:49,069 is the last piece I have for this 868 00:40:46,940 --> 00:40:52,819 presentation and it has to do with the 869 00:40:49,069 --> 00:40:55,730 condition number so up to now what I say 870 00:40:52,819 --> 00:40:58,430 these either problems come from rounding 871 00:40:55,730 --> 00:41:02,059 input I mean not giving enough detail or 872 00:40:58,430 --> 00:41:05,089 because essentially the limits of the 873 00:41:02,059 --> 00:41:07,760 computer numbers imply that not 874 00:41:05,089 --> 00:41:10,700 everything can be done in exact or 875 00:41:07,760 --> 00:41:13,190 Eataly but not all the problems really 876 00:41:10,700 --> 00:41:16,220 come from that some problems really come 877 00:41:13,190 --> 00:41:17,569 from the geometry and that has to do 878 00:41:16,220 --> 00:41:21,950 with the condition number what I was 879 00:41:17,569 --> 00:41:24,770 showing before as Kappa so the usual 880 00:41:21,950 --> 00:41:28,700 definition is while the condition number 881 00:41:24,770 --> 00:41:31,480 of an optimization problem is trying to 882 00:41:28,700 --> 00:41:34,309 capture the idea of how sensitive your 883 00:41:31,480 --> 00:41:38,180 optimization problem is or the optimal 884 00:41:34,309 --> 00:41:41,779 solution of it is to small changes in 885 00:41:38,180 --> 00:41:45,200 different parts of the input a common 886 00:41:41,779 --> 00:41:47,750 thing that is used as a reference for 887 00:41:45,200 --> 00:41:50,240 this number for linear system at least 888 00:41:47,750 --> 00:41:52,339 is known as Cavafy which is defined as 889 00:41:50,240 --> 00:41:54,560 the norm of the matrix time the norm of 890 00:41:52,339 --> 00:41:57,140 the inverse of the matrix but I 891 00:41:54,560 --> 00:41:59,840 prefer really the geometry 892 00:41:57,140 --> 00:42:03,010 interpretation which has to do with how 893 00:41:59,840 --> 00:42:05,870 rounded the feasible region is and 894 00:42:03,010 --> 00:42:09,560 provided sets this is really simple 895 00:42:05,870 --> 00:42:13,010 essentially what you can try to imagine 896 00:42:09,560 --> 00:42:15,890 is what's the ratio between the smallest 897 00:42:13,010 --> 00:42:19,400 ball containing the feasible solution 898 00:42:15,890 --> 00:42:22,580 and the largest ball contained in the 899 00:42:19,400 --> 00:42:25,700 feasible solution so let me give you two 900 00:42:22,580 --> 00:42:28,970 examples so you may think okay I can 901 00:42:25,700 --> 00:42:31,430 work in optimizing something like behind 902 00:42:28,970 --> 00:42:33,860 with many many many many linear 903 00:42:31,430 --> 00:42:36,530 inequalities at the border so in this 904 00:42:33,860 --> 00:42:38,590 situation the geometry of the condition 905 00:42:36,530 --> 00:42:42,110 number will be the ratio between that 906 00:42:38,590 --> 00:42:44,630 slightly gray circle outside and the 907 00:42:42,110 --> 00:42:47,810 darker gray circle inside and that will 908 00:42:44,630 --> 00:42:50,150 be fairly close to one and what the 909 00:42:47,810 --> 00:42:53,570 condition number says is that well if 910 00:42:50,150 --> 00:42:56,900 that ratio is small then small changes 911 00:42:53,570 --> 00:42:59,870 in the input data should produce small 912 00:42:56,900 --> 00:43:01,640 changes in the outer solution and indeed 913 00:42:59,870 --> 00:43:04,760 that's the case so I have a script here 914 00:43:01,640 --> 00:43:07,340 you'll get it later on that does exactly 915 00:43:04,760 --> 00:43:09,790 that I define a certain weight I don't 916 00:43:07,340 --> 00:43:12,200 know in this case a thousand 917 00:43:09,790 --> 00:43:15,050 inequalities to find the circle and then 918 00:43:12,200 --> 00:43:17,330 I move a little bit here and there some 919 00:43:15,050 --> 00:43:19,550 coefficients and reoptimize and see how 920 00:43:17,330 --> 00:43:24,110 much the primary smidgen changes and if 921 00:43:19,550 --> 00:43:27,770 I do that I get these so out of 500 922 00:43:24,110 --> 00:43:29,750 iterations changing small coefficients 923 00:43:27,770 --> 00:43:32,630 on the order of ten to the minus six 924 00:43:29,750 --> 00:43:35,180 only change the primary solution about 925 00:43:32,630 --> 00:43:37,790 in the same order of magnitude one point 926 00:43:35,180 --> 00:43:42,470 seven ten to the minus six so this 927 00:43:37,790 --> 00:43:47,240 really chose that small changes as the 928 00:43:42,470 --> 00:43:49,610 geometry idea was wind resolving small 929 00:43:47,240 --> 00:43:51,980 changes in the optimal solution which 930 00:43:49,610 --> 00:43:54,920 makes sense so these problems are easy 931 00:43:51,980 --> 00:43:58,610 to solve but what if I try to optimize 932 00:43:54,920 --> 00:44:00,770 over a very thin triangle like here so 933 00:43:58,610 --> 00:44:03,830 if you imagine if I stretch this 934 00:44:00,770 --> 00:44:05,390 triangle more and more the circle thing 935 00:44:03,830 --> 00:44:08,450 we delete will get smaller and smaller 936 00:44:05,390 --> 00:44:11,390 whereas a circuit that contains it 937 00:44:08,450 --> 00:44:14,780 we'll grow more and more and the ratio 938 00:44:11,390 --> 00:44:17,000 will get to be pretty bad and what I can 939 00:44:14,780 --> 00:44:19,040 do is essentially define this 940 00:44:17,000 --> 00:44:22,550 optimization problem tree constrain 941 00:44:19,040 --> 00:44:25,430 solve it and then see what happen if I 942 00:44:22,550 --> 00:44:29,599 change a little bit a little tiny bit 943 00:44:25,430 --> 00:44:32,660 the objective function if I do that 944 00:44:29,599 --> 00:44:34,400 right away I mean even changes in the 945 00:44:32,660 --> 00:44:37,460 objective functions in the order of 10 946 00:44:34,400 --> 00:44:39,859 to the minus 6 translate themselves into 947 00:44:37,460 --> 00:44:42,230 changes in the primal solution of 16 948 00:44:39,859 --> 00:44:43,609 order of magnitude and this is not 949 00:44:42,230 --> 00:44:45,530 because it's always getting the wrong 950 00:44:43,609 --> 00:44:49,550 answer actually these three numbers are 951 00:44:45,530 --> 00:44:52,040 showing how big the primal-dual and all 952 00:44:49,550 --> 00:44:54,380 the other bound violations are which is 953 00:44:52,040 --> 00:44:56,180 essentially zero so it's always getting 954 00:44:54,380 --> 00:44:59,839 the right solution for this horrible 955 00:44:56,180 --> 00:45:02,030 problem the problem is that if you 956 00:44:59,839 --> 00:45:06,290 change the tiny major objective function 957 00:45:02,030 --> 00:45:09,380 you really may get very different 958 00:45:06,290 --> 00:45:12,530 solutions from the primal side and this 959 00:45:09,380 --> 00:45:15,200 is not to do with numerical issues is 960 00:45:12,530 --> 00:45:17,020 really to do with the geometry which 961 00:45:15,200 --> 00:45:19,910 again take us back to scaling 962 00:45:17,020 --> 00:45:22,190 essentially if you have a good scaling a 963 00:45:19,910 --> 00:45:24,950 problem that looks more round or square 964 00:45:22,190 --> 00:45:27,770 if you wish then everything should 965 00:45:24,950 --> 00:45:31,160 behave much better whereas if you have 966 00:45:27,770 --> 00:45:34,310 very thin areas you are exposing 967 00:45:31,160 --> 00:45:39,859 yourself to have being very unstable 968 00:45:34,310 --> 00:45:43,819 solutions so remember that ok recap to 969 00:45:39,859 --> 00:45:46,040 end up essentially in stability it's not 970 00:45:43,819 --> 00:45:49,069 always coming from the numbers it also 971 00:45:46,040 --> 00:45:51,109 comes from the geometry Robbie will 972 00:45:49,069 --> 00:45:53,960 never second-guess your input will 973 00:45:51,109 --> 00:45:57,530 really trust it and the best way to 974 00:45:53,960 --> 00:45:59,720 handle these numerical issues is most of 975 00:45:57,530 --> 00:46:02,390 the time think hard and rescale 976 00:45:59,720 --> 00:46:05,349 everything and maybe even change the way 977 00:46:02,390 --> 00:46:05,349 you formulate the problem 978 00:46:05,400 --> 00:46:10,340 thank for your attention your time