1 00:00:00,720 --> 00:00:01,590 ‫Welcome back. 2 00:00:02,220 --> 00:00:06,120 ‫So let's now review how we apply constraints to our car. 3 00:00:06,630 --> 00:00:09,900 ‫First of all, we took a very easy case. 4 00:00:10,530 --> 00:00:19,980 ‫We imagined that our horizon period equal to and that our cost function j only depended on Delta Delta 5 00:00:19,980 --> 00:00:22,050 ‫zero and Delta Delta One. 6 00:00:22,620 --> 00:00:26,820 ‫So J was a function of Delta Delta Zero and Delta Delta One. 7 00:00:27,390 --> 00:00:29,760 ‫We had a two dimensional domain. 8 00:00:30,390 --> 00:00:33,510 ‫In other words, two independent variables. 9 00:00:34,230 --> 00:00:36,690 ‫Delta Delta zero and Delta Delta One. 10 00:00:37,290 --> 00:00:41,580 ‫And then the third dimension was for the cost function J. 11 00:00:42,180 --> 00:00:49,650 ‫Since our cost function was quadratic, just like in this course, we had a simple 3D parabola there. 12 00:00:50,250 --> 00:00:56,250 ‫We depicted that 3D parabola on a 2D plane with contour lines. 13 00:00:56,940 --> 00:01:04,110 ‫If we forget about constraints, then the minimum point of the cost function would be the lowest points 14 00:01:04,620 --> 00:01:07,770 ‫of that parabola or on a 2D plane. 15 00:01:08,310 --> 00:01:10,620 ‫It would be the center of the circle. 16 00:01:11,370 --> 00:01:20,160 ‫And then you would find the corresponding Delta Delta zero and Delta Delta One for that minimum point. 17 00:01:20,760 --> 00:01:30,210 ‫That would be the solution of your unconstrained NPC, which in that particular case would be zero and 18 00:01:30,210 --> 00:01:30,630 ‫zero. 19 00:01:31,230 --> 00:01:42,630 ‫So Delta You Global equals zero and zero because Delta Delta zero equals zero and Delta Delta One equals 20 00:01:43,170 --> 00:01:43,650 ‫zero. 21 00:01:44,280 --> 00:01:49,440 ‫So that was unconstrained optimization or unconstrained minimization. 22 00:01:50,010 --> 00:01:55,170 ‫But then we said that, OK, what if we impose certain conditions? 23 00:01:55,800 --> 00:02:03,060 ‫What if we say that Delta Delta Zero must be greater than or equal to zero point zero five? 24 00:02:03,060 --> 00:02:10,140 ‫Radiance and Delta Delta One must be greater than, or equal to 0.07 radiance. 25 00:02:10,830 --> 00:02:15,510 ‫Our goal was still to minimize our cost function, J. 26 00:02:16,140 --> 00:02:21,630 ‫But we had to find the smallest j possible that would respect these conditions as well. 27 00:02:22,230 --> 00:02:31,350 ‫And so we divided this two dimensional domain into several areas based on those constraints. 28 00:02:31,950 --> 00:02:38,400 ‫For example, the orange area here, it satisfied the first constraint here. 29 00:02:38,400 --> 00:02:45,360 ‫Delta Delta zero is greater than 0.05 radiance, but it violated the second constraint. 30 00:02:45,990 --> 00:02:48,030 ‫It's below Delta. 31 00:02:48,030 --> 00:02:51,930 ‫Delta One equals zero point zero seven radiance. 32 00:02:52,590 --> 00:02:55,320 ‫The purple area was vice versa. 33 00:02:55,920 --> 00:03:00,210 ‫The second condition was OK, but the first condition was not OK. 34 00:03:00,810 --> 00:03:02,120 ‫Then the red area. 35 00:03:02,790 --> 00:03:10,860 ‫It violated both constraints and only the green area was feasible because it respected both constraints. 36 00:03:11,520 --> 00:03:19,950 ‫And so this original G men that we found earlier, it was no longer valid because it would be in the 37 00:03:19,950 --> 00:03:22,920 ‫region that violated both constraints. 38 00:03:23,520 --> 00:03:32,190 ‫We had to find a new J Min, a new smallest value for the cost function that would be somewhere in the 39 00:03:32,190 --> 00:03:33,390 ‫green region here. 40 00:03:33,960 --> 00:03:39,630 ‫We found that in the green area, J was smallest in this corner here. 41 00:03:40,200 --> 00:03:42,070 ‫We called that point J. 42 00:03:42,090 --> 00:03:42,450 ‫Min. 43 00:03:42,870 --> 00:03:46,320 ‫Superscript C C stands for constraints. 44 00:03:46,980 --> 00:03:55,710 ‫Essentially, if you're dealing with a perfect 3D parabola, then has a circular contour lines from 45 00:03:55,710 --> 00:04:03,750 ‫the top view, then a good way to know that this is the point here that gives you the smallest value 46 00:04:03,750 --> 00:04:04,380 ‫for your cost. 47 00:04:04,380 --> 00:04:12,610 ‫Function in the green region is by going to the place where you have your S'mores J without the constraints. 48 00:04:13,260 --> 00:04:19,140 ‫And then you draw a straight line from that point till the green region. 49 00:04:19,770 --> 00:04:23,670 ‫And since in this corner, this line is the shortest. 50 00:04:24,360 --> 00:04:32,700 ‫That's why we concluded that that's the place where your cost function took the smallest value in the 51 00:04:32,700 --> 00:04:33,600 ‫green region. 52 00:04:34,230 --> 00:04:42,240 ‫Obviously, your G means superscript C is greater than your G men when you don't take your constraints 53 00:04:42,240 --> 00:04:43,050 ‫into account. 54 00:04:43,620 --> 00:04:50,490 ‫So in general, J means C is greater than or equal to G men. 55 00:04:51,150 --> 00:04:53,560 ‫And in this case, it's definitely greater. 56 00:04:54,180 --> 00:04:59,760 ‫But at least the solution that you have now, which is this. 57 00:04:59,890 --> 00:05:00,580 ‫On here. 58 00:05:01,270 --> 00:05:08,260 ‫So your Delta Delta zero is zero point zero five in Delta, Delta One is zero point zero seven. 59 00:05:08,950 --> 00:05:16,150 ‫This solution minimizes your cost function and honors your constraints at the same time. 60 00:05:16,720 --> 00:05:20,830 ‫So that's the main idea behind constrained optimization. 61 00:05:21,430 --> 00:05:28,300 ‫We wanted to minimize our cost function, but at the same time we had additional conditions, we had 62 00:05:28,300 --> 00:05:32,770 ‫constraints and we had to make sure that we respect those constraints.