QyYtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJLkxpbmVhckFsZ2VicmFHRiUhIiItRiQ2I0kkTVRNR0YnRis=QyQ+SSJBRzYiLUknTWF0cml4R0YlNiMvSSQlaWRHRiUiKyV5Kik0SyUiIiI=QyQ+SSJiRzYiLUknVmVjdG9yRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiM3KCEiJiEiKiEjNSIiIUYwRjAiIiI=QyQ+SSJjRzYiLUknVmVjdG9yRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiM3JSIiJiIiJCIiIyIiIg==We start with the roof B={4,5,6} and compute the vertex (0,0,0) of the roof. The first inequality is violated by this vertex. We denote the sub-matrix consisting of the rows in B by ABQyQ+SSNBQkc2Ii1JKlN1Yk1hdHJpeEdGJTYlSSJBR0YlNyUiIiUiIiYiIic3JSIiIiIiIyIiJEYvWe have x* = (5,3,2,0) and y*=(-2,-3,-4,1). The index set J is {1,2,3} and the minimum of the formula 3.15 is attined at index 6 which means that the new roof is B = {1,4,5} We compute the new vertex first. (START SECOND ITERATION) QyY+SSJCRzYiNyUiIiIiIiUiIiZGJz5JI0FCR0YlLUkqU3ViTWF0cml4R0YlNiVJIkFHRiVGJDclRiciIiMiIiRGJw==QyQ+SSNiQkc2Ii1JKlN1YlZlY3RvckdGJTYkSSJiR0YlSSJCR0YlIiIiQyQ+SSJ2RzYiLSZJLExpbmVhclNvbHZlR0YlNiNJIlFHRiU2JEkjQUJHRiVJI2JCR0YlIiIiDoes the new vertex violate an inequality? LUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJUkoX3N5c2xpYkdGKDYlSSJBR0YoSSJ2R0YoSSV0cnVlR0YlThe second inequality is violated. We bring index 2 into the roof. And compute x* and y*: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=QyQ+SSJ4RzYiLSZJLExpbmVhclNvbHZlR0YlNiNJIlFHRiU2JC1JKnRyYW5zcG9zZUdGJTYjSSNBQkdGJUkiY0dGJSIiIg==QyQ+SSJ5RzYiLSZJLExpbmVhclNvbHZlR0YlNiNJIlFHRiU2JC1JKnRyYW5zcG9zZUdGJTYjSSNBQkdGJSwkLUYtNiMtSSpTdWJNYXRyaXhHRiU2JUkiQUdGJTcjIiIjNyUiIiJGOCIiJCEiIkY6The two negative components of y are the ones corresponding to index 1 and 5. The minimum is attained at the first index and thus the new roof is {2,4,5} (START THIRD ITERATION) QyY+SSJCRzYiNyUiIiMiIiUiIiYiIiI+SSNBQkdGJS1JKlN1Yk1hdHJpeEdGJTYlSSJBR0YlRiQ3JUYqRiciIiRGKg==QyQ+SSNiQkc2Ii1JKlN1YlZlY3RvckdGJTYkSSJiR0YlSSJCR0YlIiIiQyQ+SSJ2RzYiLSZJLExpbmVhclNvbHZlR0YlNiNJIlFHRiU2JEkjQUJHRiVJI2JCR0YlIiIiDoes the new vertex violate an inequality? LUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJUkoX3N5c2xpYkdGKDYlSSJBR0YoSSJ2R0YoSSV0cnVlR0YlThe third inequality is violated. We bring index 3 into the roof. And compute x* and y*: JSFHQyQ+SSJ4RzYiLSZJLExpbmVhclNvbHZlR0YlNiNJIlFHRiU2JC1JKnRyYW5zcG9zZUdGJTYjSSNBQkdGJUkiY0dGJSIiIg==QyQ+SSJ5RzYiLSZJLExpbmVhclNvbHZlR0YlNiNJIlFHRiU2JC1JKnRyYW5zcG9zZUdGJTYjSSNBQkdGJSwkLUYtNiMtSSpTdWJNYXRyaXhHRiU2JUkiQUdGJTcjIiIkNyUiIiIiIiNGOCEiIkY6The two negative components of y are the ones corresponding to index 2 and 5. The minimum is attained at the first index and thus the new roof is {3,4,5} (START THIRD ITERATION) QyY+SSJCRzYiNyUiIiQiIiUiIiYiIiI+SSNBQkdGJS1JKlN1Yk1hdHJpeEdGJTYlSSJBR0YlRiQ3JUYqIiIjRidGKg==QyQ+SSNiQkc2Ii1JKlN1YlZlY3RvckdGJTYkSSJiR0YlSSJCR0YlIiIiQyQ+SSJ2RzYiLSZJLExpbmVhclNvbHZlR0YlNiNJIlFHRiU2JEkjQUJHRiVJI2JCR0YlIiIiDoes the new vertex violate an inequality? LUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJUkoX3N5c2xpYkdGKDYlSSJBR0YoSSJ2R0YoSSV0cnVlR0YlNo inequaity is violated. The roof is optimal. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=QyQ+SSJ4RzYiLSZJLExpbmVhclNvbHZlR0YlNiNJIlFHRiU2JC1JKnRyYW5zcG9zZUdGJTYjSSNBQkdGJUkiY0dGJSIiIg==We checked that c is a conic combination of the roof-vectors and this is indeed the case. TTdSMApJN1JUQUJMRV9TQVZFLzQzMjA5ODk3ODRYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISMzIiciJCIiIyIiJCIiIkYpIiIhRipGKEYpRidGKkYpRioiIiVGK0YnRipGKkYpRiY=TTdSMApJN1JUQUJMRV9TQVZFLzQzNzU1MTYxMTJYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISInIichIiYhIiohIzUiIiFGKkYqRiY=TTdSMApJN1JUQUJMRV9TQVZFLzQzNzU1MTYyMzJYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISIkIiQiIiYiIiQiIiNGJg==TTdSMApJN1JUQUJMRV9TQVZFLzQzNzU1MTYzNTJYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISMqIiQiJCIiIiIiIUYoRihGJ0YoRihGKEYnRiY=TTdSMApJN1JUQUJMRV9TQVZFLzQzNzU1MTY0NzJYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISMqIiQiJCIiIyIiIiIiISIiJEYpRigiIiVGKUYpRiY=TTdSMApJN1JUQUJMRV9TQVZFLzQzNzU1MTY1OTJYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISIkIiQhIiYiIiFGKEYmTTdSMApJN1JUQUJMRV9TQVZFLzQzNzU1MTY5NTJYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISIkIiQiIiFGJyMhIiYiIiVGJg==TTdSMApJN1JUQUJMRV9TQVZFLzQzNzU1MTc1NTJYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISInIichIiZGJyNGJyIiIyIiIUYqI0YnIiIlRiY=TTdSMApJN1JUQUJMRV9TQVZFLzQzNzU1MTgyNzJYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISIkIiQjIiIiIiIjIiIlIyIiJEYpRiY=TTdSMApJN1JUQUJMRV9TQVZFLzQzNzU1MjIyMzJYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISMkIiQiIiEiIkYnIiIjRiY=TTdSMApJN1JUQUJMRV9TQVZFLzQzNzU1MjI0NzJYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISMqIiQiJCIiJCIiIiIiIUYoRilGKCIiJUYpRilGJg==TTdSMApJN1JUQUJMRV9TQVZFLzQzNzk4MDgzODRYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISIkIiQhIioiIiFGKEYmTTdSMApJN1JUQUJMRV9TQVZFLzQzNzU1MjI4MzJYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISIkIiQiIiFGJyMhIioiIiVGJg==TTdSMApJN1JUQUJMRV9TQVZFLzQzNzU1MjM0MzJYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISInIichIipGJyNGJyIiIyIiIUYqI0YnIiIlRiY=TTdSMApJN1JUQUJMRV9TQVZFLzQzNzU1MDk4NDBYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISIkIiQjIiIiIiIjIyIiKEYpIyIiJkYpRiY=TTdSMApJN1JUQUJMRV9TQVZFLzQzNzU1MTM2ODBYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISMkIiQiIiMhIiIiIiMjIiIiRikjISIkRilGJg==TTdSMApJN1JUQUJMRV9TQVZFLzQzOTIyMjQxNTJYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISMqIiQiJCIiIkYnIiIhIiIjRihGJ0YpRihGKEYmTTdSMApJN1JUQUJMRV9TQVZFLzQzOTIyMjQyNzJYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISIkIiQhIzUiIiFGKEYmTTdSMApJN1JUQUJMRV9TQVZFLzQzOTIyMjQ2MzJYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISIkIiQiIiFGJyEiJkYmTTdSMApJN1JUQUJMRV9TQVZFLzQzOTIyMjUyMzJYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISInIichIz9GJyEjNSIiIUYpISImRiY=TTdSMApJN1JUQUJMRV9TQVZFLzQzOTIyMjU3MTJYKiUpYW55dGhpbmdHNiI2IltnbCEjJSEhISIkIiQiIiIiIiVGJ0Ym