Simplex (quick recap) Replace all the inequality
Description: Simplex (quick recap) Replace all the inequality constraints by equalities, using slack variables 8 constraints 5 variables (i.e., 5 dimensional space) any vertex solution corresponds to the intersection of 5 tight constraints. 3 of these
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slide1. Simplex (quick recap)<br>
slide2. Replace all the inequality constraints by equalities, using slack variables<br>
slide3. 8 constraints 5 variables (i.e., 5 dimensional space) any vertex solution corresponds to the intersection of 5 tight constraints. 3 of these are three equalities. other 2 must be xi = 0 and xj = 0 for some i,j x2 x1 x3 x4 x5 (x1 = 0, x2 = 0, x3 = 1, x4 = 3, x5 = 2) rewrite constraints to move slack variables to the left. and write the objective function as z = …<br>
slide4. non-basic vars basic
vars non-basic vars are set to 0. basic vars values are then fixed Pick a non-basic variable (say x2) with positive coefficient in objective increase its value until some other variable (x3) goes to 0 x2 enters basis, x3 exits basis. x2 x1 x3 x4 x5 (x1 = 0, x2 = 0, x3 = 1, x4 = 3, x5 = 2) which one?
(pivot rule) constraint for x3 >= 0 now tight constraint x2 >= 0 no longer tight Now rewrite, replacing x3 by x2 using the tight constraint. (x1 = 0, x2 = 1, x3 = 0, x4 = 3, x5 = 1)<br>
slide5. non-basic vars are set to 0. basic vars values are then fixed (x1 = 0, x2 = 1, x3 = 0, x4 = 3, x5 = 1) x2 x1 x3 x4 x5 Pick a non-basic variable (say x2) with positive coefficient in objective increase its value until some other variable (x3) goes to 0 x2 enters basis, x3 exits basis. Now rewrite, replacing x3 by x2 using the tight constraint.<br>
slide6. non-basic vars are set to 0. basic vars values are then fixed Pick a non-basic variable (say x1) with positive coefficient in objective increase its value until some other variable (x5) goes to 0 x1 enters basis, x5 exits basis. (x1 = 0, x2 = 1, x3 = 0, x4 = 3, x5 = 1) x2 x1 x3 x4 x5 (x1 = 1, x2 = 2, x3 = 0, x4 = 2, x5 = 0) Now rewrite, replacing x5 by x1 using the tight constraint.<br>
slide7. non-basic vars are set to 0. basic vars values are then fixed Pick a non-basic variable (say x3) with positive coefficient in objective increase its value until some other variable (x4) goes to 0 x3 enters basis, x4 exits basis. x2 x1 x3 x4 x5 (x1 = 1, x2 = 2, x3 = 0, x4 = 2, x5 = 0) (x1 = 3, x2 = 2, x3 = 2, x4 = 0, x5 = 0) Now rewrite, replacing x4 by x3 using the tight constraint.<br>
slide8. non-basic vars are set to 0. basic vars values are then fixed Pick a non-basic variable with positive coefficient in objective if no such variable, must be at optimum. (x1 = 3, x2 = 2, x3 = 2, x4 = 0, x5 = 0) x2 x1 x3 x4 x5<br>
slide9. Pick a non-basic variable (say x2) with positive coefficient in objective non-basic vars are set to 0. basic vars values are then fixed If it can be increased arbitrarily, LP is unbounded<br>
slide10. Pick a non-basic variable (say x2) with positive coefficient in objective non-basic vars are set to 0. basic vars values are then fixed If it can be increased by only zero, degenerate vertex.<br>
slide11. Pick a non-basic variable (say x1) with positive coefficient in objective non-basic vars are set to 0. basic vars values are then fixed now it can be increased… What if we keep cycling in this degenerate vertices (or degenerate faces)? Solution: anti-cycling pivot rules.<br>
slide12. Pivot rules largest coefficient largest increase steepest edge Bland’s rule (non-cycling) Random edge often best in practice<br>
slide2. Replace all the inequality constraints by equalities, using slack variables<br>
slide3. 8 constraints 5 variables (i.e., 5 dimensional space) any vertex solution corresponds to the intersection of 5 tight constraints. 3 of these are three equalities. other 2 must be xi = 0 and xj = 0 for some i,j x2 x1 x3 x4 x5 (x1 = 0, x2 = 0, x3 = 1, x4 = 3, x5 = 2) rewrite constraints to move slack variables to the left. and write the objective function as z = …<br>
slide4. non-basic vars basic
vars non-basic vars are set to 0. basic vars values are then fixed Pick a non-basic variable (say x2) with positive coefficient in objective increase its value until some other variable (x3) goes to 0 x2 enters basis, x3 exits basis. x2 x1 x3 x4 x5 (x1 = 0, x2 = 0, x3 = 1, x4 = 3, x5 = 2) which one?
(pivot rule) constraint for x3 >= 0 now tight constraint x2 >= 0 no longer tight Now rewrite, replacing x3 by x2 using the tight constraint. (x1 = 0, x2 = 1, x3 = 0, x4 = 3, x5 = 1)<br>
slide5. non-basic vars are set to 0. basic vars values are then fixed (x1 = 0, x2 = 1, x3 = 0, x4 = 3, x5 = 1) x2 x1 x3 x4 x5 Pick a non-basic variable (say x2) with positive coefficient in objective increase its value until some other variable (x3) goes to 0 x2 enters basis, x3 exits basis. Now rewrite, replacing x3 by x2 using the tight constraint.<br>
slide6. non-basic vars are set to 0. basic vars values are then fixed Pick a non-basic variable (say x1) with positive coefficient in objective increase its value until some other variable (x5) goes to 0 x1 enters basis, x5 exits basis. (x1 = 0, x2 = 1, x3 = 0, x4 = 3, x5 = 1) x2 x1 x3 x4 x5 (x1 = 1, x2 = 2, x3 = 0, x4 = 2, x5 = 0) Now rewrite, replacing x5 by x1 using the tight constraint.<br>
slide7. non-basic vars are set to 0. basic vars values are then fixed Pick a non-basic variable (say x3) with positive coefficient in objective increase its value until some other variable (x4) goes to 0 x3 enters basis, x4 exits basis. x2 x1 x3 x4 x5 (x1 = 1, x2 = 2, x3 = 0, x4 = 2, x5 = 0) (x1 = 3, x2 = 2, x3 = 2, x4 = 0, x5 = 0) Now rewrite, replacing x4 by x3 using the tight constraint.<br>
slide8. non-basic vars are set to 0. basic vars values are then fixed Pick a non-basic variable with positive coefficient in objective if no such variable, must be at optimum. (x1 = 3, x2 = 2, x3 = 2, x4 = 0, x5 = 0) x2 x1 x3 x4 x5<br>
slide9. Pick a non-basic variable (say x2) with positive coefficient in objective non-basic vars are set to 0. basic vars values are then fixed If it can be increased arbitrarily, LP is unbounded<br>
slide10. Pick a non-basic variable (say x2) with positive coefficient in objective non-basic vars are set to 0. basic vars values are then fixed If it can be increased by only zero, degenerate vertex.<br>
slide11. Pick a non-basic variable (say x1) with positive coefficient in objective non-basic vars are set to 0. basic vars values are then fixed now it can be increased… What if we keep cycling in this degenerate vertices (or degenerate faces)? Solution: anti-cycling pivot rules.<br>
slide12. Pivot rules largest coefficient largest increase steepest edge Bland’s rule (non-cycling) Random edge often best in practice<br>