Dennis L. Bricker Dept of Mechanical & Industrial Engineering The University of Iowa i.e., 1 1 1 Minimize X X X subject to XX 4 X 1 0.5X 1 Minimize X X X X 1X X s.t. 4 1 1 1 1 4X X 1 1 1 1 0.5X X X 1 1 1 1 4/11/006 page 1 of 51 4/11/006 page of 51 Introduce a new variable X so that the signomial now appears in a constraint: Minimize X subject to X X X 4 X 1X X X 1 1 1 1 4X X 1 1 1 1 0.5X X X 1 1 1 1 Next, rewrite the signomial constraint: X X 1 X X X X X 4 1 1 1 1 X X 1 X X X X X 4 1 1 1 1 If we condense the denominator X X X X 1 1 1 into a monomial, using the Arithmetic-Geometric Mean Inequality, the result will be a posynomial approximation! 4/11/006 page of 51 4/11/006 page 4 of 51
u n n i ui i1 i1 i n for all satisfying i1 with equality if & only if 1 1, 0 i u1 u un i i n Condensing the numerator: for such that 1 X X1 X X1 1 1 1 X X X X 1 1, i 0, i 1,, X X X X 1 1 X X X 1 1 1 1 coefficient C 4/11/006 page 5 of 51 4/11/006 page 6 of 51 We choose so that, for a given X, X 1 X X1 X X1 X1 X X X1 X X1 so that 1 1 X 1 X X1 X X1 Minimize X subject to 4 X X1 1 X1 X X X X 1 1 1 1 1 1 1 1 1 4X X 1 0.5X X X 1 and the approximation is exact at X. 4/11/006 page 7 of 51 4/11/006 page 8 of 51
We will choose X,,1 as the initial point. 1 X 1 1 0.047619 1 16 X1 X 16 0.761905 1 4 X1 4 0.190476 1 So we get the posynomial approximation X X 1 X.77409 4 1 1 1.714 0.7619 0.0476 X1 X X which is the posynomial constraint: 1 1 1 1 X X X X X X 1.714 0.7619 0.0476 1 1.77409 0.64965X X X 1.714 1.81 0.0476 1 0.64965X X X.8571 0.7619 0.0476 1 0.64965X X X 1.714 0.7619 0.0476 1 0.64965X X X 1 0.8571 0.7619 0.0476 1 4/11/006 page 9 of 51 4/11/006 page 10 of 51 Posynomial GP approximation of the signomial GP: Minimize X subject to 0.64965X X X 1.714 1.81 0.0476 1 0.64965X X X.8571 0.7619 0.0476 1 0.64965X X X 1.714 0.7619 0.0476 1 0.64965X X X 1 0.8571 0.7619 0.0476 1 4X X 1 1 1 1 0.5X X X 1 1 1 1 4/11/006 page 11 of 51 4/11/006 page 1 of 51
Number of variables: Number of polynomials: 4 Total number of terms: 10 Degrees of difficulty: 6 Terms per polynomial: 1 6 1 Rosenbrock et al. t p Ct exponents 1 1 1 0 0 1 ------------ --------- 1 0 1 1 1 4 1 4 0 1 5 1 0 0 1 6 1 0 1 7 1 0 1 ------------ --------- 8 4 1 1 0 ------------ --------- 9 4 0.5 1 0 10 4 0 1 0 ------------ --------- t = term number, p = polynomial Ct = coefficient 4/11/006 page 1 of 51 4/11/006 page 14 of 51 Bounds on variables # var LB UB 1 X[1] 0.00001 100000 X[] 0.00001 100000 X[] 0.00001 100000 Current Parameters Tolerances for duality gap: 0.0001 Tolerances for constraints (maximum allowable infeasibility) = 0.0001 Tolerance "epsilon" for stopping criterion: epsilon > sum d_rho, where d_rho is the vector of changes in the weights for the terms of the polynomials: epsilon = 0.0005 Maximum # Posynomial subproblems to be solved = 5 Maximum # LPs to be solved per posynomial subproblem = 5 4/11/006 page 15 of 51 4/11/006 page 16 of 51
User-specified Grid Point 1 X[1] X[] The objective function value for this point is 5 The values of the constraint polynomials are: k P(k) 1 1.5 User-specified Grid Point 1 X[1] X[] X[] 1 The objective function value for this point is 1 The values of the constraint polynomials are: k P(k) 5 1 4 1.5 4/11/006 page 17 of 51 4/11/006 page 18 of 51 <>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<> Major Iteration # 1 1 X[1] X[] X[] 1 1 1 1.0 4.0 16.0 4 16.0 5 1.0 6 4.0 7 4.0 8 1.0 9 4 0.5 10 4 1.0 constraint Value Infeasibility 5.0 4.0 1.0 0.0 4 1.5 0.5 poly term value 0.761905 5 0.190476 Objective function = 1 4/11/006 page 19 of 51 4/11/006 page 0 of 51
Condensation of Signomial GP Number of variables: Number of posynomials: 10 Total number of terms: 14 Degrees of difficulty: 11 Terms per posynomial: 1 4 1 1 1 1 1 1 1 (includes bounds on variables to ensure dual feasibility) 0.64965 1.7149 1.81 0.047619 0.64965.8571 0.761905 0.047619 4 0.64965 1.7149 0.761905 0.047619 5 0.64965 0.85714 0.761905 0.047619 6 4 1 1 0 7 4 0.5 1 0 8 4 0 1 0 1 8 0.00001 1 0 0 1 9 0.00001 0 1 0 4/11/006 page 1 of 51 4/11/006 page of 51 1 X[1] 1.597846 X[].66614 X[] 0.6564 Primal: 0.6564 Dual: 0.6564 <>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<> Major Iteration # 1 X[1] 1.597846 X[].66614 X[] 0.6564 1 1 0.6564 19.0171 6.8048 4 17.8718 5.74911 6 8.799 7 6.985 8 0.949464 9 4 0.408 10 4 0.758549 Objective function = 0.6564 4/11/006 page of 51 4/11/006 page 4 of 51
0.996581 0.000000000 4.0059 0.949464 0.000000000 0.0000 4 1.00061 0.0006115 0.0000 small infeasibility! poly term value change 0.790811 0.0890610 5 0.18771 0.007645 Condensation of Signomial GP 0.8551 1.769 1.0919 0.014775 0.8551.067 0.790811 0.014775 4 0.8551 1.769 0.790811 0.014775 5 0.8551 0.0667 0.790811 0.014775 6 4 1 1 0 7 4 0.5 1 0 8 4 0 1 0 1 8 0.00001 1 0 0 1 9 0.00001 0 1 0 4/11/006 page 5 of 51 4/11/006 page 6 of 51 1 X[1] 1.561004 X[].60590 X[] 0.8918 Primal: 0.8918 Dual: 0.8918 <>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<> Major Iteration # 1 X[1] 1.561004 X[].60590 X[] 0.8918 1 1 0.8918.48061 4.91675 4 0.5668 5.4580 6 10.796007 7 8.4606 8 0.9816 9 4 0.768 10 4 0.767480 Objective function = 0.8918 4/11/006 page 7 of 51 4/11/006 page 8 of 51
1.1876 0.18761.4648 large infeasibility! 0.9816 0.00000000 0.0000 4 1.00148 0.001487 46.5604 poly term value change 0.78871 0.005994 5 0.19780 0.00606819 Condensation of Signomial GP 0.84107 1.770 1.117 0.017949 0.84107.968 0.78871 0.017949 4 0.84107 1.770 0.78871 0.017949 5 0.84107 0.9678 0.78871 0.017949 6 4 1 1 0 7 4 0.5 1 0 8 4 0 1 0 1 8 0.00001 1 0 0 1 9 0.00001 0 1 0 4/11/006 page 9 of 51 4/11/006 page 0 of 51 1 X[1] 1.54998 X[].600550 X[] 0.470 Primal: 0.470 Dual: 0.470 <>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<> Major Iteration # 4 1 X[1] 1.54998 X[].600550 X[] 0.470 1 1 0.470 19.47818 5.986660 4 16.61670 5.880168 6 8.9816 7 6.919047 8 0.9986 9 4 0.094 10 4 0.769068 Objective function = 0.470 4/11/006 page 1 of 51 4/11/006 page of 51
0.98446 0.0000000000 11.9815 0.9986 0.0000000000 0.0000 4 1.000010 0.0000100608 47.088 small infeasibility! poly term value change 0.78770 0.0045009 5 0.194451 0.000670688 Notice that the solution seems to alternate between one with small infeasibility and objective approximately 0.47, and one with large infeasibility and objective approximately 0.84 Condensation of Signomial GP 0.8061 1.76199 1.16 0.017795 0.8061.801 0.7877 0.017795 4 0.8061 1.76199 0.7877 0.017795 5 0.8061 0.8009 0.7877 0.017795 6 4 1 1 0 7 4 0.5 1 0 8 4 0 1 0 1 8 0.00001 1 0 0 1 9 0.00001 0 1 0 4/11/006 page of 51 4/11/006 page 4 of 51 1 X[1] 1.548186 X[].595916 X[] 0.84417 Primal: 0.84417 Dual: 0.84417 <>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<> Major Iteration # 5 1 X[1] 1.548186 X[].595916 X[] 0.84417 1 1 0.84417.698 4.759 4 0.19998 5.515961 6 10.88671 7 8.47 8 0.9958 9 4 0.08 10 4 0.770441 Objective function = 0.84417 4/11/006 page 5 of 51 4/11/006 page 6 of 51
1.195860 0.19585984 4.05055 large infeasibility! 0.9958 0.00000000 0.00000 4 1.0017 0.001779 45.9148 poly term value change 0.78664 0.005980 5 0.195664 0.001198 Condensation of Signomial GP 0.84 1.7689 1.164 0.017977 0.84.161 0.78664 0.017977 4 0.84 1.7689 0.78664 0.017977 5 0.84 0.1609 0.78664 0.017977 6 4 1 1 0 7 4 0.5 1 0 8 4 0 1 0 1 8 0.00001 1 0 0 1 9 0.00001 0 1 0 4/11/006 page 7 of 51 4/11/006 page 8 of 51 1 X[1] 1.549449 X[].600175 X[] 0.47004 Primal: 0.47004 Dual: 0.47004 <>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<> Major Iteration # 6 1 X[1] 1.549449 X[].600175 X[] 0.47004 1 1 0.47004 19.4868 5.9790 4 16.610155 5.881808 6 8.9046 7 6.918618 8 0.9984 9 4 0.080 10 4 0.769179 Objective function = 0.47004 4/11/006 page 9 of 51 4/11/006 page 40 of 51
0.98456 0.0000000000 5.9064 0.9984 0.0000000000 0.00000 4 1.000009 0.000008895 54.008 small infeasibility! poly term value change 0.78696 0.0066745 5 0.1945 0.00114178 Condensation of Signomial GP 0.8058 1.76191 1.16 0.017819 0.8058.809 0.78696 0.017819 4 0.8058 1.76191 0.78696 0.017819 5 0.8058 0.8086 0.78696 0.017819 6 4 1 1 0 7 4 0.5 1 0 8 4 0 1 0 1 8 0.00001 1 0 0 1 9 0.00001 0 1 0 4/11/006 page 41 of 51 4/11/006 page 4 of 51 1 X[1] 1.54807 X[].5958 X[] 0.849 Primal: 0.849 Dual: 0.849 <>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<> Major Iteration # 7 1 X[1] 1.54807 X[].5958 X[] 0.849 1 1 0.8490.6989 4.74974 4 0.195068 5.51601 6 10.886941 7 8.46858 8 0.99594 9 4 0.0805 10 4 0.770468 Objective function = 0.849 4/11/006 page 4 of 51 4/11/006 page 44 of 51
1.195904 0.1959058 4.0566 large infeasibility! 0.99594 0.00000000 0.00000 4 1.0017 0.00170 45.90966 poly term value change 0.78645 0.0064891 5 0.195681 0.00115910 Condensation of Signomial GP 0.8 1.7687 1.165 0.017979 0.8.16 0.78645 0.017979 4 0.8 1.7687 0.78645 0.017979 5 0.8 0.169 0.78645 0.017979 6 4 1 1 0 7 4 0.5 1 0 8 4 0 1 0 1 8 0.00001 1 0 0 1 9 0.00001 0 1 0 4/11/006 page 45 of 51 4/11/006 page 46 of 51 1 X[1] 1.54946 X[].600165 X[] 0.47001 Primal: 0.47001 Dual: 0.47001 <>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<> Major Iteration # 8 1 X[1] 1.54946 X[].600165 X[] 0.47001 1 1 0.47001 19.48698 5.978894 4 16.60979 5.88188 6 8.90448 7 6.918579 8 0.99854 9 4 0.087 10 4 0.76918 Objective function = 0.47001 4/11/006 page 47 of 51 4/11/006 page 48 of 51
0.984567 0.00000000000 5.87 0.99854 0.00000000000 0.00000 4 1.000009 0.0000088645 54.06786 small infeasibility! <>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<> 4/11/006 page 49 of 51 4/11/006 page 50 of 51 Dropping two initial points from the plot: zig-zagging behavior is apparent! 4/11/006 page 51 of 51