i.e., into a monomial, using the Arithmetic-Geometric Mean Inequality, the result will be a posynomial approximation!
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1 Dennis L. Bricker Dept of Mechanical & Industrial Engineering The University of Iowa i.e., Minimize X X X subject to XX 4 X 1 0.5X 1 Minimize X X X X 1X X s.t X X X X X /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 X X X X X Next, rewrite the signomial constraint: X X 1 X X X X X X X 1 X X X X X If we condense the denominator X X X X 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
2 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 X X X X X 1 1, i 0, i 1,, X X X X 1 1 X X X 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 X 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
3 We will choose X,,1 as the initial point. 1 X X1 X X So we get the posynomial approximation X X 1 X X1 X X which is the posynomial constraint: X X X X X X X X X X X X X X X X X X /11/006 page 9 of 51 4/11/006 page 10 of 51 Posynomial GP approximation of the signomial GP: Minimize X subject to X X X X X X X X X X X X X X X X X /11/006 page 11 of 51 4/11/006 page 1 of 51
4 Number of variables: Number of polynomials: 4 Total number of terms: 10 Degrees of difficulty: 6 Terms per polynomial: Rosenbrock et al. t p Ct exponents 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] X[] X[] Current Parameters Tolerances for duality gap: Tolerances for constraints (maximum allowable infeasibility) = 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 = 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
5 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) 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) /11/006 page 17 of 51 4/11/006 page 18 of 51 <>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<> Major Iteration # 1 1 X[1] X[] X[] constraint Value Infeasibility poly term value Objective function = 1 4/11/006 page 19 of 51 4/11/006 page 0 of 51
6 Condensation of Signomial GP Number of variables: Number of posynomials: 10 Total number of terms: 14 Degrees of difficulty: 11 Terms per posynomial: (includes bounds on variables to ensure dual feasibility) /11/006 page 1 of 51 4/11/006 page of 51 1 X[1] X[] X[] Primal: Dual: <>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<> Major Iteration # 1 X[1] X[] X[] Objective function = /11/006 page of 51 4/11/006 page 4 of 51
7 small infeasibility! poly term value change Condensation of Signomial GP /11/006 page 5 of 51 4/11/006 page 6 of 51 1 X[1] X[] X[] Primal: Dual: <>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<> Major Iteration # 1 X[1] X[] X[] Objective function = /11/006 page 7 of 51 4/11/006 page 8 of 51
8 large infeasibility! poly term value change Condensation of Signomial GP /11/006 page 9 of 51 4/11/006 page 0 of 51 1 X[1] X[] X[] Primal: Dual: <>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<> Major Iteration # 4 1 X[1] X[] X[] Objective function = /11/006 page 1 of 51 4/11/006 page of 51
9 small infeasibility! poly term value change 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 /11/006 page of 51 4/11/006 page 4 of 51 1 X[1] X[] X[] Primal: Dual: <>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<> Major Iteration # 5 1 X[1] X[] X[] Objective function = /11/006 page 5 of 51 4/11/006 page 6 of 51
10 large infeasibility! poly term value change Condensation of Signomial GP /11/006 page 7 of 51 4/11/006 page 8 of 51 1 X[1] X[] X[] Primal: Dual: <>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<> Major Iteration # 6 1 X[1] X[] X[] Objective function = /11/006 page 9 of 51 4/11/006 page 40 of 51
11 small infeasibility! poly term value change Condensation of Signomial GP /11/006 page 41 of 51 4/11/006 page 4 of 51 1 X[1] X[].5958 X[] Primal: Dual: <>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<> Major Iteration # 7 1 X[1] X[].5958 X[] Objective function = /11/006 page 4 of 51 4/11/006 page 44 of 51
12 large infeasibility! poly term value change Condensation of Signomial GP /11/006 page 45 of 51 4/11/006 page 46 of 51 1 X[1] X[] X[] Primal: Dual: <>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<>-<> Major Iteration # 8 1 X[1] X[] X[] Objective function = /11/006 page 47 of 51 4/11/006 page 48 of 51
13 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
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