1. 3 RD NSF Workshop on Imprecise Probability in Engineering Analysis & Design February 20-22, 2008 | Georgia Institute of Technology, Savannah, USA On using global optimization for approximating hull solution of parametric interval systems Iwona Skalna AGH University of Science & Technology, Poland Andrzej Pownuk The University of Texas, El Paso, USA

2. 3 RD NSF Workshop on Imprecise Probability in Engineering Analysis & Design February 20-22, 2008 | Georgia Institute of Technology, Savannah, USA 2 Outline  Parametric interval linear systems  Global optimization  Monotonicity test  Subdivision directions  Multidivision  Examples  Concluding remarks acceleration techniques

3. 3 RD NSF Workshop on Imprecise Probability in Engineering Analysis & Design February 20-22, 2008 | Georgia Institute of Technology, Savannah, USA 3 Parametric linear systems Parametric solution set Interval hull solution Parametric linear system with parameters p, q Coefficients: affine linear functions of parameters

4. 3 RD NSF Workshop on Imprecise Probability in Engineering Analysis & Design February 20-22, 2008 | Georgia Institute of Technology, Savannah, USA 4 Optimization problem (hull solution) Let A(p) be regular, p  IR k, and x i min, x i max denote the global solutions of the i-th minimization and, respectively, maximization problem Then the interval vector Theorem

5. 3 RD NSF Workshop on Imprecise Probability in Engineering Analysis & Design February 20-22, 2008 | Georgia Institute of Technology, Savannah, USA 5 Global optimization – general assumptions Define r  IR k+l with r i = p i for i = 1, , k, and r j = q j-k, j = 1, , k. Then x(p,q) can be written in a compact form as x(r). w(f(x))  0 as w(x)  0 Inclusion function f = x i (p,q) is calculated using a Direct Method for solving parametric linear systems. It can be easily shown that the Direct Method preserves the isotonicity property, i.e. x  y implies f(x)  f(y). It is also assumed that for all inclusion functions holds:

6. 3 RD NSF Workshop on Imprecise Probability in Engineering Analysis & Design February 20-22, 2008 | Georgia Institute of Technology, Savannah, USA 6 Global optimization - algorithm Step 0. Set y = r and f = min x(y). Initialize list L = {f, y} and cutoff level z = max x(y) Step 1. Choose a coordinate v  {1, , k+l} for subdivision using one of the subdivision rules Step 2.Bisect (multisect) y in direction v : y 1  y 2, int ( y 1 )  int(y 2 ) =  Step 3. Calculate x(y 1 ) and x(y 2 ) and set f i = min x(y i ) and z = min { z, max x(y 1 ), max x(y 1 ) } Step 4. Remove (f, y) from the list L Step 5.Cutoff test: discard any pair (f i, y i ) if f i > z Step 6.Monotonicity test Step 7.Add any remaining pairs to the list, if the list become empty then STOP Step 8.Denote the the pair with the smallest first element by (f *, y * ) Step 9.If w(y * ) <  then STOP, else goto 1.

7. 3 RD NSF Workshop on Imprecise Probability in Engineering Analysis & Design February 20-22, 2008 | Georgia Institute of Technology, Savannah, USA 7 Monotonicity test The monotonicity test is used to figure out whether the function f is strictly monotone in a whole subbox y  x. Then, y cannot contain a global minimizer in its interior. Therefore, if f satisfies Monotonicity test is pefromed using the Method for Checking the Monotonicity (MCM). The MCM method is based on a Direct Method for solving parametric linear systems. Let f = x(p, q). Approximations of partial derivates are obtained by solving the following k+l parametric linear systems where x * is an approximation of the solution set S calculated using Direct method, and

8. 3 RD NSF Workshop on Imprecise Probability in Engineering Analysis & Design February 20-22, 2008 | Georgia Institute of Technology, Savannah, USA 8 Subdivision direction selection The following merit function (Ratz, Csendes) is used to define subdivision rules: where D(i) is determined by a given rule. Rule A (Hansen): Rule B (Casado): Rule C (Walster, Hansen): D(i) = w(y i ) D(i) = w(F(y i )w(y i )) D(i) = p(f k, f i )

9. 3 RD NSF Workshop on Imprecise Probability in Engineering Analysis & Design February 20-22, 2008 | Georgia Institute of Technology, Savannah, USA 9 Example 1 Young’s modulus Y = 7.0  10 10 [Pa] Cross section aresaC = 0.003[m 2 ] LengthL = 2[m] LoadsP 1 = P 2 = P 3 = 30[kN] The stiffness of all bars is uncertain by 5%

10. 3 RD NSF Workshop on Imprecise Probability in Engineering Analysis & Design February 20-22, 2008 | Georgia Institute of Technology, Savannah, USA 10 Results – nodes displacements n x lb[  10 -5 ]x ub[  10 -5 ] n x lb[  10 -5 ]x lb[  10 -5 u 1-32.53-29.27123.904.67 2-1.61-1,4513-16.23-14.67 3-26.45-23.93143.183.87 4-2.41-2.1715-3.63-2.96 5-15.78-14.27163.183.87 6-1.69-1.3717-0.050.05 7-4.08-3.37182.353.02 8-0.96-0.5719-0.46-0.40 90.360.50200.851.47 103.904.6721-2.78-2.09 11-26.45-23.93 result of the Global Optimization = result of the Evolutionary Algorithm

11. 3 RD NSF Workshop on Imprecise Probability in Engineering Analysis & Design February 20-22, 2008 | Georgia Institute of Technology, Savannah, USA 11 Example 2 result of the Global Optimization = result of the Evolutionary Algorithm Young’s modulus Y = 2.1  10 11 [Pa] Cross section aresaC = 0.004[m 2 ] LengthL = 2[m] LoadsP 1 = 80[kN], P 2 = 120[kN] The stiffness of thick bars is uncertain by 5%

12. 3 RD NSF Workshop on Imprecise Probability in Engineering Analysis & Design February 20-22, 2008 | Georgia Institute of Technology, Savannah, USA 12 Conclusions  Global Optimization Method can be succesfully used to approximate hull solution of parametric linear systems  Monotonicity test significantly improves the convergence of Global Optimization  Multidivision and different rules for subdivision directions have no influence on the convergence of the Global Optimization, they are computationally very expensive  In future work we will try combine different methods for solving parametric linear systems