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1. Kliknij, aby edytować styl wzorca tytułu Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions A global optimization method for solving parametric linear system whose input data are rational functions of interval parameters Iwona Skalna AGH University of Science and Technology Krakow, Poland Iwona Skalna, Poland Small Workshop on Interval Methods’09, Lausanne

2. Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions Kliknij, aby edytować styl wzorca tytułu • Parametric linear systems • Optimization problem • Interval global optmization • Monotonicity test • Revised affine arithmetic • Evolutionary optimization • Examples • Conclusions Iwona Skalna, Krakow, Poland Small Workshop on Interval Methods’09, Lausanne

3. Kliknij, aby edytować styl wzorca tytułu Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions Parametric linear system is defined as a family of real linear systems with coefficients where are nonlinear continuous functions of parameters Iwona Skalna, Krakow, Poland Small Workshop on Interval Methods’09, Lausanne

4. Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions Kliknij, aby edytować styl wzorca tytułu Parametric (united) solution set is define as The goal is to find the thightest interval enclosure for S, possibly the interval hull solution defined as If the solution is monotone with respect to all parameters, then the interval hull solution can be calculated by solving at most 2n real linear systems with coefficients being the respective endpoints of interval parameters Iwona Skalna, Krakow, Poland Small Workshop on Interval Methods’09, Lausanne

5. Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions Kliknij, aby edytować styl wzorca tytułu In general case, to calculate the hull solution, the following 2n constrained optimization problems must be solved where is an objective function Iwona Skalna, Krakow, Poland Small Workshop on Interval Methods’09, Lausanne

6. Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions Kliknij, aby edytować styl wzorca tytułu The optimizations problems are solved using an interval global optimization. The interval global optimization algorithm has the following steps: Step 1. Initialize the list L =(pq, x(pq)) Step 2. Remove (pq, x(pq)) from the list L Step 3. Bisect pq= pq1pq2 Step 4. Calculate x(pqi), pqi Step 5. Perform the monotonicity test Step 6. If w(pq) < STOP else GOTO 2 Various acceleration techniques are used to speed up the convergence of global optimization. The monotonicity test is the most important one for the considered problem. Iwona Skalna, Krakow, Poland Small Workshop on Interval Methods’09, Lausanne

7. Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions Kliknij, aby edytować styl wzorca tytułu The monotonicicty test is performed using the Direct Method solving parametric linear systems. To check the sign of derivatives, the following parametric linear systems must be solved: If a devirative has constant sing, then the corresponding interval can be reduced to one of its edges. or The Direct Method is also used to calculate inclusion function for the objective function x(p,q) is Iwona Skalna, Krakow, Poland Small Workshop on Interval Methods’09, Lausanne

8. Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions Kliknij, aby edytować styl wzorca tytułu Direct Method requires affine-linear dependencies. The nonlinear functions must be transformed into affine-linear forms. This is acheived using the revised affine arithmetic. Revised affine form Arithmetic operations used in this work are defined as follows: Iwona Skalna, Krakow, Poland Small Workshop on Interval Methods’09, Lausanne

9. Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions Kliknij, aby edytować styl wzorca tytułu multiplication where Iwona Skalna, Krakow, Poland Small Workshop on Interval Methods’09, Lausanne

10. Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions Kliknij, aby edytować styl wzorca tytułu reciprocal where is a range of an affine form division where Iwona Skalna, Krakow, Poland Small Workshop on Interval Methods’09, Lausanne

11. Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions Kliknij, aby edytować styl wzorca tytułu Interval global optimization method produces hull solution for parametric linear systems with affine-linear dependencies which is en enclosure for the solution set of the original system with non-affine dependencies. The amount of the overestimation is verified using an evolutionary optimization method. Each evolutionary algorith has the following steps: Step 1. Initialize population P(t : 0) Step 2. Crossover P(t) Step 3. Mutation P(t) Step 4. Select P(t+1) from P(t) Step 5. t :t + 1 Step 6. If done then STOP else GOTO 2 Iwona Skalna, Krakow, Poland Small Workshop on Interval Methods’09, Lausanne

12. Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions Kliknij, aby edytować styl wzorca tytułu The results of evolutionary optimization depends strongly on parameters. Here, the following parameters: Population size: 16 Number of generations: 80 Crossover probability: 0.1 Mutation probability: 0.9 and the following genetic operators are used : non-uniform mutation arithmetic crossover Iwona Skalna, Krakow, Poland Small Workshop on Interval Methods’09, Lausanne

13. Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions Kliknij, aby edytować styl wzorca tytułu Example 1. Two dimensional systems with 5 parameters Iwona Skalna, Krakow, Poland Small Workshop on Interval Methods’09, Lausanne

14. Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions Kliknij, aby edytować styl wzorca tytułu Example 1. Two dimensional systems with 5 parameters Iwona Skalna, Krakow, Poland Small Workshop on Interval Methods’09, Lausanne

15. Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions Kliknij, aby edytować styl wzorca tytułu Example 3. Real-life problem of structure mechanics One-bay structural steel frame Iwona Skalna, Krakow, Poland Small Workshop on Interval Methods’09, Lausanne

16. Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions Kliknij, aby edytować styl wzorca tytułu Iwona Skalna, Krakow, Poland Small Workshop on Interval Methods’09, Lausanne

17. Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions Kliknij, aby edytować styl wzorca tytułu Global optimization method can be succesfully used for solving parametric linear systems whose input data are rational functions of interval parameters The main drawback of global optimization is the complexity. This deficiency can be overcome by parallel programming techniques The parallelism can be introduced both in the process of the monotonicity check and in the optimization process. This will be the subject of future work Iwona Skalna, Krakow, Poland Small Workshop on Interval Methods’09, Lausanne

18. Kliknij, aby edytować styl wzorca tytułu Thank you for your attention Iwona Skalna, Krakow, Poland Small Workshop on Interval Methods’09, Lausanne