Kliknij, aby edytować styl wzorca tytułu 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 Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Iwona Skalna, PolandSmall Workshop on Interval Methods09, Lausanne 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, PolandSmall Workshop on Interval Methods09, Lausanne 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 whereare nonlinear continuous functions of parameters with coefficients Kliknij, aby edytować styl wzorca tytułu Small Workshop on Interval Methods09, LausanneIwona Skalna, Krakow, Poland Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions

The goal is to find the thightest interval enclosure for S, possibly the interval hull solution defined as Parametric (united) solution set is define as Kliknij, aby edytować styl wzorca tytułu Small Workshop on Interval Methods09, LausanneIwona Skalna, Krakow, Poland 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 Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions

In general case, to calculate the hull solution, the following 2n constrained optimization problems must be solved where is an objective function Kliknij, aby edytować styl wzorca tytułu Small Workshop on Interval Methods09, LausanneIwona Skalna, Krakow, Poland Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions

The optimizations problems are solved using an interval global optimization. The interval global optimization algorithm has the following steps: 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. Step 1. Initialize the list L =(pq, x(pq)) Step 2. Remove (pq, x(pq)) from the list L Step 3. Bisect pq = pq 1 pq 2 Step 4. Calculate x(pq i ), pq i Step 5. Perform the monotonicity test Step 6. If w(pq) < STOP else GOTO 2 Kliknij, aby edytować styl wzorca tytułu Small Workshop on Interval Methods09, LausanneIwona Skalna, Krakow, Poland Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions

If a devirative has constant sing, then the corresponding interval can be reduced to one of its edges. or 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: Kliknij, aby edytować styl wzorca tytułu Small Workshop on Interval Methods09, LausanneIwona Skalna, Krakow, Poland Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions The Direct Method is also used to calculate inclusion function for the objective function x(p,q) is

Direct Method requires affine-linear dependencies. The nonlinear functions must be transformed into affine-linear forms. This is acheived using the revised affine arithmetic. Kliknij, aby edytować styl wzorca tytułu Arithmetic operations used in this work are defined as follows: Small Workshop on Interval Methods09, LausanneIwona Skalna, Krakow, Poland Revised affine form Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions

multiplication Kliknij, aby edytować styl wzorca tytułu Small Workshop on Interval Methods09, LausanneIwona Skalna, Krakow, Poland Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions where

reciprocal division Kliknij, aby edytować styl wzorca tytułu Small Workshop on Interval Methods09, LausanneIwona Skalna, Krakow, Poland Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions whereis a range of an affine form where

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 Kliknij, aby edytować styl wzorca tytułu Small Workshop on Interval Methods09, LausanneIwona Skalna, Krakow, Poland Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions

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 Kliknij, aby edytować styl wzorca tytułu Small Workshop on Interval Methods09, LausanneIwona Skalna, Krakow, Poland and the following genetic operators are used : non-uniform mutation arithmetic crossover Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions

Example 1. Two dimensional systems with 5 parameters Kliknij, aby edytować styl wzorca tytułu Small Workshop on Interval Methods09, LausanneIwona Skalna, Krakow, Poland Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions

Example 1. Two dimensional systems with 5 parameters Kliknij, aby edytować styl wzorca tytułu Small Workshop on Interval Methods09, LausanneIwona Skalna, Krakow, Poland Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions

Example 3. Real-life problem of structure mechanics Kliknij, aby edytować styl wzorca tytułu Small Workshop on Interval Methods09, LausanneIwona Skalna, Krakow, Poland Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions One-bay structural steel frame

Kliknij, aby edytować styl wzorca tytułu Small Workshop on Interval Methods09, LausanneIwona Skalna, Krakow, Poland Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions

Kliknij, aby edytować styl wzorca tytułu Small Workshop on Interval Methods09, LausanneIwona Skalna, Krakow, Poland 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 Outline Parametric linear systems Optimization problem Global optmization Monotonicity test Revised affine arithmetic Evolutionary optimization Examples Conclusions

Kliknij, aby edytować styl wzorca tytułu Small Workshop on Interval Methods09, LausanneIwona Skalna, Krakow, Poland