multiobj_2009_10_21_02_2up

multiobj_2009_10_21_02_2up - 9-1 Multiobjective...

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9 - 1 Multiobjective Least-Squares S. Lall, Stanford 2009.10.21.02 9. Multiobjective Least-Squares Multi-objective least squares Trade-off curves Weighted-sum objectives Example: trade-off curve 9 - 2 Multiobjective Least-Squares S. Lall, Stanford 2009.10.21.02 Multi-Objective Least Squares in many problems we have two (or more) objectives we want J 1 = Ax y 2 small and also J 2 = Fx g 2 small usually the objectives are competing we can make one smaller at the expense of making the other larger common example: F = I, g = 0 : we want Ax y small, with small x
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x (1) x (2) x (3) J 1 J 2 9 - 3 Multiobjective Least-Squares S. Lall, Stanford 2009.10.21.02 trade-off curve shaded area shows ( J 2 , J 1 ) achieved by some x R n clear area shows ( J 2 , J 1 ) not achieved by any x R n boundary of region is called optimal trade-off curve corresponding x called Pareto optimal (for the objectives Ax y 2 , Fx g 2 ) three example choices of x : x (1) , x (2) , x (3) x (3) is worse than x (2) on both counts ( J 2 and J 1
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