Optimization of Linear Discrete Systems An Approach to the Staircase Problem
Optimization of Linear Discrete Systems An Approach to the Staircase Problem
Hubert Tavernier
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Assuming that we are given an objective function separable over time, we have to deal with infinite sequences. Two alternatives are open. 1) Introduce a discount factor such that any considered sequence will converge. 2) Use the criteria of optimality suggested by Gale. * a) A program x is said to be strongly optimal if it overtakes any other program, i. E. For the minimization case N * * Z[c(x)-c(x)]>0 V N>N t^ n n n n — — * b) A program x is said to be optimal if it catches up any othe progra...m, i. E. In the minimization case 17 V e>0, 3T s. T. E[c (x^) - c (x^)] > - e N > T . Nn nn — — e When dealing with such infinite horizon program, one has to be very careful for there does not necessarily exist an optimal program. A very good example of this phenomenon is provided by Gale with the sharing of a pie and a utility function c = c which is convex. Let e. = ( ^' 7. -.. Ao.... , o, ... ) n n n n n times The sequence (e } gives an increasing return and nevertheless, for z the uniform convergence topology converges toward (0, .
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