Algorithms and Complexity for Continuous Problems

s of Talks (in alphabetical order by speaker’s surname) Probabilistic Analysis of Interior Point Methods for Linear Programming Karl Heinz Borgwardt joint work with Petra Huhn It is our aim to make a fair comparison of the Simplex Method and Interior Point Methods on the basis of their average case behavior in the solution of Linear Programming problems possible. Therefore we use the same stochastic model (the Rotation Symmetry Model ) for the average case analysis of both solution methods. Interior Point Methods run in three Phases. In Phase I it is the aim to get close to the analytic center of the feasible region. In Phase IIa we perform an iteration process which reduces the distance to the optimum at least with linear convergence. And in Phase IIb one starts a search for a closeby vertex from the last iteration point of Phase IIa. We demonstrate that for worst-case and for average-case behavior the following geometric figures determine the number of iterations: 1. The maximal distance of a vertex to the origin (the maximal vertex norm) is the crucial measure for the effort in Phase I. 2. The difference of the objective values at the best and the second best vertex determines the number of iterations required in Phase IIa. In worst case polynomiality-proofs these two figures can only be bounded by use of the encoding length L of the problem. And this is the reason why worst case bounds contain this extremely high factor and that polynomiality but not strong polynomiality can be shown. In the average case analysis we calculate the distribution functions of the two geometric figures mentioned above. And this can be via the evaluation of integral formulas used to achieve bounds on the expected number of iterations for all Phases. Our result of that stochastic analysis leads to a proof that the expected number of iterations for the whole method is not only polynomial in the encoding length but also strongly polynomial in the dimensions of the problem. Impossibility of Exponential Convergence Rate for Optimization on the Wiener Space Jim Calvin It is possible to approximate the minimum of a unimodal function on an interval with the worst case error converging to zero at an exponential rate (for example, using the Fibonacci search algorithm). Without strong assumptions such as unimodality, no such worst case bound is available. In a stochastic setting we can obtain probabilistic bounds for large classes of functions. Let P be the Wiener measure, and choose a sequence of numbers cn 0 arbitrarily slowly. Then there exists an algorithm such that P ∆n exp cnn 1 where ∆n is the approximation error after n observations. The purpose of this talk is to show that P ∆n exp cn 0 for any algorithm and any c 0.