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We describe a potential reduction method for convex optimization problems involving matrix inequalities. The method is based on the theory developed by Nesterov and Nemirovsky and generalizes Gonzaga and Todd's method for linear programming. A worst-case analysis shows that the number of iterations grows as the square root of the problem size, but in practice it appears to grow more slowly. As in other interior-point methods the overall computational effort is therefore dominated by the least-squares system that must be solved in each iteration. A type of conjugate-gradient algorithm can be used for this purpose, which results in important savings for two reasons. First, it allows us to take advantage of the special structure the problems often have (e.g., Lyapunov or algebraic Riccati inequalities). Second, we show that the polynomial bound on the number of iterations remains valid even if the conjugate-gradient algorithm is not run until completion, which in practice can greatly reduce the computational effort per iteration.

Many problems in the field of computational biology consist of the analysis of so-called gene-expression data. The successful application of approximation and optimization techniques, dynamical systems, algorithms and the utilization of the underlying combinatorial structures lead to a better understanding in that field. For the concrete example of gene-expression data we extend an algorithm, which exploits discrete information. This is lying in extremal points of polyhedra, which grow step by step, up to a possible stopping. We study gene-expression data in time, mathematically model it by a time-continuous system, and time-discretize this system. By our algorithm we compute the regions of stability and instability. We give an motivating introduction from genetics, present biological and mathematical interpretations of (in)stability, point out structural frontiers and give an outlook to future research.

We describe several line search strategies in recent potential reduction algorithms for linear programming. We clarify some concerns about the step size of the original algorithm. In particular, we illustrate that the dual step of the algorithm can be sufficiently "long". We also discuss some other implementation issues for the algorithm. Keywords: Linear programming, line search, potential reduction algorithms. Abbreviated title: Line Search in Potential Reduction Algorithms y Department of Management Sciences, The University of Iowa, Iowa City, Iowa 52242 Since Karmarkar proposed his projective algorithm [5], various primal-dual potential reduction algorithms for linear programming have been developed by Anstreicher and Bosch [1], Freund [2], Gonzaga and Todd [4], Kojima, Mizuno and Yoshise [6], Liu and Goldfarb [7], McShane, Monma and Shanno [8], and Ye [10][11] among others. All of these algorithms are based on reducing a primal-dual potential function that is first appeared in T...

There are several classes of interior point algorithms that solve linear programming problems in O(Vin L) iterations, but it is not known whether this bound is tight for any interior point algorithm. Among interior point algorithms, several potential reduction algorithms combine both theoretical (O(+/E L) iterations) and practical efficiency as they allow the flexibility of line searches in the potential function and thus can lead to practical implementations. It is a significant open question whether interior point algorithms can lead to better complexity bounds. In the present paper we give some negative answers to this question for the class of potential reduction algorithms. We show that, without line searches in the potential function, the bound O(v/i- L) is tight for several potential reduction algorithms, i.e., there is a class of examples, in which the algorithms need at least Q(v/i L) iterations to find an optimal solution. In addition, we show that for a class of potential functions, even if we allow line searches in the potential function, the bounds are still tight. We note that it is the first time that tight bounds are obtained for any interior point algorithm. III 1

: We provide a probabilistic analysis of the second order term that arises in pathfollowing algorithms for linear programming. We use this result to show that two such methods, algorithms generating a sequence of points in a neighborhood of the central path and in its relaxation, require a worst-case number of iterations that is O(nL) and an anticipated number of iterations that is O(log(n)L). The second neighborhood spreads almost all over the feasible region so that the generated points are close to the boundary rather than the central path. We also Research was supported in part by Grant-in-Aid 63490010 for General Scientific Research of the Ministry of Education, Science and Culture, Japan. y Research was supported in part by NSF grant DMS-8904406 and ONR contract N-00014-87-K0212. propose a potential reduction algorithm which requires the same order of number of iterations as the path-following algorithms. Key words: Linear Programming, interior point algorithms, path-foll...