This article considers continuous trajectories of the vector fields induced by primal-dual potential-reduction algorithms for solving linear programming problems. It is known that these trajectories converge to the analytic center of the primal-dual optimal face. We establish that this convergence may be tangential to the central path, tangential to the optimal face, or in between, … Read more
Solving systems of linear equations with “normal” matrices of the form $A D^2 A^T$ is a key ingredient in the computation of search directions for interior-point algorithms. In this article, we establish that a well-known basis preconditioner for such systems of linear equations produces scaled matrices with uniformly bounded condition numbers as $D$ varies over … Read more
We perform a smoothed analysis of the termination phase of an interior-point method. By combining this analysis with the smoothed analysis of Renegar’s interior-point algorithm by Dunagan, Spielman and Teng, we show that the smoothed complexity of an interior-point algorithm for linear programming is $O (m^{3} \log (m/\sigma ))$. In contrast, the best known bound … Read more
We survey how semidefinite programming can be used for finding good approximative solutions to hard combinatorial optimization problems. Citation Preliminary version appeared as Report PNA-R0210, CWI, Amsterdam, April 2002. To appear as Chapter in the Handbook on Discrete Optimization, K. Aardal, G. Nemhauser, R. Weismantel, eds., Elsevier Publishers. Article Download View Semidefinite programming and integer … Read more
This paper describes an active-set algorithm for large-scale nonlinear programming based on the successive linear programming method proposed by Fletcher and Sainz de la Maza. The step computation is performed in two stages. In the first stage a linear program is solved to estimate the active set at the solution. The linear program is obtained … Read more
We study the issue of updating the analytic center after multiple cutting planes have been added through the analytic center of the current polytope in Euclidean n-space. This is an important issue that arises at every `stage’ in a cutting plane algorithm. If q cuts are to be added, with q no larger than n, … Read more
There is a large number of implementational choices to be made for the primal-dual interior point method in the context of mixed semidefinite and second order cone optimization. This paper presents such implementational issues in a unified framework, and compares the choices made by different research groups. This is also the first paper to provide … Read more
This paper considers the following inverse optimization problem: given a linear program, a desired optimal objective value, and a set of feasible cost coefficients, determine a cost-coefficient vector such that the corresponding optimal objective value of the linear program is closest to the given value. The above problem, referred here as the inverse optimal value … Read more
Current mixed-integer linear programming solvers are based on linear programming routines that use floating point arithmetic. Occasionally, this leads to wrong solutions, even for problems where all coefficients and all solution components are small integers. It is shown how, using directed rounding and interval arithmetic, cheap pre- and postprocessing of the linear programs arising in … Read more
It was recently shown that, unlike in linear optimization, the central path in semidefinite optimization (SDO) does not converge to the analytic center of the optimal set in general. In this paper we analyze the limiting behavior of the central path to explain this unexpected phenomenon. This is done by deriving a new necessary and … Read more