Self-Scaled Cones and Interior-Point Methods in Nonlinear Programming

: This paper provides a theoretical foundation for efficient interior-point algorithms for nonlinear programming problems expressed in conic form, when the cone and its associated barrier are self-scaled. For such problems we devise long-step and symmetric primal-dual methods. Because of the special properties of these cones and barriers, our algorithms can take steps that go typically a large fraction of the way to the boundary of the feasible region, rather than being confined to a ball of unit radius in the local norm defined by the Hessian of the barrier. Key words: Nonlinear Programming, conical form, interior point algorithms, self-concordant barrier, self-scaled cone, self-scaled barrier, path-following algorithms, potential-reduction algorithms. AMS 1980 subject classification. Primary: 90C05, 90C25, 65Y20. CORE, Catholic University of Louvain, Louvain-la-Neuve, Belgium. E-mail: [email protected]. Part of this work was done while the author was visiting the Cornell C..

Self-Scaled Barriers and Interior-Point Methods for Convex Programming

Self-Scaled Barriers and Interior-Point Methods for Convex Programmin

Assimilation of industrial production of economic hot-rolled shapes in the 11th Five-Year Plan

SIGLEAvailable from British Library Document Supply Centre- DSC:5828.4(M--37249)T / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Comparison of Bundle and Classical Column Generation

When a column generation approach is applied to decomposable mixed integer programming problems, it is standard to formulate and solve the master problem as a linear program. Seen in the dual space, this results in the algorithm known in the nonlinear programming community as the cutting-plane algorithm of Kelley and Cheney-Goldstein. However, more stable methods with better theoretical convergence rates are known and have been used as alternatives to this standard. One of them is the bundle method; our aim is to illustrate its differences with Kelley's method. In the process we review alternative stabilization techniques used in column generation, comparing them from both primal and dual points of view. Numerical comparisons are presented for five applications: cutting stock (which includes bin packing), vertex coloring, capacitated vehicle routing, multi-item lot sizing, and traveling salesman. We also give a sketchy comparison with the volume algorithm