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Effective Finite Termination Procedures in InteriorPoint Methods for Linear Programming

, 1998
"... Due to the structure of the solution set an exact solution to... bounds are incorporated into the model to prevent the computed solution from violating the bound constraints. Theory in the literature is extended to the new model. Empirical evidence shows that the proposed model reduces the number of ..."
Abstract

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Due to the structure of the solution set an exact solution to... bounds are incorporated into the model to prevent the computed solution from violating the bound constraints. Theory in the literature is extended to the new model. Empirical evidence shows that the proposed model reduces the number of projection attempts needed to find an exact solution. When early termination is the goal, projection from a pure composite Newton step is advocated. However, the cost may exceed the benefits due to the average need of more than one projection attempt to find an exact solution. Variants of Mehrotra's predictorcorrector primaldual interiorpoint algorithm provide the foundation for most practical interiorpoint codes. To take advantage of all available algorithmic information, we investigate the behavior of the Tapia predictorcorrector indicator, which incorporates the corrector step, to identify the optimal partition. Globally, the Tapia predictorcorrector indicator behaves poorly as do all indicators, but locally exhibits fast convergence.
Optimal Face Identification Methods and Bounded Variable Linear Programs
, 1998
"... . We extend optimal face identification methods to bounded variable linear programming problems. Distance to the lower and upper bounds are incorporated into a projection model and MehrotraYe's solution technique to prevent the computed solution from violating the bound constraints. Empirical ..."
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. We extend optimal face identification methods to bounded variable linear programming problems. Distance to the lower and upper bounds are incorporated into a projection model and MehrotraYe's solution technique to prevent the computed solution from violating the bound constraints. Empirical and theoretical evidence are provided that support use of the new models to compute an exact solution. We also introduce a nonlinear weighted projection method to solve the optimal face identification problem. Key words. active set, finite termination, infeasible primaldual interiorpoint algorithms, linear programming, optimal face, optimal partition, polynomiality AMS subject classifications. 90C05, 49M35, 65K05 1. Introduction. Finite termination procedures are methods inserted into iterative algorithms to compute an exact solution in a finite number of steps. In particular, finite termination procedures can be added to interiorpoint methods to advance from an approximate solution to an ...
Evaluations of the 1997 Spend a Summer with a Scientist Program
"... There are several models that predict ambient (ground level) ozone pollution by transforming the data. However, transformations sometime increase prediction error. Our goal is to directly model ambient ozone levels in regions located away from monitoring stations. Our objective is not to predict fut ..."
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There are several models that predict ambient (ground level) ozone pollution by transforming the data. However, transformations sometime increase prediction error. Our goal is to directly model ambient ozone levels in regions located away from monitoring stations. Our objective is not to predict future ozone levels. Some Key Words: Multivariate skewnormal distribution; Bivariate skewnormal distribution; Ambient ozone. __________________________________ * Research supported by the Center for Research and Parallel Computation, Rice University, Houston, Texas ** Graduate Student, Department of Statistics, Rice University, Houston, Texas Introduction Ambient ozone (O 3 ) pollution is a major environmental problem in many metropolitan areas in this country. In particular, the Houston, Texas area is second to Los Angeles in nationwide ambient air quality (Carroll, Chen, George, Li, Newton, Schmiediche and Wang, 1997). There are several models for estimating O 3 , however, many modeling...
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"... In this thesis modelling and solution methods for portfolio optimisation are presented. The investigations reported in this thesis extend the Markowitz meanvariance model to the domain of quadratic mixed integer programming (QMIP) models which are 'NPhard ' discrete optimisation problems ..."
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In this thesis modelling and solution methods for portfolio optimisation are presented. The investigations reported in this thesis extend the Markowitz meanvariance model to the domain of quadratic mixed integer programming (QMIP) models which are 'NPhard ' discrete optimisation problems. In addition to the modelling extensions a number of challenging aspects of solution algorithms are considered. The relative performances of sparse simplex (SSX) as well as the interior point method (IPM) are studied in detail. In particular, the roles of 'warmstart ' and dual simplex are highlighted as applied to the construction of the efficient frontier which requires processing a family of problems; that is, the portfolio planning model stated in a parametric form. The method of solving QMIP models using the branch and bound algorithm is first developed; this is followed up by heuristics which improve the performance of the (discrete) solution algorithm. Some properties of the efficient frontier with discrete constraints are considered and a method of computing the discrete efficient frontier (DEF) efficiently is proposed. The computational investigation considers the efficiency and effectiveness in respect of the scale up properties of the proposed algorithm. The extensions of the real world models and the proposed solution algorithms make contribution as new knowledge. 1