Solving a Class of LP Problems with a Primal-Dual Logarithmic Barrier Method

Applying a higher order primal-dual logarithmic barrier method for solving large real-life linear programming problems is addressed in this paper. The efficiency of interior point algorithm on these problems is compared with the one of the state-of-the-art simplex code MINOS version 5.3. Based on such experience, a wide class of LP problems is identified for which logarithmic barrier approach seems advantageous over the simplex one. Additionally, some practical rules for model builders are derived that should allow them to create problems that can easily be solved with logarithmic barrier algorithms

Adapting the interior point method for the solution of linear programs on high performance computers

In this paper we describe a unified algorithmic framework for the interior point method (IPM) of solving Linear Programs (LPs) which allows us to adapt it over a range of high performance computer architectures. We set out the reasons as to why IPM makes better use of high performance computer architecture than the sparse simplex method. In the inner iteration of the IPM a search direction is computed using Newton or higher order methods. Computationally this involves solving a sparse symmetric positive definite (SSPD) system of equations. The choice of direct and indirect methods for the solution of this system and the design of data structures to take advantage of coarse grain parallel and massively parallel computer architectures are considered in detail. Finally, we present experimental results of solving NETLIB test problems on examples of these architectures and put forward arguments as to why integration of the system within sparse simplex is beneficial

Experimental investigations in combining primal dual interior point method and simplex based LP solvers

The use of a primal dual interior point method (PD) based optimizer as a robust linear programming (LP) solver is now well established. Instead of replacing the sparse simplex algorithm (SSX), the PD is increasingly seen as complementing it. The progress of PD iterations is not hindered by the degeneracy or the stalling problem of the SSX, indeed it reaches the 'near optimum' solution very quickly. The SSX algorithm, in contrast, is not affected by the boundary conditions which slow down the convergence of the PD. If the solution to the LP problem is non unique, the PD algorithm converges to an interior point of the solution set while the SSX algorithm finds an extreme point solution. To take advantage of the attractive properties of both the PD and the SSX, we have designed a hybrid framework whereby cross over from PD to SSX can take place at any stage of the PD optimization run. The cross over to SSX involves the partition of the PD solution set to active and dormant variables. In this paper we examine the practical difficulties in partitioning the solution set, we discuss the reliability of predicting the solution set partition before optimality is reached and report the results of combining exact and inexact prediction with SSX basis recovery

Numerical Methods for Model Predictive Control

This thesis mainly deals with the extended linear quadratic control problem, that is a special case of equality constrained quadratic program. A number of other problems in optimal control and estimation of linear systems can be reduced to this form. Furthermore, it arises as sub-problem in sequential quadratic programs methods and interior-point methods for the solution of optimal control and estimation in case of non-linear systems and in presence of inequality constraints. This thesis can be divided into two parts. In the first part, a number of methods for the solution of the extended linear quadratic control problem are presented and analyzed. These methods have been implemented in efficient C code and compared each other. In the second part, this problem is expanded taking into account also inequality constraints. Two interior-point methods are presented and analyzed. Both methods have been implemented in C code and compared each othe

Computational analysis of real-time convex optimization for control systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2000.Includes bibliographical references (p. 177-189).Computational analysis is fundamental for certification of all real-time control software. Nevertheless, analysis of on-line optimization for control has received little attention to date. On-line software must pass rigorous standards in reliability, requiring that any embedded optimization algorithm possess predictable behavior and bounded run-time guarantees. This thesis examines the problem of certifying control systems which utilize real-time optimization. A general convex programming framework is used, to which primal-dual path-following algorithms are applied. The set of all optimization problem instances which may arise in an on-line procedure is characterized as a compact parametric set of convex programming problems. A method is given for checking the feasibility and well-posedness of this compact set of problems, providing certification that every problem instance has a solution and can be solved in finite time. The thesis then proposes several algorithm initialization methods, considering the fixed and time-varying constraint cases separately. Computational bounds are provided for both cases. In the event that the computational requirements cannot be met, several alternatives to on-line optimization are suggested. Of course, these alternatives must provide feasible solutions with minimal real-time computational overhead. Beyond this requirement, these methods approximate the optimal solution as well as possible. The methods explored include robust table look-up, functional approximation of the solution set, and ellipsoidal approximation of the constraint set. The final part of this thesis examines the coupled behavior of a receding horizon control scheme for constrained linear systems and real-time optimization. The driving requirement is to maintain closed-loop stability, feasibility and well-posedness of the optimal control problem, and bounded iterations for the optimization algorithm. A detailed analysis provides sufficient conditions for meeting these requirements. A realistic example of a small autonomous air vehicle is furnished, showing how a receding horizon control law using real-time optimization can be certified.by Lawrence Kent McGovern.Ph.D

HOPDM Modular Solver for LP Problems User's Guide to version 2.12

The paper provides a description of HOPDM, a library of routines for solving large scale linear programming problems and its implementation at IIASA. HOPDM stands for Higher Order Primal Dual Method. The algorithm implemented in HOPDM is a new variant of a primal-dual logarithmic barrier method that uses multiple correctors of centrality. The newest version of the library -- HOPDM 2.12 -- is a robust and efficient LP code that compares favorably with the up to date commercial solvers. The paper contains an outline of the algorithm implemented in HOPDM and information about results of tests done with large LP problems developed at IIASA. Finally, the paper provides with details of the implementation of HOPDM and its use at IIASA, as well as with information about availability of the portable version of the HOPDM library

A primal-dual decomposition based interior point approach to two-stage stochastic linear programming

Decision making under uncertainty is a challenge faced by many decision makers. Stochastic programming is a major tool developed to deal with optimization with uncertainties that has found applications in, e.g. finance, such as asset-liability and bond-portfolio management. Computationally however, many models in stochastic programming remain unsolvable because of overwhelming dimensionality. For a model to be well solvable, its special structure must be explored. Most of the solution methods are based on decomposing the data. In this paper we propose a new decomposition approach for two-stage stochastic programming, based on a direct application of the path-following method combined with the homogeneous self-dual technique. Numerical experiments show that our decomposition algorithm is very efficient for solving stochastic programs. In particular, we apply our deompostition method to a two-period portfolio selection problem using options on a stock index. In this model the investor can invest in a money-market account, a stock index, and European options on this index with different maturities. We experiment our model with market prices of options on the S&P500