Interior Point Methods for Nonconvex Nonlinear Programming

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The work will continue to develop the algorithm of the LOQO interior point code for general continuous mathematical programming problems, and expand the range of problems to which it can be applied. Areas of research include betterdetection of infeasibility and unboundedness, better detection of linear dependence of the constraints, and better estimates of initial estimates to the optimum point. Work on higher order methods, which can greatly improve algorithmic efficiency, will be continued, with particular interest in applying filter methods. In an attempt to expand the scope of the types of problems that the algorithm can solve, the research will be concerned with solving discrete problems using interior point methods. On type of problem considered will be mixed integer quadratic programming. Problems of this type commonly arise in finance. Mixed integer linear programming problems will also be studied, with an initial focus on quickly finding feasible solutions. Further work will be concerned with general complementarity problems, especially those that arise in economics and engineering. In all cases, appropriate collections of problems arising from real applications will be modeled in AMPL. These will be made available via the internet, as will extensions to the LOQO program.The LOQO code for solving a variety of types of mathematical programming problems has been under development for the past four years as a joint research effort of the principal investigator and Professor Robert Vanderbei of Princeton University. The code is made freely available via the internet, and to date has been downloaded several thousand times. It is used to solve problems as diverse as engineering design, portfolio optimization, airline scheduling problems, and is currently being adapted for various projects in medical research. The proposed research is concentrated both on improving the efficiency of the code and increasing the set of problems for which the code can be used. The new types of problems for which it is hoped the code can be adapted efficiently are problems with variables that must be an integer, generally 0 or 1. These problems are extremely common, and of great practical value. For example, when assigning aircraft to routes, a whole plane must be assigned to a specific route, not some fraction of a plane. Including integer variables within the context an algorithm of this type is very new, and will be approached initially through important specific applications.