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SWP-3453-26776383

# SWP-3453-26776383 - Following a"Balanced Trajectory from an...

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Following a "Balanced" Trajectory from an Infeasible Point to an Optimal Linear Programming Solution with a Polynomial-time Algorithm Robert M. Freund WP# 3453-92-MSA July, 1992

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Following a "Balanced" Trajectory from an Infeasible Point to an Optimal Linear Programming Solution with a Polynomial-time Algorithm July 1992 Robert M. Freund Research supported in part by NSF, AFOSR, and ONR through NSF grant DMS- 8920550, and in part by the MIT-NTU Collaboration Research Fund. Part of this research was performed when the author was a Visiting Scientist at the Center for Applied Mathematics, Cornell University, in 1991.
Abstract: This paper is concerned with the problem of following a trajectory from an infeasible "warm start" solution of a linear programming problem, directly to an optimal solution of the linear programming problem. A class of trajectories for the problem is defined, based on the notion of a -balanced solution to the "warm start" problem. Given a prespecified positive balancing constant , an infeasible solution x is said to be P-balanced if the optimal value gap is less than or equal to [3 times the infeasibility gap. Mathematically, this can be written as cTx - z* < 34Tx, where the linear form t x is the Phase I objective function. The concept of a -balanced solution is used to define a class of trajectories from an infeasible points to an optimal solution of a given linear program. Each trajectory has the property that all points on or near the trajectory (in a suitable metric) are -balanced. The main thrust of the paper is the development of an algorithm that traces a given -balanced trajectory from a starting point near the trajectory to an optimal solution to the given linear programming problem in polynomial-time. More specifically, the algorithm allows for fixed improvement in the bound on the Phase I and the Phase II objectives in O(n) iterations of Newton steps. Key Words: Linear program, interior-point algorithm, polynomial-time complexity, trajectory method, Newton's method. Running Header: Balanced Trajectory for Linear Programming 2

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III 1. Introduction: This paper is concerned with following a trajectory from an infeasible "warm start" solution of a linear programming problem, directly to an optimal solution of the linear programming problem. By a "warm start" solution, we mean a solution that is hopefully close to being feasible and is hopefully close to being optimal. Like other research on the "warm start" problem, this paper is motivated by the need in practice to solve many amended versions of the same base-case linear programming problem. In this case, it makes sense to use the optimal solution to the previous version of the problem as a "warm start" solution to the current amended version of the problem. Whereas this strategy has been successfully employed in simplex-based algorithms for solving linear programming problems, there is no guarantee that it will improve solution times, due to the inherent combinatorial nature of the simplex algorithm. However, in the case of interior-point algorithms, there is no
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SWP-3453-26776383 - Following a"Balanced Trajectory from an...

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