Every iteration of an interiorpoint method of large scale linearprogramming requires computing at least one orthogonal projection. In practice, Cholesky decomposition seems to be the most efficient and sufficiently ...
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Every iteration of an interiorpoint method of large scale linearprogramming requires computing at least one orthogonal projection. In practice, Cholesky decomposition seems to be the most efficient and sufficiently stable method. We studied the 'column orientated' or 'left looking 'sparse variant of the Cholesky decomposition, which is a very popular method in large scale optimization. We show some techniques such as using supernodes and loop unrolling for improving the speed of computation. We show numerical results on a wide variety of large scale, real-life linearprogramming problems.
In this work, we first study in detail the formulation of the primal-dual interior-point method for linearprogramming. We show that, contrary to popular belief, it cannot be viewed as a damped Newton method applied t...
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In this work, we first study in detail the formulation of the primal-dual interior-point method for linearprogramming. We show that, contrary to popular belief, it cannot be viewed as a damped Newton method applied to the Karush-Kuhn-Tucker conditions for the logarithmic barrier function problem. Next, we extend the formulation to general nonlinearprogramming, and then validate this extension by demonstrating that this algorithm can be implemented so that it is locally and Q-quadratically convergent under only the standard Newton method assumptions. We also establish a global convergence theory for this algorithm and include promising numerical experimentation.
A fundamental homotopy-based linearprogramming algorithm, which utilizes Euler-predictor Newton-corrector steps with restarts. is formulated and investigated numerically on problems representative of linear programs ...
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A fundamental homotopy-based linearprogramming algorithm, which utilizes Euler-predictor Newton-corrector steps with restarts. is formulated and investigated numerically on problems representative of linear programs that arise in practice. A rich array of refinements of this basic algorithm are possible within the homotopy framework. Such refinements are needed in any practical implementation and are discussed in detail. Implications for the design of integrated large-scale mathematical programming software are also briefly considered.
Recently, Todd has analyzed in detail the primal-dual affine-scaling method for linearprogramming, which is close to what is implemented in practice, and proved that it may take at least n(1/3) iterations to improve ...
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Recently, Todd has analyzed in detail the primal-dual affine-scaling method for linearprogramming, which is close to what is implemented in practice, and proved that it may take at least n(1/3) iterations to improve the initial duality gap by a constant factor. He also showed that this lower bound holds for some polynomial variants of primal-dual interior-pointmethods, which restrict all iterates to certain neighborhoods of the central path. In this paper, we further extend his result to long-step primal-dual variants that restrict the iterates to a wider neighborhood. This neigh-borhood seems the least restrictive one to guarantee polynomiality for primal-dual path-following methods, and the variants are also even closer to what is implemented in practice.
The Thevenin theorem, one of the most celebrated results of electric circuit theory, provides a two-parameter characterization of the behavior of an arbitrarily large circuit, as seen from two of its terminals. We int...
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The Thevenin theorem, one of the most celebrated results of electric circuit theory, provides a two-parameter characterization of the behavior of an arbitrarily large circuit, as seen from two of its terminals. We interpret the theorem as a sensitivity result in an associated minimum energy/network flow problem, and we abstract its main idea to develop a decomposition method for convex quadratic programming problems with linear equality constraints, of the type arising in a variety of contexts such as the Newton method, interiorpointmethods, and least squares estimation. Like the Thevenin theorem, our method is particularly useful in problems involving a system consisting of several subsystems, connected to each other with a small number of coupling variables.
We present a simplification and generalization of the recent homogeneous and self-dual linearprogramming (LP) algorithm. The algorithm does not use any Big-M initial point and achieves O(root nL)-iteration complexity...
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We present a simplification and generalization of the recent homogeneous and self-dual linearprogramming (LP) algorithm. The algorithm does not use any Big-M initial point and achieves O(root nL)-iteration complexity, where n and L are the number of variables and the length of data of the LP problem. It also detects LP infeasibility based on a provable criterion. Its preliminary implementation with a simple predictor and corrector technique results in an efficient computer code in practice. In contrast to other interior-pointmethods, our code solves NETLIB problems, feasible or infeasible, starting simply from x = e (primal variables), y = 0 (dual variables), z = e (dual slack variables), where e is the vector of all ones. We describe our computational experience in solving these problems, and compare our results with OB1.60, a state-of-the-art implementation of interior-point algorithms.
The introduction of Karmarkar's polynomial algorithm for linearprogramming (LP) in 1984 has influenced wide areas in the field of optimization. While in the 1980s emphasis was on developing and implementing effic...
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The introduction of Karmarkar's polynomial algorithm for linearprogramming (LP) in 1984 has influenced wide areas in the field of optimization. While in the 1980s emphasis was on developing and implementing efficient variants of interiorpointmethods for LP, the 1990s have shown applicability to certain structured nonlinearprogramming and combinatorial problems. We will give a historical account of the developments and illustrate the typical results by analyzing a new method for computing the smallest eigenvalue of a matrix. We formulate this latter problem as a so-called semidefinite optimization problem. Semidefinite optimization has recently gained much attention since it has a lot of applications in various fields (like control and system theory, combinatorial optimization, algebra, statistics, structural design) and semidefinite problems can be efficiently solved with interiorpointmethods.
My Ph. D, research concerned the algorithmic issues of the interiorpointmethods, My basic idea at the very beginning was to break with the "traditional" normal equations approarli and elaborate the increas...
My Ph. D, research concerned the algorithmic issues of the interiorpointmethods, My basic idea at the very beginning was to break with the "traditional" normal equations approarli and elaborate the increased freedom of the augrriented syslciri. An important point in my research was to realize that these appruar-lies r-mnplenu-nt each other. To exploit the advantages of both the allgrriented systeiu and normal equations ap1n'uaL~l1es, a ncw heuristics was developed which is able to remgnize the structure of the linearprogramming problems and to make algorithmical decisions. Both in qualitative sense and computationally. my method shows its advantage i11 comparison with any other method applied so far. The efficient use of the augmented system approach raised special questions. Actually, the interiorpointmethods of linearprogramming with upper/ lower bounds on variables and ranges on constraints can be handled using quasidcfinitc matrices [72]. Unfortunately, linear programs. as formulated usually do not fit into this form. The other method used in context of solving indefinite system of equations is the Bunch-Parlett factorization, but it tend to be more computationally burdensome. I developed a new approach for computing a symmetric decomposition of an indefinite system. It does not require the quasidefinite property (and the bounded variable and range constraint formulation of the linearprogramming problem), but it uses 1 X 1 pivots during the decomposition. As such. my approach can be regarded as the common generalization of the Bunch-Parlett decomposition and the quasidefinite theory, where the good properties of both approaches are preserved. I view this as main contribution of my Ph. D. research. I investigated all important parts of the efficient implementation of the "linear algebra kernel" of interiorpointmethods. A new technique was proposed for symbolical ordering. It gives a connection between the minimum degree and minimum local fill-in heuristics throug
In this paper, we briefly describe two interior-point algorithms for semidefinite programming. At each iteration, both these algorithms compute search directions by solving a linear system. We discuss some preliminary...
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In this paper, we briefly describe two interior-point algorithms for semidefinite programming. At each iteration, both these algorithms compute search directions by solving a linear system. We discuss some preliminary experiments for moderately sized, block diagonal semidefinite programs, comparing direct and iterative methods for solving the linear systems.
Solving a linearprogramming problem by perturbing its primal objective functiun with a barrier or penalty function for the development of interior-pointmethods has attracted much attention recently. However, the ide...
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Solving a linearprogramming problem by perturbing its primal objective functiun with a barrier or penalty function for the development of interior-pointmethods has attracted much attention recently. However, the idea of perturbing the feasible region has not been fully explored. In this paper , we propose such an approach. Given a linear program with inequality constraints, aperturbed feasible region based on a measure of “constraint violation” inlp-norm is defined . Th e optimal so lution to the perturbed program is shown to converge to an optimal solution of the given linear program as thelp-norm tends to thel∞, -norm. A simple formula which converts the optimal solution of a perturbed program to a dual feasible solution, which in turn converges to a dual optimal solution, of the given linear program is provided . In addition, an ε-optimality theory that specifies the perturbation parameter in terms of required accuracy level ε and other program parameters is studied
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