A geometric method in nonlinear programming

Tanabe, Kunio

doi:10.1007/bf00934495

Cited by 66 publications

(77 citation statements)

References 24 publications

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“…Due to condition C 2 we have the complimentarity condition x i * v i * = 0, 1 ≤ i ≤ n. Hence we conclude that the pair x * , u * defined by (51) forms Kuhn-Tucker point in Problem (18).…”

Section: Barrier-newton Methods For Linear Programmingmentioning

confidence: 72%

“…In this section we apply barrier-Newton method (15) for solving linear programming Problem (18). In this case we have…”

Section: Barrier-newton Methods For Linear Programmingmentioning

confidence: 99%

“…This means that if τ > 0, then method (20) has a remarkable property: all its trajectories approach the feasible set as t tends to infinity and the polyhedra X is an asymptotically stable attractor for the system (see [6,11,18]). …”

Section: Introductionmentioning

confidence: 99%

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Stable barrier-projection and barrier-Newton methods in linear programming

Evtushenko

Жадан

1994

Comput Optim Applic

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Dedicated to Professor George B. Dantzig on the occasion of his eightieth birthday.Abstract. The present paper is devoted to the application of the space transformation techniques for solving linear programming problems. By using a surjective mapping the original constrained optimization problem is transformed to a problem in a new space with only equality constraints. For the numerical solution of the latter problem the stable version of the gradient-projection and Newton's methods are used. After an inverse transformation to the original space a family of numerical methods for solving optimization problems with equality and inequality constraints is obtained. The proposed algorithms are based on the numerical integration of the systems of ordinary differential equations. These algorithms do not require feasibility of the starting and current points, but they preserve feasibility. As a result of a space transformation the vector fields of differential equations are changed and additional terms are introduced which serve as a barrier preventing the trajectories from leaving the feasible set. A proof of a convergence is given.

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Section: Barrier-newton Methods For Linear Programmingmentioning

confidence: 72%

“…In this section we apply barrier-Newton method (15) for solving linear programming Problem (18). In this case we have…”

Section: Barrier-newton Methods For Linear Programmingmentioning

confidence: 99%

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Stable barrier-projection and barrier-Newton methods in linear programming

Evtushenko

Жадан

1994

Comput Optim Applic

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“…But their concerns were far removed from the complexity of optimization algorithms. Also, differential and Riemannian geometry has been studied in optimization in relation to the need to remain feasible -see, e.g., Tanabe [11] and Edelman et al [1]. In this paper, however (at least in feasible methods), the algorithms move in the intersection of an affine manifold and the interior of the constraint set, and thus maintaining feasibility is not an issue.…”

Section: Introductionmentioning

confidence: 97%

On the Riemannian Geometry Defined by Self-Concordant Barriers and Interior-Point Methods

Nesterov¹,

Todd²

2002

Foundations of Computational Mathematics

107

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We consider the Riemannian geometry defined on a convex set by the Hessian of a selfconcordant barrier function, and its associated geodesic curves. These provide guidance for the construction of efficient interior-point methods for optimizing a linear function over the intersection of the set with an affine manifold. We show that algorithms that follow the primal-dual central path are in some sense close to optimal. The same is true for methods that follow the shifted primal-dual central path among certain infeasible-interior-point methods. We also compute the geodesics in several simple sets.

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“…Most of the available algorithms, such as the widely used distributed algorithms based on subgradient [14] and projected subgradient [15], are developed in discrete-time mainly due to the overwhelming ability of digital computers to execute the algorithms discretely. Recently, more and more distributed convex optimization algorithms are explored in continuous-time since continuous-time set up is favored for utilizing more techniques (the elegant Lyapunov argument in [4] for example) to prove the algorithm convergence, and is beneficial for adopting differential geometry viewpoint which is extremely powerful when the optimization is constrained (see for example [21]). …”

Section: Introductionmentioning

confidence: 99%