This paper proposes a globally convergent predictor-corrector infeasible-interior-point algorithm for the monotone semidefinite linear complementarity problem using the Alizadeh-Haeberly-Overton search direction, and ...
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This paper proposes a globally convergent predictor-corrector infeasible-interior-point algorithm for the monotone semidefinite linear complementarity problem using the Alizadeh-Haeberly-Overton search direction, and shows its quadratic local convergence under the strict complementarity condition.
Sufficient conditions are given for the Q-superlinear convergence of the iterates produced by primal-dual interior-point methods for linear complementarity problems. It is shown that those conditions are satisfied by ...
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Sufficient conditions are given for the Q-superlinear convergence of the iterates produced by primal-dual interior-point methods for linear complementarity problems. It is shown that those conditions are satisfied by several well known interior-point methods. In particular it is shown that the iteration sequences produced by the simplified predictor-corrector method of Gonzaga and Tapia, the simplified largest step method of Gonzaga and Bonnans, the LPF+ algorithm of Wright, the higher order methods of Wright and Zhang. Potra and Sheng, and Stoer, Wechs and Mizuno are Q-superlinearly convergent.
Two nonsymmetric search directions for semidefinite programming, the XZ and ZX search directions, are proposed. They are derived from a nonsymmetric formulation of the semidefinite programming problem. The XZ directio...
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Two nonsymmetric search directions for semidefinite programming, the XZ and ZX search directions, are proposed. They are derived from a nonsymmetric formulation of the semidefinite programming problem. The XZ direction corresponds to the direct linearization of the central path equation XZ = nu I;while the ZX direction corresponds to ZX = nu I. The XZ and ZX directions are well defined if both X and Z are positive definite matrices, where X may be nonsymmetric. We present an algorithm using the XZ and ZX directions alternately following the Mehrotra predictor-corrector framework. Numerical results show that the XZ/ZX algorithm, in many cases, requires less CPU time than the XZ+ZX method of Alizadeh, Overton, and Haeberly [SIAM J. Optim., 8 (1998), pp. 746-768] while achieving similar accuracy.
Various search directions used in interior-point algorithms for the semidefinite program (SDP) and the monotone semidefinite linear complementarity problem (SDLCP) are characterized by the intersection of a maximal mo...
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Various search directions used in interior-point algorithms for the semidefinite program (SDP) and the monotone semidefinite linear complementarity problem (SDLCP) are characterized by the intersection of a maximal monotone affine subspace and a maximal and strictly antitone affine subspace. This observation provides a unified geometric view over the existence of those search directions.
Some interior-point algorithms have superlinear convergence. When solving an LCP (linear complementarity problem), superlinear convergence had been achieved under the assumption that a strictly complementary solution ...
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Some interior-point algorithms have superlinear convergence. When solving an LCP (linear complementarity problem), superlinear convergence had been achieved under the assumption that a strictly complementary solution exists, whether starting from a feasible or an infeasible interiorpoint. In this paper, we propose an algorithm for solving monotone geometrical LCPs, and we prove its superlinear convergence without the strictly complementary condition. The algorithm can start from an infeasible interiorpoint and has globally linear convergence. When we use a big initial point or an almost feasible initial point, the algorithm has polynomial time convergence.
The Euclidean Steiner tree problem is to find the tree with minimal Euclidean length spanning a set of fixed points in the plane, allowing the addition of auxiliary points to the set (Steiner points). The problem is N...
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The Euclidean Steiner tree problem is to find the tree with minimal Euclidean length spanning a set of fixed points in the plane, allowing the addition of auxiliary points to the set (Steiner points). The problem is NP-hard, so polynomial-time heuristics are desired. We present two such heuristics, both of which utilize an efficient method for computing a locally optimal tree with a given topology. The first systematically inserts Steiner points between edges of the minimal spanning tree meeting at angles less than 120 degrees, performing a local optimization at the end. The second begins by finding the Steiner tree for three of the fixed points. Then, at each iteration, it introduces a new fixed point to the tree, connecting it to each possible edge by inserting a Steiner point, and minimizes over all connections, performing a local optimization for each. We present a variety of test cases that demonstrate the strengths and weaknesses of both algorithms.
This paper proposes a Multiobjective Linear Programming (MLP) model on injection oilfield recovery system. A modified interior-point algorithm to MLP problems has been constructed by using concepts of Kamarkar's i...
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This paper proposes a Multiobjective Linear Programming (MLP) model on injection oilfield recovery system. A modified interior-point algorithm to MLP problems has been constructed by using concepts of Kamarkar's interiorpointalgorithm and the Analytic Hierarchy Process (AHP). This algorithm is shown to likely be more efficient than other MLP's algorithms in the application of decision making on the petroleum industry through the demonstration of a numerical example. The MLP model's optimal solution allows decision makers to optimally design the developing plan of the injection oilfield recovery system. (C) 1998 Elsevier Science Ltd. All rights reserved.
A generalized class of infeasible-interior-point methods for solving horizontal linear complementarity problems is analyzed and sufficient conditions are given for the convergence of the sequence of iterates produced ...
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A generalized class of infeasible-interior-point methods for solving horizontal linear complementarity problems is analyzed and sufficient conditions are given for the convergence of the sequence of iterates produced by methods in this class, In particular it is shown that the largest step path following algorithms generates convergent iterates er en when starting from infeasible points. The computational complexity of the latter method is discussed in detail and its local convergent rate is analyzed. The primal-dual gap of the iterates produced by this method is superlinearly convergent to zero. A variant of the method has quadratic convergence.
In this paper we show that the primal-dual Dikin affine scaling algorithm for linear programming of Jansen, Roos and Terlaky enhances an asymptotical O(root nL) complexity by using corrector steps, We also show that t...
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In this paper we show that the primal-dual Dikin affine scaling algorithm for linear programming of Jansen, Roos and Terlaky enhances an asymptotical O(root nL) complexity by using corrector steps, We also show that the result remains valid when the method is applied to positive semi-definite linear complementarity problems.
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