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Tamás Terlaky - One of the best experts on this subject based on the ideXlab platform.
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A rounding procedure for semidefinite optimization
Operations Research Letters, 2019Co-Authors: Ali Mohammad-nezhad, Tamás TerlakyAbstract:Abstract Mohammad-Nezhad and Terlaky studied the identification of the optimal partition for semidefinite optimization. An approximation of the optimal partition was obtained from a bounded sequence of Solutions on, or in a neighborhood of the central path. We use the approximation of the optimal partition in a rounding procedure to generate an approximate maximally Complementary Solution. From an interior Solution, sufficiently close to the optimal set, the procedure rounds to a conic feasible Solution, while the equality constraints are slightly violated.
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a strongly polynomial rounding procedure yielding a maximally Complementary Solution for p_ kappa linear complementarity problems
Siam Journal on Optimization, 2000Co-Authors: Tibor Illés, Jiming Peng, Cornelis Roos, Tamás TerlakyAbstract:We deal with linear complementarity problems (LCPs) with $P_*(\kappa)$ matrices. First we establish the convergence rate of the Complementary variables along the central path. The central path is parameterized by the barrier parameter $\mu$, as usual. Our elementary proof reproduces the known result that the variables on or close to the central path fall apart in three classes in which these variables are ${\cal O}(1), {\cal O}(\mu),$ and ${\cal O}(\sqrt{\mu})$, respectively. The constants hidden in these bounds are expressed in or bounded by the input data. All this is preparation for our main result: a strongly polynomial rounding procedure. Given a point with sufficiently small complementarity gap and which is close enough to the central path, the rounding procedure produces a maximally Complementary Solution in at most ${\cal O} (n^3)$ arithmetic operations. The result implies that interior point methods (IPMs) not only converge to a Complementary Solution of $P_*(\kappa)$ LCPs, but, when furnished with our rounding procedure, they can also produce a maximally Complementary (exact) Solution in polynomial time.
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A Strongly Polynomial Rounding Procedure Yielding a Maximally Complementary Solution for $P_*(\kappa)$ Linear Complementarity Problems
SIAM Journal on Optimization, 2000Co-Authors: Tibor Illés, Jiming Peng, Cornelis Roos, Tamás TerlakyAbstract:We deal with linear complementarity problems (LCPs) with $P_*(\kappa)$ matrices. First we establish the convergence rate of the Complementary variables along the central path. The central path is parameterized by the barrier parameter $\mu$, as usual. Our elementary proof reproduces the known result that the variables on or close to the central path fall apart in three classes in which these variables are ${\cal O}(1), {\cal O}(\mu),$ and ${\cal O}(\sqrt{\mu})$, respectively. The constants hidden in these bounds are expressed in or bounded by the input data. All this is preparation for our main result: a strongly polynomial rounding procedure. Given a point with sufficiently small complementarity gap and which is close enough to the central path, the rounding procedure produces a maximally Complementary Solution in at most ${\cal O} (n^3)$ arithmetic operations. The result implies that interior point methods (IPMs) not only converge to a Complementary Solution of $P_*(\kappa)$ LCPs, but, when furnished with our rounding procedure, they can also produce a maximally Complementary (exact) Solution in polynomial time.
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Basis- and partition identification for quadratic programming and linear complementarity problems
Mathematical Programming, 1999Co-Authors: Arjan Berkelaar, B. Jansen, Kees Roos, Tamás TerlakyAbstract:Optimal Solutions of interior point algorithms for linear and quadratic programming and linear complementarity problems provide maximally Complementary Solutions. Maximally Complementary Solutions can be characterized by optimal partitions. On the other hand, the Solutions provided by simplex–based pivot algorithms are given in terms of Complementary bases. A basis identification algorithm is an algorithm which generates a Complementary basis, starting from any Complementary Solution. A partition identification algorithm is an algorithm which generates a maximally Complementary Solution (and its corresponding partition), starting from any Complementary Solution. In linear programming such algorithms were respectively proposed by Megiddo in 1991 and Balinski and Tucker in 1969. In this paper we will present identification algorithms for quadratic programming and linear complementarity problems with sufficient matrices. The presented algorithms are based on the principal pivot transform and the orthogonality property of basis tableaus.
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Basis- and tripartition identification for quadratic programming and lineair Complementary problems; From an interior Solution to an optimal basis and viceversa
1996Co-Authors: Arjan Berkelaar, B. Jansen, Kees Roos, Tamás TerlakyAbstract:Optimal Solutions of interior point algorithms for linear and quadratic programming and linear complementarity problems provide maximal Complementary Solutions. Maximal Complementary Solutions can be characterized by optimal (tri)partitions. On the other hand, the Solutions provided by simplex--based pivot algorithms are given in terms of Complementary bases. A basis identification algorithm is an algorithm which generates a Complementary basis, starting from any Complementary Solution. A tripartition identification algorithm is an algorithm which generates a maximal Complementary Solution (and its corresponding tripartition), starting from any Complementary Solution. In linear programming such algorithms were respectively proposed by Megiddo in 1991 and Balinski and Tucker in 1969. In this paper we will present identification algorithms for quadratic programming and linear complementarity problems with sufficient matrices. The presented algorithms are based on the principal pivot transform and the orthogonality property of basis tableaus.
Zheng-hai Huang - One of the best experts on this subject based on the ideXlab platform.
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A Smoothing-Type Algorithm for Solving Linear Complementarity Problems with Strong Convergence Properties
Applied Mathematics and Optimization, 2007Co-Authors: Zheng-hai HuangAbstract:In this paper, we construct an augmented system of the standard monotone linear complementarity problem (LCP), and establish the relations between the augmented system and the LCP. We present a smoothing-type algorithm for solving the augmented system. The algorithm is shown to be globally convergent without assuming any prior knowledge of feasibility/infeasibility of the problem. In particular, if the LCP has a Solution, then the algorithm either generates a maximal Complementary Solution of the LCP or detects correctly solvability of the LCP, and in the latter case, an existing smoothing-type algorithm can be directly applied to solve the LCP without any additional assumption and it generates a maximal Complementary Solution of the LCP; and that if the LCP is infeasible, then the algorithm detect correctly infeasibility of the LCP. To the best of our knowledge, such properties have not appeared in the existing literature for smoothing-type algorithms.
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Convergence properties of a non-interior-point smoothing algorithm for the P*NCP
Journal of Industrial & Management Optimization, 2007Co-Authors: Zheng-hai HuangAbstract:In this paper, a non-interior-point smoothing algorithm is applied to solve the $P_*$ nonlinear complementarity problem (NCP). The algorithm is proved to be globally convergent under an assumption that the $P_*$ NCP has a nonempty Solution set. In particular, the Solution obtained by the algorithm is shown to be a maximally Complementary Solution of the $P_*$ NCP. The results we obtained strictly generalize the relative results appeared in the literature.
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Smoothing-type algorithm for solving linear programs by using an augmented complementarity problem
Applied Mathematics and Computation, 2006Co-Authors: Zheng-hai Huang, Hui WangAbstract:We present a smoothing-type algorithm for solving the linear program (LP) by making use of an augmented system of its optimality conditions. The algorithm is shown to be globally convergent without requiring any assumption. It only needs to solve one system of linear equations and to perform one line search at each iteration. In particular, if the LP has a Solution (and hence it has a strictly Complementary Solution), then the algorithm will generate a strictly Complementary Solution of the LP; and if the LP is infeasible, then the algorithm will correctly detect infeasibility of the LP. To the best of our knowledge, this is the first smoothing-type algorithm for solving the LP having the above desired convergence features.
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A smoothing Newton algorithm for the LCP with a sufficient matrix that terminates finitely at a maximally Complementary Solution
Optimization Methods and Software, 2006Co-Authors: Jie Sun, Zheng-hai HuangAbstract:By using a smoothing function, the linear complementarity problem (LCP) can be reformulated as a parameterized smooth equation. A Newton method with a projection-type testing procedure is proposed to solve this equation. We show that, for the LCP with a sufficient matrix, the iteration sequence generated by the proposed algorithm is bounded as long as the LCP has a Solution. This assumption is weaker than the ones used in most existing smoothing algorithms. Moreover, we show that the proposed algorithm can find a maximally Complementary Solution to the LCP in a finite number of iterations.
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Locating a maximally Complementary Solution of the monotone NCP by using non-interior-point smoothing algorithms
Mathematical Methods of Operations Research, 2005Co-Authors: Zheng-hai HuangAbstract:In this paper we propose a non-interior-point smoothing algorithm for solving the monotone nonlinear complementarity problem (NCP). The proposed algorithm is simpler than many existing non-interior-point smoothing algorithms in the sense that it only needs to solve one system of linear equations and to perform one line search at each iteration. We show that the proposed algorithm is globally convergent under the assumption that the NCP concerned has a nonempty Solution set. Such assumption is weaker than those required by most other non-interior-point smoothing algorithms. In particular, we prove that the Solution obtained by the proposed algorithm is a maximally Complementary Solution of the NCP concerned. Preliminary numerical results are reported.
Thanh Tran - One of the best experts on this subject based on the ideXlab platform.
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browsing oriented semantic faceted search
Database and Expert Systems Applications, 2011Co-Authors: Andreas Wagner, Gunter Ladwig, Thanh TranAbstract:Faceted search enables users to browse and discover relevant items from a large collection such as the Web of data. Existing faceted search Solutions assume a precise information need, and thus optimise relevance, interestingness, and costs of fulfilling an information need. In this paper, we propose a Complementary Solution. Instead of assuming a search scenario (i.e., a user has a precise information need), our Solution targets a browsing scenario (i.e., a user has a fuzzy need). We aimto support users in exploring an unknown collection of items, thereby allowing them to discover new or unfamiliar items of interest. Our approach comprises mechanisms for grouping facets and facet values and facet ranking. Via a task-based evaluation, we demonstrate that the proposed Solution enables more effective browsing compared to the state-of-the-art, given fuzzy information needs.
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DEXA (1) - Browsing-oriented semantic faceted search
Lecture Notes in Computer Science, 2011Co-Authors: Andreas Wagner, Gunter Ladwig, Thanh TranAbstract:Faceted search enables users to browse and discover relevant items from a large collection such as the Web of data. Existing faceted search Solutions assume a precise information need, and thus optimise relevance, interestingness, and costs of fulfilling an information need. In this paper, we propose a Complementary Solution. Instead of assuming a search scenario (i.e., a user has a precise information need), our Solution targets a browsing scenario (i.e., a user has a fuzzy need). We aimto support users in exploring an unknown collection of items, thereby allowing them to discover new or unfamiliar items of interest. Our approach comprises mechanisms for grouping facets and facet values and facet ranking. Via a task-based evaluation, we demonstrate that the proposed Solution enables more effective browsing compared to the state-of-the-art, given fuzzy information needs.
Andreas Wagner - One of the best experts on this subject based on the ideXlab platform.
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browsing oriented semantic faceted search
Database and Expert Systems Applications, 2011Co-Authors: Andreas Wagner, Gunter Ladwig, Thanh TranAbstract:Faceted search enables users to browse and discover relevant items from a large collection such as the Web of data. Existing faceted search Solutions assume a precise information need, and thus optimise relevance, interestingness, and costs of fulfilling an information need. In this paper, we propose a Complementary Solution. Instead of assuming a search scenario (i.e., a user has a precise information need), our Solution targets a browsing scenario (i.e., a user has a fuzzy need). We aimto support users in exploring an unknown collection of items, thereby allowing them to discover new or unfamiliar items of interest. Our approach comprises mechanisms for grouping facets and facet values and facet ranking. Via a task-based evaluation, we demonstrate that the proposed Solution enables more effective browsing compared to the state-of-the-art, given fuzzy information needs.
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DEXA (1) - Browsing-oriented semantic faceted search
Lecture Notes in Computer Science, 2011Co-Authors: Andreas Wagner, Gunter Ladwig, Thanh TranAbstract:Faceted search enables users to browse and discover relevant items from a large collection such as the Web of data. Existing faceted search Solutions assume a precise information need, and thus optimise relevance, interestingness, and costs of fulfilling an information need. In this paper, we propose a Complementary Solution. Instead of assuming a search scenario (i.e., a user has a precise information need), our Solution targets a browsing scenario (i.e., a user has a fuzzy need). We aimto support users in exploring an unknown collection of items, thereby allowing them to discover new or unfamiliar items of interest. Our approach comprises mechanisms for grouping facets and facet values and facet ranking. Via a task-based evaluation, we demonstrate that the proposed Solution enables more effective browsing compared to the state-of-the-art, given fuzzy information needs.
Florian A. Potra - One of the best experts on this subject based on the ideXlab platform.
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Corrector-predictor methods for sufficient linear complementarity problems
Computational Optimization and Applications, 2011Co-Authors: Filiz Gurtuna, Florian A. Potra, Cosmin Petra, Olena Shevchenko, Adrian VanceaAbstract:We present a new corrector-predictor method for solving sufficient linear complementarity problems for which a sufficiently centered feasible starting point is available. In contrast with its predictor-corrector counterpart proposed by Miao, the method does not depend on the handicap κ of the problem. The method has $O((1+\kappa)\sqrt{n}L)$ -iteration complexity, the same as Miao’s method, but our error estimates are sightly better. The algorithm is quadratically convergent for problems having a strictly Complementary Solution. We also present a family of infeasible higher order corrector-predictor methods that are superlinearly convergent even in the absence of strict complementarity. The algorithms of this class are globally convergent for general positive starting points. They have $O((1+\kappa)\sqrt{n}L)$ -iteration complexity for feasible, or “almost feasible”, starting points and O ((1+ κ )^2 nL )-iteration complexity for “sufficiently large” infeasible starting points.
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A superlinearly convergent predictor-corrector method for degenerate LCP in a wide neighborhood of the central path with $$O(\sqrt{n}L)$$-iteration complexity
Mathematical Programming, 2006Co-Authors: Florian A. PotraAbstract:An interior point method for monotone linear complementarity problems acting in a wide neighborhood of the central path is presented. The method has *-iteration complexity and is superlinearly convergent even when the problem does not possess a strictly Complementary Solution.
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A superlinearly convergent predictor-corrector method for degenerate LCP in a wide neighborhood of the central path with [InlineMediaObject not available: see fulltext.]-iteration complexity
Mathematical Programming, 2004Co-Authors: Florian A. PotraAbstract:An interior point method for monotone linear complementarity problems acting in a wide neighborhood of the central path is presented. The method has [InlineMediaObject not available: see fulltext.]-iteration complexity and is superlinearly convergent even when the problem does not possess a strictly Complementary Solution.