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PrimalDual TargetFollowing Algorithms for Linear Programming
 ANNALS OF OPERATIONS RESEARCH
, 1993
"... In this paper we propose a method for linear programming with the property that, starting from an initial noncentral point, it generates iterates that simultaneously get closer to optimality and closer to centrality. The iterates follow paths that in the limit are tangential to the central path. Al ..."
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Cited by 24 (1 self)
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In this paper we propose a method for linear programming with the property that, starting from an initial noncentral point, it generates iterates that simultaneously get closer to optimality and closer to centrality. The iterates follow paths that in the limit are tangential to the central path. Along with the convergence analysis we provide a general framework which enables us to analyze various primaldual algorithms in the literature in a short and uniform way.
The Linear Complementarity Problem, Sufficient Matrices and the CrissCross Method
, 1990
"... Specially structured Linear Complementarity Problems (LCP's) and their solution by the crisscross method are examined in this paper. The crisscross method is known to be finite for LCP's with positive semidefinite bisymmetric matrices and with Pmatrices. It is also a simple finite algorithm for o ..."
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Cited by 6 (4 self)
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Specially structured Linear Complementarity Problems (LCP's) and their solution by the crisscross method are examined in this paper. The crisscross method is known to be finite for LCP's with positive semidefinite bisymmetric matrices and with Pmatrices. It is also a simple finite algorithm for oriented matroid programming problems. Recently Cottle, Pang and Venkateswaran identified the class of (column, row) sufficient matrices. They showed that sufficient matrices are a common generalization of P and PSDmatrices. Cottle also showed that the principal pivoting method (with a clever modification) can be applied to row sufficient LCP's. In this paper the finiteness of the crisscross method for sufficient LCP's is proved. Further it is shown that a matrix is sufficient if and only if the crisscross method processes all the LCP's defined by this matrix and all the LCP's defined by the transpose of this matrix and any parameter vector.
A Variant of the TopkisVeinott Method for Solving Inequality Constrained Optimization Problems
 J. Appl. Math. Optim
, 1997
"... . In this paper, we give a variant of the TopkisVeinott method for solving inequality constrained optimization problems. This method uses a linearly constrained positive semidefinite quadratic problem to generate a feasible descent direction at each iteration. Under mild assumptions, the algorithm ..."
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Cited by 4 (0 self)
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. In this paper, we give a variant of the TopkisVeinott method for solving inequality constrained optimization problems. This method uses a linearly constrained positive semidefinite quadratic problem to generate a feasible descent direction at each iteration. Under mild assumptions, the algorithm is shown to be globally convergent in the sense that every accumulation point of the sequence generated by the algorithm is a FritzJohn point of the problem. We introduce a FritzJohn (FJ) function, an FJ1 strong secondorder sufficiency condition (FJ1SSOSC) and an FJ2 strong secondorder sufficiency condition (FJ2SSOSC), and then show, without any constraint qualification (CQ), that (i) if an FJ point z satisfies the FJ1SSOSC, then there exists a neighborhood N(z) of z such that for any FJ point y 2 N(z) n fzg, f 0 (y) 6= f 0 (z), where f 0 is the objective function of the problem; (ii) if an FJ point z satisfies the FJ2SSOSC, then z is a strict local minimum of the problem. The resu...