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On a Homogeneous Algorithm for the Monotone Complementarity Problem
 Mathematical Programming
, 1995
"... We present a generalization of a homogeneous selfdual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "bigM" parameter or twophase method, and it generates either a solution converging towards feasibility an ..."
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Cited by 41 (3 self)
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We present a generalization of a homogeneous selfdual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "bigM" parameter or twophase method, and it generates either a solution converging towards feasibility and complementarity simultaneously or a certificate proving infeasibility. Moreover, if the MCP is polynomially solvable with an interior feasible starting point, then it can be polynomially solved without using or knowing such information at all. To our knowledge, this is the first interiorpoint and infeasiblestarting algorithm for solving the MCP that possesses these desired features. Preliminary computational results are presented. Key words: Monotone complementarity problem, homogeneous and selfdual, infeasiblestarting algorithm. Running head: A homogeneous algorithm for MCP. Department of Management, Odense University, Campusvej 55, DK5230 Odense M, Denmark, email: eda@busieco.ou.dk. y De...
A Computational Study of the Homogeneous Algorithm for LargeScale Convex Optimization
, 1997
"... Recently the authors have proposed a homogeneous and selfdual algorithm for solving the monotone complementarity problem (MCP) [5]. The algorithm is a single phase interiorpoint type method, nevertheless it yields either an approximate optimal solution or detects a possible infeasibility of th ..."
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Cited by 22 (2 self)
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Recently the authors have proposed a homogeneous and selfdual algorithm for solving the monotone complementarity problem (MCP) [5]. The algorithm is a single phase interiorpoint type method, nevertheless it yields either an approximate optimal solution or detects a possible infeasibility of the problem. In this paper we specialize the algorithm to the solution of general smooth convex optimization problems that also possess nonlinear inequality constraints and free variables. We discuss an implementation of the algorithm for largescale sparse convex optimization. Moreover, we present computational results for solving quadratically constrained quadratic programming and geometric programming problems, where some of the problems contain more than 100,000 constraints and variables. The results indicate that the proposed algorithm is also practically efficient. Department of Management, Odense University, Campusvej 55, DK5230 Odense M, Denmark. Email: eda@busieco.ou.dk y ...
A computational study of the homogeneous algorithm for largescale convex optimization
, 1996
"... Key words: Monotone complementarity problem, homogeneous and selfdual model, interiorpoint algorithms, largescale convex optimization. 1 Introduction In 1984 Karmarkar [31] presented an interiorpoint method for linear programming (LP) and since then interiorpoint algorithms enjoyed great public ..."
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Key words: Monotone complementarity problem, homogeneous and selfdual model, interiorpoint algorithms, largescale convex optimization. 1 Introduction In 1984 Karmarkar [31] presented an interiorpoint method for linear programming (LP) and since then interiorpoint algorithms enjoyed great publicity for two reasons. First, these algorithms solve LP problems in polynomial time, as proved by Karmarkar and many others. Secondly, interiorpoint algorithms have demonstrated excellent practical performance when solving largescale LP problems, see Lustig et al. [37]. It was soon realized (see Gill et al. [25]) that Karmarkar's method was closely related to the logarithmic barrier algorithm for general nonlinear programming studied by Fiacco and McCormick [23] and others in the sixties. Hence, it is natural to investigate the efficiency of the interiorpoint methods for solving more general classes of problems. In general good complexity results could only be expected for solving convex optimization problems.