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79
LOQO: An interior point code for quadratic programming
, 1994
"... ABSTRACT. This paper describes a software package, called LOQO, which implements a primaldual interiorpoint method for general nonlinear programming. We focus in this paper mainly on the algorithm as it applies to linear and quadratic programming with only brief mention of the extensions to convex ..."
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Cited by 191 (10 self)
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ABSTRACT. This paper describes a software package, called LOQO, which implements a primaldual interiorpoint method for general nonlinear programming. We focus in this paper mainly on the algorithm as it applies to linear and quadratic programming with only brief mention of the extensions to convex and general nonlinear programming, since a detailed paper describing these extensions were published recently elsewhere. In particular, we emphasize the importance of establishing and maintaining symmetric quasidefiniteness of the reduced KKT system. We show that the industry standard MPS format can be nicely formulated in such a way to provide quasidefiniteness. Computational results are included for a variety of linear and quadratic programming problems. 1.
Solving LargeScale Linear Programs by InteriorPoint Methods Under the MATLAB Environment
 Optimization Methods and Software
, 1996
"... In this paper, we describe our implementation of a primaldual infeasibleinteriorpoint algorithm for largescale linear programming under the MATLAB 1 environment. The resulting software is called LIPSOL  Linearprogramming InteriorPoint SOLvers. LIPSOL is designed to take the advantages of M ..."
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Cited by 97 (1 self)
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In this paper, we describe our implementation of a primaldual infeasibleinteriorpoint algorithm for largescale linear programming under the MATLAB 1 environment. The resulting software is called LIPSOL  Linearprogramming InteriorPoint SOLvers. LIPSOL is designed to take the advantages of MATLAB's sparsematrix functions and external interface facilities, and of existing Fortran sparse Cholesky codes. Under the MATLAB environment, LIPSOL inherits a high degree of simplicity and versatility in comparison to its counterparts in Fortran or C language. More importantly, our extensive computational results demonstrate that LIPSOL also attains an impressive performance comparable with that of efficient Fortran or C codes in solving largescale problems. In addition, we discuss in detail a technique for overcoming numerical instability in Cholesky factorization at the endstage of iterations in interiorpoint algorithms. Keywords: Linear programming, PrimalDual infeasibleinteriorp...
X.: Implementation of interior point methods for large scale linear programming
 Interior Point Methods in Mathematical Programming. Kluwer Acad Pub
, 1996
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Symmetric quasidefinite matrices
 SIAM Journal on Optimization
, 1995
"... We say that a symmetric matrix K is quasidefinite if it has the form ..."
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Cited by 70 (4 self)
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We say that a symmetric matrix K is quasidefinite if it has the form
A PrimalDual Interior Point Method Whose Running Time Depends Only on the Constraint Matrix
, 1995
"... We propose a primaldual "layeredstep " interior point (LIP) algorithm for linear programming with data given by real numbers. This algorithm follows the central path, either with short steps or with a new type of step called a "layered least squares " (LLS) ste ..."
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Cited by 58 (8 self)
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We propose a primaldual &quot;layeredstep &quot; interior point (LIP) algorithm for linear programming with data given by real numbers. This algorithm follows the central path, either with short steps or with a new type of step called a &quot;layered least squares &quot; (LLS) step. The algorithm returns an exact optimum after a finite number of stepsin particular, after O(n3:5c(A)) iterations, where c(A) is a function of the
Algorithms For Complementarity Problems And Generalized Equations
, 1995
"... Recent improvements in the capabilities of complementarity solvers have led to an increased interest in using the complementarity problem framework to address practical problems arising in mathematical programming, economics, engineering, and the sciences. As a result, increasingly more difficult pr ..."
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Cited by 48 (5 self)
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Recent improvements in the capabilities of complementarity solvers have led to an increased interest in using the complementarity problem framework to address practical problems arising in mathematical programming, economics, engineering, and the sciences. As a result, increasingly more difficult problems are being proposed that exceed the capabilities of even the best algorithms currently available. There is, therefore, an immediate need to improve the capabilities of complementarity solvers. This thesis addresses this need in two significant ways. First, the thesis proposes and develops a proximal perturbation strategy that enhances the robustness of Newtonbased complementarity solvers. This strategy enables algorithms to reliably find solutions even for problems whose natural merit functions have strict local minima that are not solutions. Based upon this strategy, three new algorithms are proposed for solving nonlinear mixed complementarity problems that represent a significant improvement in robustness over previous algorithms. These algorithms have local Qquadratic convergence behavior, yet depend only on a pseudomonotonicity assumption to achieve global convergence from arbitrary starting points. Using the MCPLIB and GAMSLIB test libraries, we perform extensive computational tests that demonstrate the effectiveness of these algorithms on realistic problems. Second, the thesis extends some previously existing algorithms to solve more general problem classes. Specifically, the NE/SQP method of Pang & Gabriel (1993), the semismooth equations approach of De Luca, Facchinei & Kanz...
Symmetric indefinite systems for interior point methods
, 1993
"... We present a unified framework for solving linear and convex quadratic programs via interior point methods. At each iteration, this method solves an indefinite system whose matrix is [_~2 A v] instead of reducing to obtain the usual AD2A v system. This methodology affords two advantages: (1) it avo ..."
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Cited by 44 (2 self)
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We present a unified framework for solving linear and convex quadratic programs via interior point methods. At each iteration, this method solves an indefinite system whose matrix is [_~2 A v] instead of reducing to obtain the usual AD2A v system. This methodology affords two advantages: (1) it avoids the fill created by explicitly forming the product AD2A v when A has dense columns; and (2) it can easily be used to solve nonseparable quadratic programs since it requires only that D be symmetric. We also present a procedure for converting nonseparable quadratic programs to separable ones which yields computational savings when the matrix of quadratic oefficients is dense.
A QMRbased interiorpoint algorithm for solving linear programs
 Mathematical Programming
, 1997
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Implementation of interior point methods for mixed semidefinite and second order cone optimization problems
 Optimization Methods and Software
"... There is a large number of implementational choices to be made for the primaldual interior point method in the context of mixed semidefinite and second order cone optimization. This paper presents such implementational issues in a unified framework, and compares the choices made by different resear ..."
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Cited by 40 (0 self)
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There is a large number of implementational choices to be made for the primaldual interior point method in the context of mixed semidefinite and second order cone optimization. This paper presents such implementational issues in a unified framework, and compares the choices made by different research groups. This is also the first paper to provide an elaborate discussion of the implementation in SeDuMi.
Solving RealWorld Linear Ordering Problems . . .
, 1995
"... Cutting plane methods require the solution of a sequence of linear programs, where the solution to one provides a warm start to the next. A cutting plane algorithm for solving the linear ordering problem is described. This algorithm uses the primaldual interior point method to solve the linear prog ..."
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Cited by 30 (8 self)
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Cutting plane methods require the solution of a sequence of linear programs, where the solution to one provides a warm start to the next. A cutting plane algorithm for solving the linear ordering problem is described. This algorithm uses the primaldual interior point method to solve the linear programming relaxations. A point which is a good warm start for a simplexbased cutting plane algorithm is generally not a good starting point for an interior point method. Techniques used to improve the warm start include attempting to identify cutting planes early and storing an old feasible point, which is used to help recenter when cutting planes are added. Computational results are described for some realworld problems; the algorithm appears to be competitive with a simplexbased cutting plane algorithm.