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22
A PathFollowing InfeasibleInteriorPoint Algorithm for Linear Complementarity Problems
 Optimization Methods and Software
, 1993
"... We describe an infeasibleinteriorpoint algorithm for monotone linear complementarity problems that has polynomial complexity, global linear convergence, and local superlinear convergence with a Qorder of 2. Only one matrix factorization is required per iteration, and the analysis assumes only tha ..."
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Cited by 50 (10 self)
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We describe an infeasibleinteriorpoint algorithm for monotone linear complementarity problems that has polynomial complexity, global linear convergence, and local superlinear convergence with a Qorder of 2. Only one matrix factorization is required per iteration, and the analysis assumes only that a strictly complementary solution exists. 1 Introduction The monotone linear complementarity problem is to find a vector pair (x; y) 2 IR n \Theta IR n such that y = Mx+ h; (x; y) (0; 0); x T y = 0; (1) where h 2 IR n and M is an n \Theta n positive semidefinite matrix. A vector pair (x ; y ) is called a strictly complementary solution of (1) if it satisfies the three conditions in (1) and, in addition, x i + y i ? 0 for each component i = 1; 2; \Delta \Delta \Delta ; n. We denote the solution set for (1) by S and the set of strictly complementary solutions by S c . A number of interior point methods have been proposed for (1). Among recent papers are the predictor...
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 41 (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...
Local Convergence of InteriorPoint Algorithms for Degenerate Monotone LCP
 Computational Optimization and Applications
, 1993
"... Most asymptotic convergence analysis of interiorpoint algorithms for monotone linear complementarity problems assumes that the problem is nondegenerate, that is, the solution set contains a strictly complementary solution. We investigate the behavior of these algorithms when this assumption is remo ..."
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Cited by 34 (4 self)
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Most asymptotic convergence analysis of interiorpoint algorithms for monotone linear complementarity problems assumes that the problem is nondegenerate, that is, the solution set contains a strictly complementary solution. We investigate the behavior of these algorithms when this assumption is removed. 1 Introduction In the monotone linear complementarity problem (LCP), we seek a vector pair (x; y) 2 IR n \Theta IR n that satisfies the conditions y = Mx+ q; x 0; y 0; x T y = 0; (1) where q 2 IR n , and M 2 IR n\Thetan is positive semidefinite. We use S to denote the solution set of (1). An assumption that is frequently made in order to prove superlinear convergence of interiorpoint algorithms for (1) is the nondegeneracy assumption: Assumption 1 There is an (x ; y ) 2 S such that x i + y i ? 0 for all i = 1; \Delta \Delta \Delta ; n. In general, we can define three subsets B, N , and J of the index set f1; \Delta \Delta \Delta ; ng by B = fi = 1; \Delta ...
A PathFollowing InteriorPoint Algorithm for Linear and Quadratic Problems
 Preprint MCSP4011293, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439
, 1995
"... We describe an algorithm for the monotone linear complementarity problem (LCP) that converges from any positive, not necessarily feasible, starting point and exhibits polynomial complexity if some additional assumptions are made on the starting point. If the problem has a strictly complementary solu ..."
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Cited by 21 (4 self)
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We describe an algorithm for the monotone linear complementarity problem (LCP) that converges from any positive, not necessarily feasible, starting point and exhibits polynomial complexity if some additional assumptions are made on the starting point. If the problem has a strictly complementary solution, the method converges subquadratically. We show that the algorithm and its convergence properties extend readily to the mixed monotone linear complementarity problem and, hence, to all the usual formulations of the linear programming and convex quadratic programming problems. 1 Introduction The monotone linear complementarityproblem (LCP) is to find a vector pair (x; y) 2 IR n \ThetaIR n such that y = Mx+ q; (x; y) 0; x T y = 0; (1) where q 2 IR n and M is an n \Theta n positive semidefinite (p.s.d.) matrix. The mixed monotone linear complementarity problem (MLCP) is to find a vector triple (x; y; z) 2 IR n \Theta IR n \Theta IR m such that " y 0 # = " M 11 M 12 ...
InfeasibleInteriorPoint PrimalDual PotentialReduction Algorithms For Linear Programming
 SIAM Journal on Optimization
, 1995
"... . In this paper, we propose primaldual potentialreduction algorithms which can start from an infeasible interior point. We first describe two such algorithms and show that both are polynomialtime bounded. One of the algorithms decreases the TanabeToddYe primaldual potential function by a const ..."
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Cited by 20 (4 self)
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. In this paper, we propose primaldual potentialreduction algorithms which can start from an infeasible interior point. We first describe two such algorithms and show that both are polynomialtime bounded. One of the algorithms decreases the TanabeToddYe primaldual potential function by a constant at each iteration under the condition that the duality gap decreases by at most the same ratio as the infeasibility. The other reduces a new potential function, which has one more term in the TanabeToddYe potential function, by a fixed constant at each iteration without any other conditions on the step size. Finally, we describe modifications of these methods (incorporating centering steps) which dramatically decrease their computational complexity. Our algorithms also extend to the case of monotone linear complementarity problems. Key words. Polynomial Time, Linear Programming, PrimalDual, InfeasibleInteriorPoint Algorithm, Potential Function. AMS subject classifications. 90C05, ...
A Computational View of InteriorPoint Methods for Linear Programming
 IN: ADVANCES IN LINEAR AND INTEGER PROGRAMMING
, 1994
"... Many issues that are crucial for an efficient implementation of an interior point algorithm are addressed in this paper. To start with, a prototype primaldual algorithm is presented. Next, many tricks that make it so efficient in practice are discussed in detail. Those include: the preprocessing te ..."
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Cited by 15 (10 self)
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Many issues that are crucial for an efficient implementation of an interior point algorithm are addressed in this paper. To start with, a prototype primaldual algorithm is presented. Next, many tricks that make it so efficient in practice are discussed in detail. Those include: the preprocessing techniques, the initialization approaches, the methods of computing search directions (and lying behind them linear algebra techniques), centering strategies and methods of stepsize selection. Several reasons for the manifestations of numerical difficulties like e.g.: the primal degeneracy of optimal solutions or the lack of feasible solutions are explained in a comprehensive way. A motivation for obtaining an optimal basis is given and a practicable algorithm to perform this task is presented. Advantages of different methods to perform postoptimal analysis (applicable to interior point optimal solutions) are discussed. Important questions that still remain open in the implementations of i...
Approximate Farkas Lemmas and Stopping Rules for Iterative InfeasiblePoint Algorithms for Linear Programming
 Mathematical Programming
, 1994
"... In exact arithmetic, the simplex method applied to a particular linear programming problem instance either shows that it is infeasible, shows that its dual is infeasible, or generates optimal solutions to both problems. Interiorpoint methods do not provide such clearcut information. We provide gene ..."
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Cited by 13 (1 self)
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In exact arithmetic, the simplex method applied to a particular linear programming problem instance either shows that it is infeasible, shows that its dual is infeasible, or generates optimal solutions to both problems. Interiorpoint methods do not provide such clearcut information. We provide general tools (extensions of the Farkas Lemma) for concluding that a problem or its dual is likely (in a certain welldefined sense) to be infeasible, and apply them to develop stopping rules for a generic infeasibleinteriorpoint method and for the homogeneous selfdual algorithm for linear programming. These rules allow precise conclusions to be drawn about the linear programming problem and its dual: either nearoptimal solutions are produced, or we obtain "certificates" that all optimal solutions, or all feasible solutions to the primal or dual, must have large norm. Our rules thus allow more definitive interpretation of the output of such an algorithm than previous termination criteria. We...
An Interior Point Potential Reduction Method for Constrained Equations
, 1995
"... We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In gen ..."
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Cited by 11 (3 self)
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We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In general, constrained equations provide a unified formulation for many mathematical programming problems, including complementarity problems of various kinds and the KarushKuhnTucker systems of variational inequalities and nonlinear programs. Combining ideas from the damped Newton and interior point methods, we present an iterative algorithm for solving a constrained system of equations and investigate its convergence properties. Specialization of the algorithm and its convergence analysis to complementarity problems of various kinds and the KarushKuhnTucker systems of variational inequalities are discussed in detail. We also report the computational results of the implementation of the algo...
Probabilistic Analysis of an InfeasibleInteriorPoint Algorithm for Linear Programming
, 1998
"... We consider an infeasibleinteriorpoint algorithm, endowed with a finite termination scheme, applied to random linear programs generated according to a model of Todd. Such problems have degenerate optimal solutions, and possess no feasible starting point. We use no information regarding an optimal ..."
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Cited by 11 (3 self)
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We consider an infeasibleinteriorpoint algorithm, endowed with a finite termination scheme, applied to random linear programs generated according to a model of Todd. Such problems have degenerate optimal solutions, and possess no feasible starting point. We use no information regarding an optimal solution in the initialization of the algorithm. Our main result is that the expected number of iterations before termination with an exact optimal solution is O(n ln(n)). Keywords: Linear Programming, AverageCase Behavior, InfeasibleInteriorPoint Algorithm. Running Title: Probabilistic Analysis of an LP Algorithm 1 Dept. of Management Sciences, University of Iowa. Supported by an Interdisciplinary Research Grant from the Center for Advanced Studies, University of Iowa. 2 Dept. of Mathematics, Valdosta State University. Supported by an Interdisciplinary Research Grant from the Center for Advanced Studies, University of Iowa. 3 Dept. of Mathematics, University of Iowa. Supported by ...
Equivalence between Different Formulations of the Linear Complementarity Problem
, 1995
"... One shows that different formulations of the linear complementarity problem (LCP), such as the horizontal LCP, the mixed LCP and the geometric LCP can be transformed into a standard LCP. The P ()property of the corresponding formulations as well as the convergence properties of a large class of in ..."
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Cited by 11 (10 self)
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One shows that different formulations of the linear complementarity problem (LCP), such as the horizontal LCP, the mixed LCP and the geometric LCP can be transformed into a standard LCP. The P ()property of the corresponding formulations as well as the convergence properties of a large class of interiorpoint algorithms are invariant with respect to the transformations. Therefore it is sufficient to study the algorithms only for the standard LCP. Key Words:P matrix, linear complementarity problems, predictorcorrector, infeasibleinterior point algorithm, polynomiality, quadratic convergence. Abbreviated Title: Equivalence of different LCP Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA. y Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA. z Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA. The work of this author was supported in part by NSF, Grant DMS 9305760. 1 Introduction The linear complementarity...