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45
Interiorpoint Methods
, 2000
"... The modern era of interiorpoint methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadrati ..."
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Cited by 603 (15 self)
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The modern era of interiorpoint methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semidefinite programming, and nonconvex and nonlinear problems, have reached varying levels of maturity. We review some of the key developments in the area, including comments on both the complexity theory and practical algorithms for linear programming, semidefinite programming, monotone linear complementarity, and convex programming over sets that can be characterized by selfconcordant barrier functions.
On the formulation and theory of newton interior point methods for nonlinear programming
 Journal of Optimization Theory and Applications
, 1996
"... Abstract. In this work, we first study in detail the formulation of the primaldual interiorpoint method for linear programming. We show that, contrary to popular belief, it cannot be viewed.as adamped Newton method applied to the KarushKuhnTucker conditions for the logarithmic barrier function ..."
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Cited by 113 (5 self)
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Abstract. In this work, we first study in detail the formulation of the primaldual interiorpoint method for linear programming. We show that, contrary to popular belief, it cannot be viewed.as adamped Newton method applied to the KarushKuhnTucker conditions for the logarithmic barrier function problem. Next, we extend the formulation to general nonlinear programming, and then validate this extension by demonstrating that this algorithm can be implemented sothat it is locally and Qquadratically convergent under only the standard Newton method assumptions. We also establish a global convergence theory for this algorithm and include promising numerical experimentation. Key Words. Interiorpoint methods, primaldual methods, nonlinear programming, superlinear and quadratic convergence, global convergence. 1.
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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On The Accurate Identification Of Active Constraints
, 1996
"... We consider nonlinear programs with inequality constraints, and we focus on the problem of identifying those constraints which will be active at an isolated local solution. The correct identification of active constraints is important from both a theoretical and a practical point of view. Such an id ..."
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Cited by 63 (9 self)
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We consider nonlinear programs with inequality constraints, and we focus on the problem of identifying those constraints which will be active at an isolated local solution. The correct identification of active constraints is important from both a theoretical and a practical point of view. Such an identification removes the combinatorial aspect of the problem and locally reduces the inequality constrained minimization problem to an equality constrained one which can be more easily dealt with. We present a new technique which identifies active constraints in a neighborhood of a solution and which requires neither complementary slackness nor uniqueness of the multipliers. As an example of application of the new technique we present a local active set Newtontype algorithm for the solution of general inequality constrained problems for which Qquadratic convergence of the primal variables can be proved under very weak conditions. We also present extensions to variational inequalities.
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.
INTERIOR POINT METHODS FOR COMBINATORIAL OPTIMIZATION
, 1995
"... Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivale ..."
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Cited by 16 (9 self)
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Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivalent nonconvex quadratic programming problem, interior point methods for solving network flow problems, and methods for solving multicommodity flow problems, including an interior point column generation algorithm.
Using an Interior Point Method in a Branch and Bound Algorithm for Integer Programming.
, 1992
"... This paper describes an experimental code that has been developed to solve zeroone mixed integer linear programs. The experimental code uses a primaldual interior point method to solve the linear programming subproblems that arise in the solution of mixed integer linear programs by the branch and ..."
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Cited by 12 (7 self)
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This paper describes an experimental code that has been developed to solve zeroone mixed integer linear programs. The experimental code uses a primaldual interior point method to solve the linear programming subproblems that arise in the solution of mixed integer linear programs by the branch and bound method. Computational results for a number of test problems are provided. Introduction Mixed integer linear programming problems are often solved by branch and bound methods. Branch and bound codes, such as the ones described in [7, 11, 12], normally use the simplex algorithm to solve linear programming subproblems that arise. In this paper, we describe an experimental branch and bound code for zeroone mixed integer linear programming problems that uses an interior point method to solve the LP subproblems. This project was motivated by the observation that interior point methods tend to quickly find feasible solutions with good objective values, but take a relatively long time to ...
Qsuperlinear convergence of the iterates in primaldual interiorpoint methods
 MATH. PROGRAM., SER. A 91: 99–115 (2001)
, 2001
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Computational Experience of an InteriorPoint SQP Algorithm in a Parallel BranchandBound Framework
"... An interiorpoint algorithm within a parallel branchandbound framework for solving nonlinear mixed integer programs is described. The nonlinear programming relaxations at each node are solved using an interior point SQP method. In contrast to solving the relaxation to optimality at each tree node ..."
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Cited by 10 (3 self)
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An interiorpoint algorithm within a parallel branchandbound framework for solving nonlinear mixed integer programs is described. The nonlinear programming relaxations at each node are solved using an interior point SQP method. In contrast to solving the relaxation to optimality at each tree node, the relaxation is only solved to nearoptimality. Analogous to employing advanced bases in simplexbased linear MIP solvers, a “dynamic” collection of warmstart vectors is kept to provide “advanced warmstarts” at each branchandbound node. The code has the capability to run in both sharedmemory and distributedmemory parallel environments. Preliminary computational results on various classes of linear mixed integer programs and quadratic portfolio problems are presented.
A new iterationcomplexity bound for the MTY predictorcorrector algorithm
 SIAM Journal on Optimization
"... Abstract. In this paper we present a new iterationcomplexity bound for the Mizuno–Todd–Ye predictorcorrector (MTY PC) primaldual interiorpoint algorithm for linear programming. The analysis of the paper is based on the important notion of crossover events introduced by Vavasis and Ye. For a sta ..."
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Cited by 10 (4 self)
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Abstract. In this paper we present a new iterationcomplexity bound for the Mizuno–Todd–Ye predictorcorrector (MTY PC) primaldual interiorpoint algorithm for linear programming. The analysis of the paper is based on the important notion of crossover events introduced by Vavasis and Ye. For a standard form linear program min{cT x: Ax = b, x ≥ 0} with decision variable x ∈ n, we show that the MTY PC algorithm, started from a wellcentered interiorfeasible solution with duality gap nµ0, finds an interiorfeasible solution with duality gap less than nη in O(T (µ0/η)+n3.5 log(χ̄∗A)) iterations, where T (t) ≡ min{n2 log(log t), log t} for all t> 0 and χ̄∗A is a scaling invariant condition number associated with the matrix A. More specifically, χ̄∗A is the infimum of all the conditions numbers χ̄AD, where D varies over the set of positive diagonal matrices. Under the setting of the Turing machine model, our analysis yields an O(n3.5LA + min{n2 logL,L}) iterationcomplexity bound for the MTY PC algorithm to find a primaldual optimal solution, where LA and L are the input sizes of the matrix A and the data (A, b, c), respectively. This contrasts well with the classical iterationcomplexity bound for the MTY PC algorithm, which depends linearly on L instead of logL.