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18
WarmStart Strategies In InteriorPoint Methods For Linear Programming
 SIAM Journal on Optimization
, 2000
"... . We study the situation in which, having solved a linear program with an interiorpoint method, we are presented with a new problem instance whose data is slightly perturbed from the original. We describe strategies for recovering a "warmstart" point for the perturbed problem instance from the iter ..."
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Cited by 37 (1 self)
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. We study the situation in which, having solved a linear program with an interiorpoint method, we are presented with a new problem instance whose data is slightly perturbed from the original. We describe strategies for recovering a "warmstart" point for the perturbed problem instance from the iterates of the original problem instance. We obtain worstcase estimates of the number of iterations required to converge to a solution of the perturbed instance from the warmstart points, showing that these estimates depend on the size of the perturbation and on the conditioning and other properties of the problem instances. 1. Introduction. This paper describes and analyzes warmstart strategies for interiorpoint methods applied to linear programming (LP) problems. We consider the situation in which one linear program, the "original instance," has been solved by an interiorpoint method, and we are then presented with a new problem of the same dimensions, the "perturbed instance," in which ...
Multiple Cuts in the Analytic Center Cutting Plane Method
, 1998
"... We analyze the multiple cut generation scheme in the analytic center cutting plane method. We propose an optimal primal and dual updating direction when the cuts are central. The direction is optimal in the sense that it maximizes the product of the new dual slacks and of the new primal variables wi ..."
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Cited by 26 (1 self)
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We analyze the multiple cut generation scheme in the analytic center cutting plane method. We propose an optimal primal and dual updating direction when the cuts are central. The direction is optimal in the sense that it maximizes the product of the new dual slacks and of the new primal variables within the trust regions defined by Dikin's primal and dual ellipsoids. The new primal and dual directions use the variancecovariance matrix of the normals to the new cuts in the metric given by Dikin's ellipsoid. We prove that the recovery of a new analytic center from the optimal restoration direction can be done in O(p log(p + 1)) damped Newton steps, where p is the number of new cuts added by the oracle, which may vary with the iteration. The results and the proofs are independent of the specific scaling matrix primal, dual or primaldual that is used in the computations. The computation of the optimal direction uses Newton's method applied to a selfconcordant function of p variab...
The Analytic Center Cutting Plane Method with Semidefinite Cuts
 SIAM JOURNAL ON OPTIMIZATION
, 2000
"... We analyze an analytic center cutting plane algorithm for the convex feasibility problems with semidefinite cuts. At each iteration the oracle returns a pdimensional semidefinite cut at an approximate analytic center of the set of localization. The set of localization, which contains the solution s ..."
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Cited by 16 (1 self)
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We analyze an analytic center cutting plane algorithm for the convex feasibility problems with semidefinite cuts. At each iteration the oracle returns a pdimensional semidefinite cut at an approximate analytic center of the set of localization. The set of localization, which contains the solution set, is a compact set consists of piecewise algebraic surfaces. We prove that the analytic center is recovered after adding a pdimensional cut in O(p log(p 1)) damped Newton's iteration. We also prove that the algorithm stops when the dimension of the accumulated block diagonal matrix cut reaches to the bound of O (p 2 m 3 =ffl 2 ), where p is the maximum dimension cut and ffl is radius of the largest ball contained in the solution set.
Polynomial interior point cutting plane methods
 Optimization Methods and Software
, 2003
"... Polynomial cutting plane methods based on the logarithmic barrier function and on the volumetric center are surveyed. These algorithms construct a linear programming relaxation of the feasible region, find an appropriate approximate center of the region, and call a separation oracle at this approxim ..."
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Cited by 15 (8 self)
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Polynomial cutting plane methods based on the logarithmic barrier function and on the volumetric center are surveyed. These algorithms construct a linear programming relaxation of the feasible region, find an appropriate approximate center of the region, and call a separation oracle at this approximate center to determine whether additional constraints should be added to the relaxation. Typically, these cutting plane methods can be developed so as to exhibit polynomial convergence. The volumetric cutting plane algorithm achieves the theoretical minimum number of calls to a separation oracle. Longstep versions of the algorithms for solving convex optimization problems are presented. 1
Inexact Cuts In Benders' Decomposition
 SIAM Journal on Optimization
, 1997
"... . Benders' decomposition is a wellknown technique for solving large linear programs with a special structure. In particular it is a popular technique for solving multistage stochastic linear programming problems. Early termination in the subproblems generated during Benders' decomposition (assumin ..."
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Cited by 6 (1 self)
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. Benders' decomposition is a wellknown technique for solving large linear programs with a special structure. In particular it is a popular technique for solving multistage stochastic linear programming problems. Early termination in the subproblems generated during Benders' decomposition (assuming dual feasibility) produces valid cuts which are inexact in the sense that they are not as constraining as cuts derived from an exact solution. We describe an inexact cut algorithm, prove its convergence under easily verifiable assumptions, and discuss a corresponding DantzigWolfe decomposition algorithm. The paper is concluded with some computational results from applying the algorithm to a class of stochastic programming problems which arise in hydroelectric scheduling. Key words. stochastic programming, Benders' decomposition, inexact cuts AMS subject classifications. 90C15, 90C05, 90C06, 90C90 1. Introduction. Many large linear programming problems have a block diagonal structure wh...
Warm Start and EpsilonSubgradients in Cutting Plane Scheme for BlockAngular Linear Programs
, 1997
"... This paper addresses the issues involved with an interior pointbased decomposition applied to the solution of linear programs with a blockangular structure. Unlike classical decomposition schemes that use the simplex method to solve subproblems, the approach presented in this paper employs a prima ..."
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Cited by 5 (3 self)
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This paper addresses the issues involved with an interior pointbased decomposition applied to the solution of linear programs with a blockangular structure. Unlike classical decomposition schemes that use the simplex method to solve subproblems, the approach presented in this paper employs a primaldual infeasible interior point method. The abovementioned algorithm offers a perfect measure of the distance to optimality, which is exploited to terminate the algorithm earlier (with a rather loose optimality tolerance) and to generate fflsubgradients. In the decomposition scheme, subproblems are sequentially solved for varying objective functions. It is essential to be able to exploit the optimal solution of the previous problem when solving a subsequent one (with a modified objective). A warm start routine is described that deals with this problem. The proposed approach has been implemented within the context of two optimization codes freely available for research use: the Analytic Ce...
Using selective orthonormalization to update the analytic center after the addition of multiple cuts
 Journal of Optimization Theory and Applications
"... We study the issue of updating the analytic center after multiple cutting planes have been added through the analytic center of the current polytope. This is an important issue that arises at every ‘stage ’ in a cutting plane algorithm. If q ≤ n cuts are to be added, we show that we can use a ‘Selec ..."
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Cited by 3 (2 self)
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We study the issue of updating the analytic center after multiple cutting planes have been added through the analytic center of the current polytope. This is an important issue that arises at every ‘stage ’ in a cutting plane algorithm. If q ≤ n cuts are to be added, we show that we can use a ‘Selective Orthonormalization ’ procedure to modify the cuts before adding them — it is then easy to identify a direction for an affine step into the interior of the new polytope, and the next analytic center is then found in O(q log q) Newton steps. Further, we show that multiple cut variants with selective orthonormalization of standard interior point cutting plane algorithms have the same complexity as the original algorithms.
On InteriorPoint Warmstarts for Linear and Combinatorial Optimization
, 2008
"... Despite the many advantages of interiorpoint algorithms over activeset methods for linear optimization, one of the remaining practical challenges is their current limitation to efficiently solve series of related problems by an effective warmstarting strategy. In its remedy, in this paper we prese ..."
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Cited by 3 (0 self)
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Despite the many advantages of interiorpoint algorithms over activeset methods for linear optimization, one of the remaining practical challenges is their current limitation to efficiently solve series of related problems by an effective warmstarting strategy. In its remedy, in this paper we present a new infeasibleinteriorpoint approach to quickly reoptimize an initial problem instance after data perturbations, or a new linear programming relaxation after adding cutting planes for discrete or combinatorial problems. Based on the detailed complexity analysis of the underlying algorithm, we perform a comparative analysis to coldstart initialization schemes and present encouraging computational results with iteration savings around 50 % on average for perturbations of the Netlib linear programs and successive LP relaxations of maxcut and the travelingsalesman problem.
Numerical Methods for LargeScale NonConvex Quadratic Programming
, 2001
"... We consider numerical methods for finding (weak) secondorder critical points for largescale nonconvex quadratic programming problems. We describe two new methods. The first is of the activeset variety. Although convergent from any starting point, it is intended primarily for the case where a goo ..."
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Cited by 2 (0 self)
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We consider numerical methods for finding (weak) secondorder critical points for largescale nonconvex quadratic programming problems. We describe two new methods. The first is of the activeset variety. Although convergent from any starting point, it is intended primarily for the case where a good estimate of the optimal active set can be predicted. The second is an interiorpoint trustregion type, and has proved capable of solving problems involving up to half a million unknowns and constraints. The solution of a key equality constrained subproblem, common to both methods, is described. The results of comparative tests on a large set of convex and nonconvex quadratic programming examples are given.