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Warm starting the homogeneous and selfdual interior point method for linear and conic quadratic problems.
 Mathematical Programming Computation,
, 2013
"... Abstract We present two strategies for warmstarting primaldual interior point methods for the homogeneous selfdual model when applied to mixed linear and quadratic conic optimization problems. Common to both strategies is their use of only the final (optimal) iterate of the initial problem and th ..."
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Abstract We present two strategies for warmstarting primaldual interior point methods for the homogeneous selfdual model when applied to mixed linear and quadratic conic optimization problems. Common to both strategies is their use of only the final (optimal) iterate of the initial problem and their negligible computational cost. This is a major advantage when compared to previously suggested strategies that require a pool of iterates from the solution process of the initial problem. Consequently our strategies are better suited for users who use optimization algorithms as blackbox routines which usually only output the final solution. Our two strategies differ in that one assumes knowledge only of the final primal solution while the other assumes the availability of both primal and dual solutions. We analyze the strategies and deduce conditions under which they result in improved theoretical worstcase complexity. We present extensive computational results showing work reductions when warmstarting compared to coldstarting in the range 30%75% depending on the problem class and magnitude of the problem perturbation. The computational experiments thus substantiate that the warmstarting strategies are useful in practice.
LP and SDP branchandcut algorithms for the minimum graph bisection problem: a computational comparison
, 2011
"... While semidefinite relaxations are known to deliver good approximations for combinatorial optimization problems like graph bisection, their practical scope is mostly associated with small dense instances. For large sparse instances, cutting plane techniques are considered the method of choice. These ..."
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While semidefinite relaxations are known to deliver good approximations for combinatorial optimization problems like graph bisection, their practical scope is mostly associated with small dense instances. For large sparse instances, cutting plane techniques are considered the method of choice. These are also applicable for semidefinite relaxations via the spectral bundle method, which allows to exploit structural properties like sparsity. In order to evaluate the relative strengths of linear and semidefinite approaches for large sparse instances, we set up a common branchandcut framework for linear and semidefinite relaxations of the minimum graph bisection problem. It incorporates separation algorithms for valid inequalities of the bisection cut polytope described in a recent study by the authors. While the problem specific cuts help to strengthen the linear relaxation significantly, the semidefinite bound profits much more from separating the cycle inequalities of the cut polytope on a slightly enlarged support. Extensive numerical experiments show that this semidefinite branchandcut approach without problem specific cuts is a superior choice to the classical simplex approach exploiting bisection specific inequalities on a clear majority of our large sparse test instances from VLSI design and numerical optimization.
A new warmstarting strategy for the primaldual column generation method.
 Mathematical Programming
, 2014
"... Abstract This paper presents a new warmstarting technique in the context of a primaldual column generation method applied to solve a particular class of combinatorial optimization problems. The technique relies on calculating an initial point and on solving auxiliary linear optimization problems t ..."
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Abstract This paper presents a new warmstarting technique in the context of a primaldual column generation method applied to solve a particular class of combinatorial optimization problems. The technique relies on calculating an initial point and on solving auxiliary linear optimization problems to determine the step direction needed to fully restore primal and dual feasibilities after new columns arrive. Conditions on the maximum size of the cuts and on a suitable initial point are discussed. Additionally, the strategy ensures that the duality gap of the warmstart is bounded by the old duality gap multiplied with a (small) constant, which depends on the relation between the old and modified problems. Computational experiments demonstrate the gains achieved when compared to a coldstart approach.
Using the primaldual interior point algorithm within the branchpriceandcut method.
 Computers & Operations Research
, 2013
"... Abstract Branchpriceandcut has proven to be a powerful method for solving integer programming problems. It combines decomposition techniques with the generation of both columns and valid inequalities and relies on strong bounds to guide the search in the branchandbound tree. In this paper, we ..."
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Abstract Branchpriceandcut has proven to be a powerful method for solving integer programming problems. It combines decomposition techniques with the generation of both columns and valid inequalities and relies on strong bounds to guide the search in the branchandbound tree. In this paper, we present how to improve the performance of a branchpriceandcut method by using the primaldual interior point algorithm. We discuss in detail how to deal with the challenges of using the interior point algorithm with the core components of the branchpriceandcut method. The effort to overcome the difficulties pays off in a number of advantageous features offered by the new approach. We present the computational results of solving wellknown instances of the Vehicle Routing Problem with Time Windows, a challenging integer programming problem. The results indicate that the proposed approach delivers the best overall performance when compared with a similar branchpriceandcut method which is based on the simplex algorithm.
A NONLINEAR FEASIBILITY PROBLEM HEURISTIC
, 2015
"... ABSTRACT. In this work we consider a region S ⊂ R n given by a finite number of nonlinear smooth convex inequalities and having nonempty interior. We assume a point x 0 is given, which is close in certain norm to the analytic center of S, and that a new nonlinear smooth convex inequality is added t ..."
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ABSTRACT. In this work we consider a region S ⊂ R n given by a finite number of nonlinear smooth convex inequalities and having nonempty interior. We assume a point x 0 is given, which is close in certain norm to the analytic center of S, and that a new nonlinear smooth convex inequality is added to those defining S (perturbed region). It is constructively shown how to obtain a shift of the righthand side of this inequality such that the point x 0 is still close (in the same norm) to the analytic center of this shifted region. Starting from this point and using the theoretical results shown, we develop a heuristic that allows us to obtain the approximate analytic center of the perturbed region. Then, we present a procedure to solve the problem of nonlinear feasibility. The procedure was implemented and we performed some numerical tests for the quadratic (random) case.
Activeset
"... prediction for interior point methods using controlled perturbations Coralia Cartis∗and Yiming Yan† We propose the use of controlled perturbations to address the challenging question of optimal activeset prediction for interior point methods. Namely, in the context of linear programming, we conside ..."
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prediction for interior point methods using controlled perturbations Coralia Cartis∗and Yiming Yan† We propose the use of controlled perturbations to address the challenging question of optimal activeset prediction for interior point methods. Namely, in the context of linear programming, we consider perturbing the inequality constraints/bounds so as to enlarge the feasible set. We show that if the perturbations are chosen appropriately, the solution of the original problem lies on or close to the central path of the perturbed problem. We also find that a primaldual pathfollowing algorithm applied to the perturbed problem is able to accurately predict the optimal active set of the original problem when the duality gap for the perturbed problem is not too small; furthermore, depending on problem conditioning, this prediction can happen sooner than predicting the activeset for the perturbed problem or for the original one if no perturbations are used. Encouraging preliminary numerical experience is reported when comparing activity prediction for the perturbed and unperturbed problem formulations.
Efficiently Solving Repeated Integer Linear Programming Problems by Learning Solutions of Similar Linear Programming Problems using Boosting Trees
, 2015
"... It is challenging to obtain online solutions of largescale integer linear programming (ILP) problems that occur frequently in slightly different forms during planning for autonomous systems. We refer to such ILP problems as repeated ILP problems. The branchandbound (BAB) algorithm is commonly use ..."
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It is challenging to obtain online solutions of largescale integer linear programming (ILP) problems that occur frequently in slightly different forms during planning for autonomous systems. We refer to such ILP problems as repeated ILP problems. The branchandbound (BAB) algorithm is commonly used to solve ILP problems, and a significant amount of computation time is expended in solving numerous relaxed linear programming (LP) problems at the nodes of the BAB trees. We observe that the relaxed LP problems, both within a particular BAB tree and across multiple trees for repeated ILP problems, are similar to each other in the sense that they contain almost the same number of constraints, similar objective function and constraint coefficients, and an identical number of decision variables. We present a boosting treebased regression technique for learning a set of functions that map the objective function and the constraints to the decision variables of such a system of similar LP problems; this enables us to efficiently infer approximately optimal solutions of the repeated ILP problems. We provide theoretical performance guarantees on the predicted values and demonstrate the effectiveness of the algorithm in four representative domains involving a library of benchmark ILP problems, aircraft carrier deck scheduling, vehicle routing,