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22
Valid inequalities for mixed integer linear programs
 Mathematical Programming B
, 2006
"... Abstract. This tutorial presents a theory of valid inequalities for mixed integer linear sets. It introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as liftandproject cuts, Gomory mixed integer cuts, mi ..."
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Cited by 31 (0 self)
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Abstract. This tutorial presents a theory of valid inequalities for mixed integer linear sets. It introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as liftandproject cuts, Gomory mixed integer cuts, mixed integer rounding cuts, split cuts and intersection cuts, and it reveals the relationships between these families. The tutorial also discusses computational aspects of generating the cuts and their strength. Key words: mixed integer linear program, liftandproject, split cut, Gomory cut, mixed integer rounding, elementary closure, polyhedra, union of polyhedra 1.
A NewtonCG augmented Lagrangian method for semidefinite programming
 SIAM J. Optim
"... Abstract. We consider a NewtonCG augmented Lagrangian method for solving semidefinite programming (SDP) problems from the perspective of approximate semismooth Newton methods. In order to analyze the rate of convergence of our proposed method, we characterize the Lipschitz continuity of the corresp ..."
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Cited by 30 (6 self)
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Abstract. We consider a NewtonCG augmented Lagrangian method for solving semidefinite programming (SDP) problems from the perspective of approximate semismooth Newton methods. In order to analyze the rate of convergence of our proposed method, we characterize the Lipschitz continuity of the corresponding solution mapping at the origin. For the inner problems, we show that the positive definiteness of the generalized Hessian of the objective function in these inner problems, a key property for ensuring the efficiency of using an inexact semismooth NewtonCG method to solve the inner problems, is equivalent to the constraint nondegeneracy of the corresponding dual problems. Numerical experiments on a variety of large scale SDPs with the matrix dimension n up to 4, 110 and the number of equality constraints m up to 2, 156, 544 show that the proposed method is very efficient. We are also able to solve the SDP problem fap36 (with n = 4, 110 and m = 1, 154, 467) in the Seventh DIMACS Implementation Challenge much more accurately than previous attempts.
Regularization methods for semidefinite programming
 SIAM Journal on Optimization
"... We introduce a new class of algorithms for solving linear semidefinite programming (SDP) problems. Our approach is based on classical tools from convex optimization such as quadratic regularization and augmented Lagrangian techniques. We study the theoretical properties and we show that practical im ..."
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Cited by 27 (4 self)
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We introduce a new class of algorithms for solving linear semidefinite programming (SDP) problems. Our approach is based on classical tools from convex optimization such as quadratic regularization and augmented Lagrangian techniques. We study the theoretical properties and we show that practical implementations behave very well on some instances of SDP having a large number of constraints. We also show that the “boundary point method ” from [PRW06] is an instance of this class. Key words: semidefinite programming, regularization methods, augmented Lagrangian method, large scale semidefinite problems
Alternating direction augmented Lagrangian methods for semidefinite programming
, 2009
"... Abstract. We present an alternating direction method based on an augmented Lagrangian framework for solving semidefinite programming (SDP) problems in standard form. At each iteration, the algorithm, also known as a twosplitting scheme, minimizes the dual augmented Lagrangian function sequentially ..."
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Cited by 21 (2 self)
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Abstract. We present an alternating direction method based on an augmented Lagrangian framework for solving semidefinite programming (SDP) problems in standard form. At each iteration, the algorithm, also known as a twosplitting scheme, minimizes the dual augmented Lagrangian function sequentially with respect to the Lagrange multipliers corresponding to the linear constraints, then the dual slack variables and finally the primal variables, while in each minimization keeping the other variables fixed. Convergence is proved by using a fixedpoint argument. A multiplesplitting algorithm is then proposed to handle SDPs with inequality constraints and positivity constraints directly without transforming them to the equality constraints in standard form. Finally, numerical results for frequency assignment, maximum stable set and binary integer quadratic programming problems are presented to demonstrate the robustness and efficiency of our algorithm.
Copositive and semidefinite relaxations of the quadratic assignment problem, Discrete Optim
"... Semidefinite relaxations of the quadratic assignment problem (QAP) have recently turned out to provide good approximations to the optimal value of QAP. We take a systematic look at various conic relaxations of QAP. We first show that QAP can equivalently be formulated as a linear program over the co ..."
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Cited by 18 (3 self)
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Semidefinite relaxations of the quadratic assignment problem (QAP) have recently turned out to provide good approximations to the optimal value of QAP. We take a systematic look at various conic relaxations of QAP. We first show that QAP can equivalently be formulated as a linear program over the cone of completely positive matrices. Since it is hard to optimize over this cone, we also look at tractable approximations and compare with several relaxations from the literature. We show that several of the wellstudied models are in fact equivalent. It is still a challenging task to solve the strongest of these models to reasonable accuracy on instances of moderate size. Key words: quadratic assignment problem, copositive programming, semidefinite relaxations, liftandproject relaxations.
Estimating Bounds for Quadratic Assignment Problems Associated with Hamming and Manhattan Distance Matrices based on Semidefinite Programming
, 2008
"... Quadratic assignment problems (QAPs) with a Hamming distance matrix for a hypercube or a Manhattan distance matrix for a rectangular grid arise frequently from communications and facility locations and are known to be among the hardest discrete optimization problems. In this paper we consider the is ..."
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Cited by 7 (2 self)
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Quadratic assignment problems (QAPs) with a Hamming distance matrix for a hypercube or a Manhattan distance matrix for a rectangular grid arise frequently from communications and facility locations and are known to be among the hardest discrete optimization problems. In this paper we consider the issue of how to obtain lower bounds for those two classes of QAPs based on semidefinite programming (SDP). By exploiting the data structure of the distance matrix B, we first show that for any permutation matrix X, the matrix Y = αE − XBX T is positive semidefinite, where α is a properly chosen parameter depending only on the associated graph in the underlying QAP and E = ee T is a rank one matrix whose elements have value 1. This results in a natural way to approximate the original QAPs via SDP relaxation based on the matrix splitting technique. Our new SDP relaxations have a smaller size compared with other SDP relaxations in the literature and can be solved efficiently by most open source SDP solvers. Experimental results show that for the underlying QAPs of size up to n=200, strong bounds can be obtained effectively.
New Convex Relaxations for Quadratic Assignment Problems
, 2008
"... Quadratic assignment problems (QAPs) are known to be among the hardest discrete optimization problems. Recent study shows that even obtaining a strong lower bound for QAPs is a computational challenge. In this paper, we first discuss how to construct new simple convex relaxations of QAPs based on va ..."
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Cited by 3 (2 self)
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Quadratic assignment problems (QAPs) are known to be among the hardest discrete optimization problems. Recent study shows that even obtaining a strong lower bound for QAPs is a computational challenge. In this paper, we first discuss how to construct new simple convex relaxations of QAPs based on various matrix splitting schemes. Then we introduce the socalled symmetric mappings that can be used to derive strong cuts for the proposed relaxation model. We show that the bounds based on the new models are comparable to the strongest bounds in the literature. Promising experimental results based on the new relaxations will be reported. Key words. Quadratic Assignment Problem (QAP), Semidefinite Programming
SDP relaxations for some combinatorial optimization problems
"... In this chapter we present recent developments on solving various combinatorial optimization problems by using semidefinite programming (SDP). We present several SDP relaxations of the quadratic assignment problem and the traveling salesman problem. Further, we show the equivalence of several known ..."
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Cited by 2 (2 self)
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In this chapter we present recent developments on solving various combinatorial optimization problems by using semidefinite programming (SDP). We present several SDP relaxations of the quadratic assignment problem and the traveling salesman problem. Further, we show the equivalence of several known SDP relaxations of the graph equipartition problem, and present recent results on the bandwidth problem. Notation The space of p × q real matrices is denoted by Rp×q, the space of k × k symmetric matrices is denoted by Sk, and the space of k×k symmetric positive semidefinite matrices by S + k. We will sometimes also use the notation X ≽ 0 instead of X ∈ S +, if the order of the matrix is clear from the context. k For index sets α, β ⊂ {1,..., n}, we denote the submatrix that contains the rows of A indexed by α and the columns indexed by β as A(α, β). If α = β,
A pCone Sequential Relaxation Procedure for 01 Integer Programs
, 2009
"... Several authors have introduced sequential relaxation techniques — based on linear and/or semidefinite programming — to generate the convex hull of 01 integer points in a polytope in at most n steps. In this paper, we introduce a sequential relaxation technique, which is based on porder cone progr ..."
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Cited by 2 (0 self)
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Several authors have introduced sequential relaxation techniques — based on linear and/or semidefinite programming — to generate the convex hull of 01 integer points in a polytope in at most n steps. In this paper, we introduce a sequential relaxation technique, which is based on porder cone programming (1 ≤ p ≤ ∞). We prove that our technique generates the convex hull of 01 solutions asymptotically. In addition, we show that our method generalizes and subsumes several existing methods. For example, when p = ∞, our method corresponds to the wellknown procedure of Lovász and Schrijver based on linear programming. Although the porder cone programs in general sacrifice some strength compared to the analogous linear and semidefinite programs, we show that for p = 2 they enjoy a better theoretical iteration complexity. Computational considerations of our technique are discussed.