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On Lagrangian relaxation of quadratic matrix constraints
 SIAM J. Matrix Anal. Appl
, 2000
"... Abstract. Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equ ..."
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Cited by 45 (17 self)
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Abstract. Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equivalent to semidefinite programming relaxations. For several special cases of QQP, e.g., convex programs and trust region subproblems, the Lagrangian relaxation provides the exact optimal value, i.e., there is a zero duality gap. However, this is not true for the general QQP, or even the QQP with two convex constraints, but a nonconvex objective. In this paper we consider a certain QQP where the quadratic constraints correspond to the matrix orthogonality condition XXT = I. For this problem we show that the Lagrangian dual based on relaxing the constraints XXT = I and the seemingly redundant constraints XT X = I has a zero duality gap. This result has natural applications to quadratic assignment and graph partitioning problems, as well as the problem of minimizing the weighted sum of the largest eigenvalues of a matrix. We also show that the technique of relaxing quadratic matrix constraints can be used to obtain a strengthened semidefinite relaxation for the maxcut problem. Key words. Lagrangian relaxations, quadratically constrained quadratic programs, semidefinite programming, quadratic assignment, graph partitioning, maxcut problems
A Conic Formulation for l_pNorm Optimization
, 2000
"... In this paper, we formulate the l p norm optimization problem as a conic optimization problem, derive its standard duality properties and show it can be solved in polynomial time. We first define an ad hoc closed convex cone L p , study its properties and derive its dual. This allows us to express ..."
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Cited by 8 (0 self)
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In this paper, we formulate the l p norm optimization problem as a conic optimization problem, derive its standard duality properties and show it can be solved in polynomial time. We first define an ad hoc closed convex cone L p , study its properties and derive its dual. This allows us to express the standard l p norm optimization primal problem as a conic problem involving L p . Using convex conic duality and our knowledge about L p , we proceed to derive the dual of this problem and prove the wellknown regularity properties of this primaldual pair, i.e. zero duality gap and primal attainment. Finally, we prove that the class of l p norm optimization problems can be solved up to a given accuracy in polynomial time, using the framework of interiorpoint algorithms and selfconcordant barriers.
A Unifying Investigation of InteriorPoint Methods for Convex Programming
 FACULTY OF MATHEMATICS AND INFORMATICS, TU DELFT, NL2628 BL
, 1992
"... In the recent past a number of papers were written that present low complexity interiorpoint methods for different classes of convex programs. Goal of this article is to show that the logarithmic barrier function associated with these programs is selfconcordant, and that the analyses of interiorp ..."
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Cited by 5 (4 self)
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In the recent past a number of papers were written that present low complexity interiorpoint methods for different classes of convex programs. Goal of this article is to show that the logarithmic barrier function associated with these programs is selfconcordant, and that the analyses of interiorpoint methods for these programs can thus be reduced to the analysis of interiorpoint methods with selfconcordant barrier functions.
SEMIDEFINITE AND LAGRANGIAN RELAXATIONS FOR HARD COMBINATORIAL PROBLEMS
"... Semidefinite Programming is currently a very exciting and active area of research. Semidefinite relaxations generally provide very tight bounds for many classes of numerically hard problems. In addition, these relaxations can be solved efficiently by interiorpoint methods. In this ..."
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Cited by 2 (2 self)
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Semidefinite Programming is currently a very exciting and active area of research. Semidefinite relaxations generally provide very tight bounds for many classes of numerically hard problems. In addition, these relaxations can be solved efficiently by interiorpoint methods. In this
Notes on Duality in Second Order and POrder Cone Optimization
, 2000
"... Recently, the socalled second order cone optimization problem has received much attention, because the problem has many applications and the problem can at least in theory be solved eciently by interiorpoint methods. In this note we treat duality for second order cone optimization problems and in ..."
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Cited by 2 (0 self)
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Recently, the socalled second order cone optimization problem has received much attention, because the problem has many applications and the problem can at least in theory be solved eciently by interiorpoint methods. In this note we treat duality for second order cone optimization problems and in particular whether a nonzero duality gap can be introduced when casting a convex quadratically constrained optimization problem as a second order cone optimization problem. Furthermore, we also discuss the porder cone optimization problem which is a natural generalization of the second order case. Specically, we suggest a new selfconcordant barrier for the porder cone optimization problem. 1 Introdution The second order cone optimization problem can be stated as (SOCP) minimize f T x subject to jjA i x b i jj c i: x d i ; i = 1; : : : ; k; Hx = h: where A i 2 R (m i 1)n and H 2 R ln and all the other quantities have conforming dimensions. c i: denotes the ith row of ...
Conic Optimization: An Elegant Framework for Convex Optimization
, 2001
"... The purpose of this survey article is to introduce the reader to a very elegant formulation of convex optimization problems called conic optimization and outline its many advantages. After a brief introduction to convex optimization, the notion of convex cone is introduced, which leads to the conic ..."
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The purpose of this survey article is to introduce the reader to a very elegant formulation of convex optimization problems called conic optimization and outline its many advantages. After a brief introduction to convex optimization, the notion of convex cone is introduced, which leads to the conic formulation of convex optimization problems. This formulation features a very symmetric dual problem, and several useful duality theorems pertaining to this conic primaldual pair are presented. The usefulness of this approach is then demonstrated with its application to a wellknown class of convex problems called l p norm optimization. A suitably defined convex cone leads to a conic formulation for this problem, which allows us to derive its dual and the associated weak and strong duality properties in a seamless manner.
Deriving Duality for l_pnorm Optimization Using Conic Optimization
, 1999
"... In this paper, we formulate the l p norm optimization problem as a conic optimization problem and derive its standard duality properties. We first define an ad hoc closed convex cone L and derive its dual. We express then the standard l p norm optimization primal problem as a conic problem involvi ..."
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In this paper, we formulate the l p norm optimization problem as a conic optimization problem and derive its standard duality properties. We first define an ad hoc closed convex cone L and derive its dual. We express then the standard l p norm optimization primal problem as a conic problem involving L. Using convex conic duality, we derive the dual of this problem and prove the wellknown regularity properties of this primaldual pair, i.e. zero duality gap and dual attainment.