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12
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
HOMOTOPY CONTINUATION METHODS FOR NONLINEAR COMPLEMENTARITY PROBLEMS
, 1991
"... A complementarity problem with a continuous mapping f from Rn into itself can be written as the system of equations F(x, y) = 0 and (x, y)> 0. Here F is the mapping from R ~ " into itself defined by F(x, y) = ( xl y,, x2yZ,..., x, ~ ye, y ffx)). Under the assumption that the mapping f is a P,,f ..."
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Cited by 33 (3 self)
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A complementarity problem with a continuous mapping f from Rn into itself can be written as the system of equations F(x, y) = 0 and (x, y)> 0. Here F is the mapping from R ~ " into itself defined by F(x, y) = ( xl y,, x2yZ,..., x, ~ ye, y ffx)). Under the assumption that the mapping f is a P,,function, we study various aspects of homotopy continuation methods that trace a trajectory consisting of solutions of the family of systems of equations F(x, y) = t(a, b) and (x, y) 8 0 until the parameter t> 0 attains 0. Here (a, b) denotes a 2ndimensional constant positive vector. We establish the existence of a trajectory which leads to a solution of the problem, and then present a numerical method for tracing the trajectory. We also discuss the global and local convergence of the method.
A Convergence Analysis of the Scalinginvariant Primaldual Pathfollowing Algorithms for Secondorder Cone Programming
 Optim. Methods Softw
, 1998
"... This paper is a continuation of our previous paper in which we studied a polynomial primaldual pathfollowing algorithm for SOCP using an analogue of the HRVW/KSH/M direction for SDP. We develop an improved and simplified complexity analysis which can be also applied to the algorithm using the NT di ..."
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Cited by 28 (4 self)
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This paper is a continuation of our previous paper in which we studied a polynomial primaldual pathfollowing algorithm for SOCP using an analogue of the HRVW/KSH/M direction for SDP. We develop an improved and simplified complexity analysis which can be also applied to the algorithm using the NT direction. Specifically, we show that the longstep algorithm using the NT direction has O(n log " 01 ) iterationcomplexity to reduce the duality gap by a factor of ", where n is the number of the secondorder cones. The complexity for the same algorithm using the HRVW/KSH/M direction is improved to O(n 3=2 log " 01 ) from O(n 3 log " 01 ) of the previous analysis. We also show that the short and semilongstep algorithms using the NT direction (and the HRVW/KSH/M direction) have O( p n log " 01 ) and O(n log " 01 ) iterationcomplexities, respectively. keywords: secondorder cone, interiorpoint methods, polynomial complexity, primaldual pathfollowing methods. 1 Introduction...
A Computational Study of the Homogeneous Algorithm for LargeScale Convex Optimization
, 1997
"... Recently the authors have proposed a homogeneous and selfdual algorithm for solving the monotone complementarity problem (MCP) [5]. The algorithm is a single phase interiorpoint type method, nevertheless it yields either an approximate optimal solution or detects a possible infeasibility of th ..."
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Cited by 14 (1 self)
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Recently the authors have proposed a homogeneous and selfdual algorithm for solving the monotone complementarity problem (MCP) [5]. The algorithm is a single phase interiorpoint type method, nevertheless it yields either an approximate optimal solution or detects a possible infeasibility of the problem. In this paper we specialize the algorithm to the solution of general smooth convex optimization problems that also possess nonlinear inequality constraints and free variables. We discuss an implementation of the algorithm for largescale sparse convex optimization. Moreover, we present computational results for solving quadratically constrained quadratic programming and geometric programming problems, where some of the problems contain more than 100,000 constraints and variables. The results indicate that the proposed algorithm is also practically efficient. Department of Management, Odense University, Campusvej 55, DK5230 Odense M, Denmark. Email: eda@busieco.ou.dk y ...
A Polynomial PrimalDual PathFollowing Algorithm for Secondorder Cone Programming
 Research Memorandum No. 649, The Institute of Statistical Mathematics
, 1997
"... Secondorder cone programming (SOCP) is the problem of minimizing linear objective function over crosssection of secondorder cones and an affine space. Recently this problem gets more attention because of its various important applications including quadratically constrained convex quadratic progr ..."
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Cited by 13 (1 self)
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Secondorder cone programming (SOCP) is the problem of minimizing linear objective function over crosssection of secondorder cones and an affine space. Recently this problem gets more attention because of its various important applications including quadratically constrained convex quadratic programming. In this paper we deal with a primaldual pathfollowing algorithm for SOCP to show many of the ideas developed for primaldual algorithms for LP and SDP carry over to this problem. We define neighborhoods of the central trajectory in terms of the "eigenvalues" of the secondorder cone, and develop an analogue of HRVW/KSH/M direction, and establish O( p n log " 01 ), O(n log " 01 ) and O(n 3 log " 01 ) iterationcomplexity bounds for shortstep, semilongstep and longstep pathfollowing algorithms, respectively, to reduce the duality gap by a factor of ". keywords: secondorder cone, interiorpoint methods, polynomial complexity, primaldual pathfollowing methods. 1 Intro...
Optimization with Semidefinite, Quadratic and Linear Constraints
 RUTCOR, RUTGERS UNIVERSITY
, 1997
"... We consider optimization problems where variables have either linear, or convex quadratic or semidefinite constraints. First, we define and characterize primal and dual nondegeneracy and strict complementarity conditions. Next, we develop primaldual interior point methods for such problems and show ..."
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Cited by 12 (2 self)
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We consider optimization problems where variables have either linear, or convex quadratic or semidefinite constraints. First, we define and characterize primal and dual nondegeneracy and strict complementarity conditions. Next, we develop primaldual interior point methods for such problems and show that in the absence of degeneracy these algorithms are numerically stable. Finally we describe an implementation of our method and present numerical experiments with both degenerate and nondegenerate problems.
A Long Step Barrier Method for Convex Quadratic Programming
 Algorithmica
, 1990
"... In this paper we propose a longstep logarithmic barrier function method for convex quadratic programming with linear equality constraints. After a reduction of the barrier parameter, a series of long steps along projected Newton directions are taken until the iterate is in the vicinity of the cent ..."
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Cited by 8 (2 self)
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In this paper we propose a longstep logarithmic barrier function method for convex quadratic programming with linear equality constraints. After a reduction of the barrier parameter, a series of long steps along projected Newton directions are taken until the iterate is in the vicinity of the center associated with the current value of the barrier parameter. We prove that the total number of iterations is O( p nL) or O(nL), dependent on how the barrier parameter is updated. Key Words: convex quadratic programming, interior point method, logarithmic barrier function, polynomial algorithm. 1 Introduction Karmarkar's [14] invention of the projective method for linear programming has given rise to active research in interior point algorithms. At this moment, the variants can roughly be categorized into four classes: projective, affine scaling, pathfollowing and potential reduction methods. Researchers have also extended interior point methods to other problems, including convex qu...
Infinite Dimensional Quadratic Optimization: InteriorPoint Methods and Control Applications
 JOURNAL OF APPLIED MATHEMATICS AND OPTIMIZATION
, 1995
"... An infinitedimensional convex optimization problem with the linearquadratic cost function and linearquadratic constraints is considered. We generalize the interiorpoint techniques of NesterovNemirovsky to this infinitedimensional situation. The obtained complexity estimates are similar to fini ..."
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Cited by 5 (3 self)
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An infinitedimensional convex optimization problem with the linearquadratic cost function and linearquadratic constraints is considered. We generalize the interiorpoint techniques of NesterovNemirovsky to this infinitedimensional situation. The obtained complexity estimates are similar to finitedimensional ones. We apply our results to the linearquadratic control problem with quadratic constraints. It is shown that for this problem the Newton step is basically reduced to the standard LQproblem.
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
A new linesearch step based on the Weierstrass function for minimizing a class of logarithmic barrier functions
, 1994
"... This article is concerned with linesearch procedures for a class of problems with certain nonlinear constraints. The class includes as special cases linear and convex quadratic programming problems, entropy programming problems and minimization problems over the cone of positive semidefinite matric ..."
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Cited by 1 (0 self)
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This article is concerned with linesearch procedures for a class of problems with certain nonlinear constraints. The class includes as special cases linear and convex quadratic programming problems, entropy programming problems and minimization problems over the cone of positive semidefinite matrices, [1, 2, 18, 7]. For solving a constrained optimization problem of the form minff 0 (x) j f i (x) 0 for 1 i mg (1.1) with four times continuously differentiable functions f i , we consider logarithmic barrier functions of the form