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Semidefinite optimization
 Acta Numerica
, 2001
"... Optimization problems in which the variable is not a vector but a symmetric matrix which is required to be positive semidefinite have been intensely studied in the last ten years. Part of the reason for the interest stems from the applicability of such problems to such diverse areas as designing the ..."
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Cited by 152 (2 self)
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Optimization problems in which the variable is not a vector but a symmetric matrix which is required to be positive semidefinite have been intensely studied in the last ten years. Part of the reason for the interest stems from the applicability of such problems to such diverse areas as designing the strongest column, checking the stability of a differential inclusion, and obtaining tight bounds for hard combinatorial optimization problems. Part also derives from great advances in our ability to solve such problems efficiently in theory and in practice (perhaps “or ” would be more appropriate: the most effective computational methods are not always provably efficient in theory, and vice versa). Here we describe this class of optimization problems, give a number of examples demonstrating its significance, outline its duality theory, and discuss algorithms for solving such problems.
Handbook of semidefinite programming
"... Semidefinite programming (or SDP) has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, con ..."
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Cited by 89 (3 self)
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Semidefinite programming (or SDP) has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, control theory, and statistics. This tremendous research activity was spurred by the discovery of important applications in combinatorial optimization and control theory, the development of efficient interiorpoint algorithms for solving SDP problems, and the depth and elegance of the underlying optimization theory. This book includes nineteen chapters on the theory, algorithms, and applications of semidefinite programming. Written by the leading experts on the subject, it offers an advanced and broad overview of the current state of the field. The coverage is somewhat less comprehensive, and the overall level more advanced, than we had planned at the start of the project. In order to finish the book in a timely fashion, we have had to abandon hopes for separate chapters on some important topics (such as a discussion of SDP algorithms in the
Semidefinite Programming Relaxations For The Quadratic Assignment Problem
, 1998
"... Semidefinite programming (SDP) relaxations for the quadratic assignment problem (QAP) are derived using the dual of the (homogenized) Lagrangian dual of appropriate equivalent representations of QAP. These relaxations result in the interesting, special, case where only the dual problem of the SDP re ..."
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Cited by 82 (23 self)
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Semidefinite programming (SDP) relaxations for the quadratic assignment problem (QAP) are derived using the dual of the (homogenized) Lagrangian dual of appropriate equivalent representations of QAP. These relaxations result in the interesting, special, case where only the dual problem of the SDP relaxation has strict interior, i.e. the Slater constraint qualification always fails for the primal problem. Although there is no duality gap in theory, this indicates that the relaxation cannot be solved in a numerically stable way. By exploring the geometrical structure of the relaxation, we are able to find projected SDP relaxations. These new relaxations, and their duals, satisfy the Slater constraint qualification, and so can be solved numerically using primaldual interiorpoint methods. For one of our models, a preconditioned conjugate gradient method is used for solving the large linear systems which arise when finding the Newton direction. The preconditioner is found by exploiting th...
Interior Point Trajectories in Semidefinite Programming
 SIAM Journal on Optimization
, 1996
"... In this paper we study interior point trajectories in semidefinite programming (SDP) including the central path of an SDP. This work was inspired by the seminal work by Megiddo on linear programming trajectories [15]. Under an assumption of primal and dual strict feasibility, we show that the primal ..."
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Cited by 38 (0 self)
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In this paper we study interior point trajectories in semidefinite programming (SDP) including the central path of an SDP. This work was inspired by the seminal work by Megiddo on linear programming trajectories [15]. Under an assumption of primal and dual strict feasibility, we show that the primal and dual central paths exist and converge to the analytic centers of the optimal faces of, respectively, the primal and the dual problems. We consider a class of trajectories that are similar to the central path, but can be constructed to pass through any given interior feasible point and study their convergence. Finally, we study the first order derivatives of these trajectories and their convergence. We also consider higher order derivatives associated with these trajectories.
A Predictor Corrector Method for Semidefinite Linear Programming
, 1995
"... In this paper we present a generalization of the predictor corrector method of linear programming problem to semidefinite linear programming problem. We consider a direction which, we show, belongs to a family of directions presented by Kojima, Shindoh and Hara, and, one of the directions analyzed b ..."
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Cited by 28 (1 self)
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In this paper we present a generalization of the predictor corrector method of linear programming problem to semidefinite linear programming problem. We consider a direction which, we show, belongs to a family of directions presented by Kojima, Shindoh and Hara, and, one of the directions analyzed by Monteiro. We show that starting with the initial complementary slackness violation of t 0 , in O(jlog( ffl t 0 )j p n) iterations of the predictor corrector method, the complementary slackness violation can be reduced to less than or equal to ffl ? 0. We also analyze a modified corrector direction in which the linear system to be solved differs from that of the predictor in only the right hand side, and obtain a similar bound. We then use this modified corrector step in an implementable method which is shown to take a total of O(jlog( ffl t 0 )j p nlog(n)) predictor and corrector steps. Key words: Linear programming, Semidefinite programming, Interior point methods, Path following, ...
Semidefinite Programming Duality and Linear TimeInvariant Systems
, 2003
"... Several important problems in control theory can be reformulated as semidefinite programming problems, i.e., minimization of a linear objective subject to Linear Matrix Inequality (LMI) constraints. From convex optimization duality theory, conditions for infeasibility of the LMIs as well as dual opt ..."
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Cited by 28 (3 self)
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Several important problems in control theory can be reformulated as semidefinite programming problems, i.e., minimization of a linear objective subject to Linear Matrix Inequality (LMI) constraints. From convex optimization duality theory, conditions for infeasibility of the LMIs as well as dual optimization problems can be formulated. These can in turn be reinterpreted in control or system theoretic terms, often yielding new results or new proofs for existing results from control theory. We explore such connections for a few problems associated with linear timeinvariant systems. 1
On Two InteriorPoint Mappings for Nonlinear Semidefinite Complementarity Problems
 Mathematics of Operations Research
, 1997
"... Extending our previous work Monteiro and Pang (1996), this paper studies properties of two fundamental mappings associated with the family of interiorpoint methods for solving monotone nonlinear complementarity problems over the cone of symmetric positive semidefinite matrices. The first of these m ..."
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Cited by 27 (9 self)
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Extending our previous work Monteiro and Pang (1996), this paper studies properties of two fundamental mappings associated with the family of interiorpoint methods for solving monotone nonlinear complementarity problems over the cone of symmetric positive semidefinite matrices. The first of these maps lead to a family of new continuous trajectories which include the central trajectory as a special case. These trajectories completely "fill up" the set of interior feasible points of the problem in the same way as the weighted central paths do the interior of the feasible region of a linear program. Unlike the approach based on the theory of maximal monotone maps taken by Shida and Shindoh (1996) and Shida, Shindoh, and Kojima (1995), our approach is based on the theory of local homeomorphic maps in nonlinear analysis. Key words: interior point methods, mixed nonlinear complementarity problems, generalized complementarity problems, maximal monotonicity, monotone mappings, continuous traj...
Duality Results For Conic Convex Programming
, 1997
"... This paper presents a unified study of duality properties for the problem of minimizing a linear function over the intersection of an affine space with a convex cone infinite dimension. Existing duality results are carefully surveyed and some new duality properties are established. Examples are give ..."
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Cited by 26 (10 self)
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This paper presents a unified study of duality properties for the problem of minimizing a linear function over the intersection of an affine space with a convex cone infinite dimension. Existing duality results are carefully surveyed and some new duality properties are established. Examples are given to illustrate these new properties. The topics covered in this paper include GordonStiemke type theorems, Farkas type theorems, perfect duality, Slater condition, regularization, Ramana's duality, and approximate dualities. The dual representations of various convex sets, convex cones and conic convex programs are also discussed.
On the closedness of the linear image of a closed convex cone
, 1992
"... informs doi 10.1287/moor.1060.0242 ..."
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ConeLP's and Semidefinite Programs: Geometry and a Simplextype Method
, 1996
"... . We consider optimization problems expressed as a linear program with a cone constraint. ConeLP's subsume ordinary linear programs, and semidefinite programs. We study the notions of basic solutions, nondegeneracy, and feasible directions, and propose a generalization of the simplex method fo ..."
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Cited by 22 (2 self)
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. We consider optimization problems expressed as a linear program with a cone constraint. ConeLP's subsume ordinary linear programs, and semidefinite programs. We study the notions of basic solutions, nondegeneracy, and feasible directions, and propose a generalization of the simplex method for a large class including LP's and SDP's. One key feature of our approach is considering feasible directions as a sum of two directions. In LP, these correspond to variables leaving and entering the basis, respectively. The resulting algorithm for SDP inherits several important properties of the LPsimplex method. In particular, the linesearch can be done in the current face of the cone, similarly to LP, where the linesearch must determine only the variable leaving the basis. 1 Introduction Consider the optimization problem Min cx s:t: x 2 K Ax = b (P ) where K is a closed cone in R k , A 2 R m\Thetak ; b 2 R m ; c 2 R k : (P ) is called a linear program over a cone, or a coneLP. It m...