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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
Cones Of Matrices And Successive Convex Relaxations Of Nonconvex Sets
, 2000
"... . Let F be a compact subset of the ndimensional Euclidean space R n represented by (finitely or infinitely many) quadratic inequalities. We propose two methods, one based on successive semidefinite programming (SDP) relaxations and the other on successive linear programming (LP) relaxations. Each ..."
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Cited by 50 (19 self)
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. Let F be a compact subset of the ndimensional Euclidean space R n represented by (finitely or infinitely many) quadratic inequalities. We propose two methods, one based on successive semidefinite programming (SDP) relaxations and the other on successive linear programming (LP) relaxations. Each of our methods generates a sequence of compact convex subsets C k (k = 1, 2, . . . ) of R n such that (a) the convex hull of F # C k+1 # C k (monotonicity), (b) # # k=1 C k = the convex hull of F (asymptotic convergence). Our methods are extensions of the corresponding LovaszSchrijver liftandproject procedures with the use of SDP or LP relaxation applied to general quadratic optimization problems (QOPs) with infinitely many quadratic inequality constraints. Utilizing descriptions of sets based on cones of matrices and their duals, we establish the exact equivalence of the SDP relaxation and the semiinfinite convex QOP relaxation proposed originally by Fujie and Kojima. Using th...
Convex sets with semidefinite representation. Optimization Online
, 2006
"... Abstract. We provide a sufficient condition on a class of compact basic semialgebraic sets K ⊂ R n for their convex hull co(K) to have a semidefinite representation (SDr). This SDr is explicitly expressed in terms of the polynomials gj that define K. Examples are provided. We also provide an approxi ..."
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Cited by 30 (1 self)
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Abstract. We provide a sufficient condition on a class of compact basic semialgebraic sets K ⊂ R n for their convex hull co(K) to have a semidefinite representation (SDr). This SDr is explicitly expressed in terms of the polynomials gj that define K. Examples are provided. We also provide an approximate SDr; that is, for every fixed ɛ> 0, there is a convex set Kɛ such that co(K) ⊆ Kɛ ⊆ co(K) + ɛB (where B is the unit ball of R n), and Kɛ has an explicit SDr in terms of the gj’s. For convex and compact basic semialgebraic sets K defined by concave polynomials, we provide a simpler explicit SDr when the nonnegative Lagrangian Lf associated with K and any linear f ∈ R[X] is a sum of squares. We also provide an approximate SDr specific to the convex case. 1.
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 18 (3 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.
Generic optimality conditions for semialgebraic convex programs
 Math. Oper. Res
"... We consider linear optimization over a nonempty convex semialgebraic feasible region F. Semidefinite programming is an example. If F is compact, then for almost every linear objective there is a unique optimal solution, lying on a unique “active ” manifold, around which F is “partly smooth”, and th ..."
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Cited by 16 (6 self)
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We consider linear optimization over a nonempty convex semialgebraic feasible region F. Semidefinite programming is an example. If F is compact, then for almost every linear objective there is a unique optimal solution, lying on a unique “active ” manifold, around which F is “partly smooth”, and the secondorder sufficient conditions hold. Perturbing the objective results in smooth variation of the optimal solution. The active manifold consists, locally, of these perturbed optimal solutions; it is independent of the representation of F, and is eventually identified by a variety of iterative algorithms such as proximal and projected gradient schemes. These results extend to unbounded sets F. Key words: convex optimization, sensitivity analysis, partial smoothness, identifiable surface, active sets, generic, secondorder sufficient conditions, semialgebraic.
Towards Implementations of Successive Convex Relaxation Methods for Nonconvex Quadratic Optimization Problems
, 1999
"... Recently Kojima and Tuncel proposed new successive convex relaxation methods and their localizeddiscretized variants for general nonconvex quadratic optimization problems. Although an upper bound of the optimal objective function value within a previously given precision can be found theoretically ..."
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Cited by 11 (6 self)
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Recently Kojima and Tuncel proposed new successive convex relaxation methods and their localizeddiscretized variants for general nonconvex quadratic optimization problems. Although an upper bound of the optimal objective function value within a previously given precision can be found theoretically by solving a finite number of linear programs, several important implementation issues remain unsolved. In this paper, we discuss those issues, present practically implementable algorithms and report numerical results.
GENERIC NONDEGENERACY IN CONVEX OPTIMIZATION
"... Abstract. We show that minimizers of convex functions subject to almost all linear perturbations are nondegenerate. An analogous result holds more generally for lowerC2 functions. 1. ..."
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Cited by 6 (3 self)
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Abstract. We show that minimizers of convex functions subject to almost all linear perturbations are nondegenerate. An analogous result holds more generally for lowerC2 functions. 1.
The GaussNewton Direction in Semidefinite Programming
, 1998
"... Primaldual interiorpoint methods have proven to be very successful for both linear programming (LP) and, more recently, for semidefinite programming (SDP) problems. Many of the techniques that have been so successful for LP have been extended to SDP. In fact, interior point methods are currently t ..."
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Cited by 5 (3 self)
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Primaldual interiorpoint methods have proven to be very successful for both linear programming (LP) and, more recently, for semidefinite programming (SDP) problems. Many of the techniques that have been so successful for LP have been extended to SDP. In fact, interior point methods are currently the only successful techniques for SDP.
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