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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
Strong duality for semidefinite programming
 SIAM J. Optim
, 1997
"... Abstract. It is well known that the duality theory for linear programming (LP) is powerful and elegant and lies behind algorithms such as simplex and interiorpoint methods. However, the standard Lagrangian for nonlinear programs requires constraint qualifications to avoid duality gaps. Semidefinite ..."
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Cited by 63 (18 self)
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Abstract. It is well known that the duality theory for linear programming (LP) is powerful and elegant and lies behind algorithms such as simplex and interiorpoint methods. However, the standard Lagrangian for nonlinear programs requires constraint qualifications to avoid duality gaps. Semidefinite linear programming (SDP) is a generalization of LP where the nonnegativity constraints are replaced by a semidefiniteness constraint on the matrix variables. There are many applications, e.g., in systems and control theory and combinatorial optimization. However, the Lagrangian dual for SDP can have a duality gap. We discuss the relationships among various duals and give a unified treatment for strong duality in semidefinite programming. These duals guarantee strong duality, i.e., a zero duality gap and dual attainment. This paper is motivated by the recent paper by Ramana where one of these duals is introduced.
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
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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Lifts of convex sets and cone factorizations
 Mathematics of OR
"... Abstract. In this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone. Such a representation or “lift ” of the convex set is especially useful if the cone admits an efficient algorithm for linear op ..."
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Cited by 21 (8 self)
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Abstract. In this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone. Such a representation or “lift ” of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over its affine slices. We show that the existence of a lift of a convex set to a cone is equivalent to the existence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual. This generalizes a theorem of Yannakakis that established a connection between polyhedral lifts of a polytope and nonnegative factorizations of its slack matrix. Symmetric lifts of convex sets can also be characterized similarly. When the cones live in a family, our results lead to the definition of the rank of a convex set with respect to this family. We present results about this rank in the context of cones of positive semidefinite matrices. Our methods provide new tools for understanding cone lifts of convex sets. 1.
Strong Duality and Minimal Representations for Cone Optimization
, 2008
"... The elegant results for strong duality and strict complementarity for linear programming, LP, can fail for cone programming over nonpolyhedral cones. One can have: unattained optimal values; nonzero duality gaps; and no primaldual optimal pair that satisfies strict complementarity. This failure is ..."
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Cited by 14 (2 self)
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The elegant results for strong duality and strict complementarity for linear programming, LP, can fail for cone programming over nonpolyhedral cones. One can have: unattained optimal values; nonzero duality gaps; and no primaldual optimal pair that satisfies strict complementarity. This failure is tied to the nonclosure of sums of nonpolyhedral closed cones. We take a fresh look at known and new results for duality, optimality, constraint qualifications, and strict complementarity, for linear cone optimization problems in finite dimensions. These results include: weakest and universal constraint qualifications, CQs; duality and characterizations of optimality that hold without any CQ; geometry of nice and devious cones; the geometric relationships between zero duality gaps, strict complementarity, and the facial structure of cones; and, the connection between theory and empirical evidence for lack of a CQand failure of strict complementarity. One theme is the notion of minimal representation of the cone and the constraints in order to regularize the problem and avoid both the theoretical and numerical difficulties that arise due to (near) loss of a CQ. We include a discussion on obtaining these representations efficiently.
A Simple Derivation of a Facial Reduction Algorithm and Extended Dual Systems
"... The Facial Reduction Algorithm (FRA) of Borwein and Wolkowicz and the Extended Dual System (EDS) of Ramana aim to better understand duality, when a conic linear system Ax K b (P) has no strictly feasible solution. We ffl provide a simple proof of the correctness of a variant of FRA. ffl show how ..."
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Cited by 10 (2 self)
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The Facial Reduction Algorithm (FRA) of Borwein and Wolkowicz and the Extended Dual System (EDS) of Ramana aim to better understand duality, when a conic linear system Ax K b (P) has no strictly feasible solution. We ffl provide a simple proof of the correctness of a variant of FRA. ffl show how it naturally leads to the validity of a family of extended dual systems. ffl Summarize, which subsets of K related to the system (P) (as the minimal cone and its dual) have an extended representation. 1 Introduction Farkas' lemma assuming a CQ Duality results for the conic linear system Ax K b (P) 1 A Facial Reduction Algorithm and Extended Dual Systems 2 are usually derived assuming some constraint qualification (CQ). The most frequently used CQ is strict feasibility, ie. assuming the existence of a x with A x ! K b. Here K is a closed convex cone, A : X ! Y a linear operator, with X and Y being euclidean spaces. We write z K y, and z ! K y to mean that y  z is in K, or in ri K...
Facial reduction algorithms for conic optimization problems
 Journal of Optimization Theory and Applications
"... To obtain a primaldual pair of conic programming problems having zero duality gap, two methods have been proposed: the facial reduction algorithm due to Borwein and Wolkowicz [1, 2] and the conic expansion method due to Luo, Sturm, and Zhang [5]. We establish a clear relationship between them. Our ..."
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Cited by 9 (3 self)
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To obtain a primaldual pair of conic programming problems having zero duality gap, two methods have been proposed: the facial reduction algorithm due to Borwein and Wolkowicz [1, 2] and the conic expansion method due to Luo, Sturm, and Zhang [5]. We establish a clear relationship between them. Our results show that although the two methods can be regarded as dual to each other, the facial reduction algorithm can produce a finer sequence of faces including the feasible region. We illustrate the facial reduction algorithm in LP, SOCP and an example of SDP. A simple proof of the convergence of the facial reduction algorithm for conic programming is also presented.