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Semidefinite programming (SDP) is one of the most exciting and
active research areas in optimization. It has and continues to
attract researchers with very diverse backgrounds, including
experts in convex programming, linear algebra, numerical
optimization, combinatorial optimization, control theory, and
statistics. This tremendous research activity has been prompted by
the discovery of important applications in combinatorial
optimization and control theory, the development of efficient
interior-point algorithms for solving SDP problems, and the depth
and elegance of the underlying optimization theory. The Handbook of
Semidefinite Programming offers an advanced and broad overview of
the current state of the field. It contains nineteen chapters
written by the leading experts on the subject. The chapters are
organized in three parts: Theory, Algorithms, and Applications and
Extensions.
In Linear Programming: A Modern Integrated Analysis, both boundary
(simplex) and interior point methods are derived from the
complementary slackness theorem and, unlike most books, the duality
theorem is derived from Farkas's Lemma, which is proved as a convex
separation theorem. The tedium of the simplex method is thus
avoided. A new and inductive proof of Kantorovich's Theorem is
offered, related to the convergence of Newton's method. Of the
boundary methods, the book presents the (revised) primal and the
dual simplex methods. An extensive discussion is given of the
primal, dual and primal-dual affine scaling methods. In addition,
the proof of the convergence under degeneracy, a bounded variable
variant, and a super-linearly convergent variant of the primal
affine scaling method are covered in one chapter. Polynomial
barrier or path-following homotopy methods, and the projective
transformation method are also covered in the interior point
chapter. Besides the popular sparse Cholesky factorization and the
conjugate gradient method, new methods are presented in a separate
chapter on implementation. These methods use LQ factorization and
iterative techniques.
Semidefinite programming (SDP) is one of the most exciting and
active research areas in optimization. It has and continues to
attract researchers with very diverse backgrounds, including
experts in convex programming, linear algebra, numerical
optimization, combinatorial optimization, control theory, and
statistics. This tremendous research activity has been prompted by
the discovery of important applications in combinatorial
optimization and control theory, the development of efficient
interior-point algorithms for solving SDP problems, and the depth
and elegance of the underlying optimization theory. The Handbook of
Semidefinite Programming offers an advanced and broad overview of
the current state of the field. It contains nineteen chapters
written by the leading experts on the subject. The chapters are
organized in three parts: Theory, Algorithms, and Applications and
Extensions.
In Linear Programming: A Modern Integrated Analysis, both boundary
(simplex) and interior point methods are derived from the
complementary slackness theorem and, unlike most books, the duality
theorem is derived from Farkas's Lemma, which is proved as a convex
separation theorem. The tedium of the simplex method is thus
avoided. A new and inductive proof of Kantorovich's Theorem is
offered, related to the convergence of Newton's method. Of the
boundary methods, the book presents the (revised) primal and the
dual simplex methods. An extensive discussion is given of the
primal, dual and primal-dual affine scaling methods. In addition,
the proof of the convergence under degeneracy, a bounded variable
variant, and a super-linearly convergent variant of the primal
affine scaling method are covered in one chapter. Polynomial
barrier or path-following homotopy methods, and the projective
transformation method are also covered in the interior point
chapter. Besides the popular sparse Cholesky factorization and the
conjugate gradient method, new methods are presented in a separate
chapter on implementation. These methods use LQ factorization and
iterative techniques.
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