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lercher@cs.uni-duesseldorf.de.

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We consider the solution of nonlinear programs with nonlinear semidefiniteness constraints. The need for an efficient exploitation of the cone of positive semidefinite matrices makes the solution of such nonlinear semidefinite programs more complicated than the solution of standard nonlinear programs. This paper studies a sequential semidefinite programming (SSP) method, which is a generalization of the well-known SQP method for standard nonlinear programs. We present a sensitivity result for nonlinear semidefinite programs, and then based on this result, we give a self-contained proof of local quadratic convergence of the SSP method. We also describe a class of nonlinear semidefinite programs that arise in passive reduced-order modeling, and we report results of some numerical experiments with the SSP method applied to problems in that class.

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We show that the maximal cp-rank of n × n completely positive matrices is attained at a positive-definite matrix on the boundary of the cone of n × n completely positive matrices, thus answering a long-standing question. We also show that the maximal cp-rank of 5 × 5 matrices equals six, which proves the famous Drew-Johnson-Loewy conjecture [Linear Multilinear Algebra, 37 (1994), pp. 303-310] for matrices of this order. In addition we present a simple scheme for generating completely positive matrices of high cp-rank and investigate the structure of a minimal cp factorization. Introduction.In this article we consider completely positive matrices M and their cp-rank. An n × n matrix M is said to be completely positive if there exists a nonnegative (not necessarily square) matrix V such that M = V V . Typically, a completely positive matrix M may have many such factorizations, and the cp-rank of M , cpr M , is the minimum number of columns in such a nonnegative factor V . (For completeness, we define cpr M = 0 if M is a square zero matrix and cpr M = ∞ if M is not completely positive.) Completely positive matrices play an increasingly important role as they form a cone dual to the cone of copositive matrices. An n × n matrix A is said to be copositive if x Ax ≥ 0 for every nonnegative vector x ∈ R n + . Both cones are central in the rapidly evolving field of copositive optimization which links discrete and continuous optimization and has numerous real-world applications. For recent surveys and structured bibliographies, we refer to [8,9,10,17], and for a fundamental text book to [6].Determining the maximum possible cp-rank of n×n completely positive matrices,