We discuss several di#erent search directions which can be used in primal-dual interior-point methods for semidefinite programming problems and investigate their theoretical properties, including scale invariance, primal-dual symmetry, and whether they always generate well-defined directions. Among the directions satisfying all but at most two of these desirable properties are the Alizadeh-Haeberly-Overton, HelmbergRendl -Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro, Nesterov-Todd, Gu, and Toh directions, as well as directions we will call the MTW and Half directions. The first five of these appear to be the best in our limited computational testing also. Key words: semidefinite programming, search direction, invariance properties. AMS Subject classification: 90C05. Abbreviated title: Search directions in SDP 1 Introduction This paper is concerned with interior-point methods for semidefinite programming (SDP) problems and in particular the various search directions they use and ...

. An exact solution method for the graph bisection problem is presented. We describe a branch-and-bound algorithm which is based on a cutting plane approach combining semidefinite programming and polyhedral relaxations. We report on extensive numerical experiments which were performed for various classes of graphs. The results indicate that the present approach solves general problem instances with 80 \Gamma 90 vertices exactly in reasonable time, and provides tight approximations for larger instances. Our approach is particularly well suited for special classes of graphs as planar graphs and graphs based on grid structures. 1. Introduction We consider the problem of partitioning the vertices of a graph into two components. Given is an undirected edge-weighted graph G(V; E), where V denotes the vertex set consisting of n vertices, and E the edge set. The weight of the edges are given by the Laplace matrix L, which is defined through the adjacency matrix A of the graph by L := Diag(Ae...

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is positive semidefinite (and symmetric) since (aa )y = (y a)(a y) = (a y) Example A = Diag(a 1 B B . . . . . . . . . . . . . . . 0 C C A is positive semidefinite if and only if a i 0. Ay = a i to prove "only if" choose y = (0; 1; 0) Cone The set of semidefinite matrices is a cone (see exercise) 6 Properties of semidefinite matrices Observations from linear programming: . . . . . . All submatrices are again positive semidefinite Diagonal elements a ii 0 (see exercise) A 1 0;A 2 0; A k 0 , B B A 1 0 0 . . . . . . . . . . . . . . . 0 0 0 0 A k C C 7 Characterisation of positive semiefinite matrices Proposition 1 The following are equivalent 2 Eigenvalues l i 0 for i = 1; 3 9C 2 R such that A =C C and rank(C) = rank(A) A 2 : eigenvector v 6= 0, eigenvalue l Av = lv A = PLP eigenvalue decomposition of A 1 ) 2 Let v 2 be an eigenvector with jvj = 1 corresponding to l. Since A is pos. semidef. v A

minimize cx subject to Ax b x 0 x Solved through branch-and-bound, using lower bounds LP-relaxation can be solved efficiently Natural way of formulating ILP models Many problems cannot be formulated as linear models, or the formulation is inefficient. 1 2 Interior point methods Interior point methods can be extended to a number of cones (self-dual homogeneous cones) n (linear programming) vectorized symmetric matrices over real numbers (semidefinite programming) vectorized Hermitian matrices over complex numbers vectorized Hermitian matrices over quaternions vectorized Hermitian 3 3 matrices over octonions