F. RENDL. Semidefinite programming and combinatorial optimization. Appl. Numer. Math., 29:255-- 281, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....bound and a semidefinite programming approach for the sparsest cut problem in Section 6. Because of space limitations, we can barely scratch the surface, and there are many aspects (e.g. computational) of the area that we will not cover. We refer the reader to Alizadeh [1] Lov asz [43] and Rendl [58] for additional coverage of the topic. 2 Preliminaries In this section, we collect several basic results about (positive semidefinite) matrices and semidefinite programming. Further results will be mentioned as needed. Most of the results on matrices quoted in this paper can be found in standard ....

.... inequalities are sufficient to describe the cut polytope for the 5 cycle (or any planar graph [8] provided we consider all triplets (i; j; k) If we denote the resulting upper bound by SDP 0 , no better bound than 0:87856 is known for the worst case ratio OPT=SDP 0 in general (see Rendl [58] for special cases) The ratio OPT=SDP 0 is known to be equal to 0:96 for the complete graph K 5 , and instances with a slightly worse gap ( 957) were obtained by Andress and Cheriyan (private communication) However, in light of Hastad s result and the polynomial solvability of semidefinite ....

F. Rendl. Semidefinite programming and combinatorial optimization. Notes for a lecture series given at the University of Pisa, 1997.

....SDP relaxation. Thus SDP provides a means of efficiently solving relaxations of quadratic approximations, though the SDPs themselves are often very large problems. Early examples of applying SDP are: the stable set problem and the max cut problem, see e.g. 24, 38] Surveys in this area are e.g. [54, 20]. 4.2 Eigenvalue Problems SDP is an eigenvalue optimization problem in the sense that the semidefinite constraint is equivalent to nonnegative bounds on eigenvalues. Solutions to SDP generally have multiple zero eigenvalues (just as solutions to LP have many nonbasic, i.e. zero, variables) ....

F. RENDL. Semidefinite programming and combinatorial optimization. Appl. Numer. Math., 29:255--281, 1999.

....result for (nonconvex) trust region optimization. 6] also extends these results to problems of the form BQP with linear equality constraints, which we do not describe here. More detailed proofs of the equivalence results in [6] using Lagrangian duality and other techniques, are provided in [8]. The purpose of this note is to give alternative proofs of the fundamental equivalence results of [7] and [6] Our goal is to minimize the number of results required to show that all the bounds are equal, and also to use proof mechanisms that are as simple as possible. As a corollary of our ....

F. Rendl (1998). Semidefinite programming and combinatorial optimization. Institut fur Mathematik, TU Graz, Graz, Austria.

....of the Hoffman Wielandt inequality. 1 Introduction Quadratic programs with quadratic constraints (QQPs) are an important modelling tool for many optimization problems; almost as important as the linear programming model. Applications for QQP include e.g. hard combinatorial problems, e.g. [25], and SQP algorithms for nonlinear programming, e.g. 17] These QQPs are often not convex and so are very hard to solve numerically. One approach is to use the Lagrangian relaxation of a QQP to obtain an approximate solution. The strength of such a relaxation depends on the duality gap, where a ....

F. RENDL. Semidefinite programming and combinatorial optimization. Technical report, University of Graz, Graz, Austria, 1997.

....Thus the problem has a bounded optimal value and a feasible solution. We also assume that the matrices fA i : i = 1; mg are linearly independent. This problem has recently generated a lot of interest. One reason is that there are many diverse applications: in discrete optimization see e.g. [3, 20, 44]; in engineering see e.g. 10, 60] for matrix completions see e.g. 24, 1] Another reason for the interest is that SDPs can be solved efficiently using interior point methods. More applications and evidence of the current high level of activity can be found in the recent theses: 2, 43, 22, 39, ....

F. RENDL. Semidefinite programming and combinatorial optimization. Technical report, University of Graz, Graz, Austria, 1997. Notes for a lecture series given at the University of Pisa.

....Thus the problem has a bounded optimal value and a feasible solution. We also assume that the matrices fA i : i = 1; mg are linearly independent. This problem has recently generated a lot of interest. One reason is that there are many diverse applications: in discrete optimization see e.g. [3, 18, 41]; in engineering see e.g. 9, 56] for matrix completions see e.g. 21, 1] Another reason for the interest is that SDPs can be solved efficiently using interior point methods. More applications and evidence of the current high level of activity can be found in the recent theses: 2, 40, 19, 36, ....

F. RENDL. Semidefinite programming and combinatorial optimization. Technical report, University of Graz, Graz, Austria, 1997.

....was shown in [1] where it appears that the term semidefinite programming first originated. Also at this time there appeared several important theoretical and algorithmic results that dealt with applications of SDP to combinatorial optimization, see e.g. 44, 45, 30, 28] and the survey papers [27, 56, 2]. The combination of the strong numerical results, theoretical breakthroughs, and diverse applications has led to an explosion of papers in this area. 1.4 Outline In this paper we present two approaches on handling ill posed SDPs, i.e. we show how to handle SDPs for which the Slater CQ (strict ....

F. RENDL. Semidefinite programming and combinatorial optimization. Technical report, University of Graz, Graz, Austria, 1997.

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F. RENDL. Semidefinite programming and combinatorial optimization. Appl. Numer. Math., 29:255-- 281, 1999.

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