Todd, M.J.: Semidefinite optimization. Acta Numer. 10, 515--560 (2001)  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... convergence 1 Introduction In this paper we consider the solution of nonlinear semidefinite programs (NLSDPs) In recent years, interior point methods for solving linear semidefinite programs (SDPs) have received a lot of attention and as a result, are now very well developed; see, e.g. [23, 25], the papers in [28] and the references given there. At each iteration of an interior point method, the complementarity condition is relaxed, symmetrized, and linearized. Various symmetrization operators are known. The choice of the symmetrization operator and of the relaxation parameter ....

....we briefly review the case of linear semidefinite programs. We remark that applications of linear semidefinite programs include relaxations of combinatorial optimization problems and problems related to Lyapunov functions or the positive real lemma in control theory; we refer the reader to [1, 4, 8, 23, 25] and the references given there. Given a linear function A : IR , a vector b 2 IR , and a matrix C 2 S , a pair of primal and dual linear semidefinite programs is as follows: maximize C ffl Y subject to Y 2 S ; Y 0; 7) and A(x) C 0: 8) We remark that this formulation ....

Todd, M.J. (2001): Semidefinite optimization. Acta Numer. 10, 515--560

.... program, a penalty approach by Kocvara and Stingl [19] low rank factorizations of Burer and Monteiro [10] and transformation to a constrained nonlinear program proposed by Burer and Monteiro [9] and Burer, Monteiro, and Zhang [11] A discussion and comparison of these methods can be found in [24]. Some of these methods are particularly well suited for large scale problems [21] In particular, the spectral bundle method and low rank factorizations have solved solved some large instances of SDP. However, these methods lack polynomial convergence in theory and sometimes exhibit slow ....

M. J. Todd. Semidefinite optimization. In Acta Numerica, pages 515--560. Cambridge University Press, 2001.

....W 11T rl12, and expressing the norm bound constraint as a linear matrix inequality (LMI) subject to W T 11T r sI W , 1TW I T, W1 1. o (16) Here the symbol denotes matrix inequality, i.e. X Y means that X Y is positive semidefinite. For background on SDP and LMIs, see, e.g. [10, 1, 40, 15, 41, 2, 39, 11]. Related background on eigenvalue optimization can be found in, e.g. 34, 9, 21] Similarly, the symmetric FDLA problem (15) can be expressed as the SDP subject to sI W 11T r sI (17) W , W W T, W1 1, with variables s G R and W G R nXn. 4 Heuristics based on the Laplacian There are ....

....of the optimal symmetric edge and node weights, found by solving the SDP (17) Note that many weights are negative. 11 6 Computational methods 6. 1 Interior point method Standard interior point algorithms for solving SDPs work well for problems with up to a thousand or so edges (see, e.g. [29, 1, 40, 42, 41, 39, 11]) The particular structure of the SDPs encountered in FDLA problems can be exploited for some gain in efficiency, but problems with more than a few thousand edges are probably beyond the capabilities of current interior point SDP solvers. We consider a simple primal barrier method, with the ....

M. Todd. Semidefinite optimization. Acta Numerica, 10:515 560, 2001.

....is not only an extension of LP but also includes convex quadratic optimization problems and some other convex optimization problems. It has a lot of applications in various fields such as combinatorial optimization [9] control theory [4] robust optimization [3, 23] and chemical physics [15] See [19, 22, 23] for a survey on SDPs and the papers in their references. The PDIPA (primal dual interior point algorithm) 10, 12, 14, 16] is known as the most powerful and practical numerical method for solving general SDPs. The method is an extension of the PDIPA [11, 18] developed for LPs. The SDPA ....

M.J. Todd, "Semidefinite optimization," Acta Numerica 10 (2001) 515--560.

....to the simplex method were made, so that now both approaches are viable for very large scale instances arising in practice: see Bixby [3] These advances are described for example in the books of Renegar and S. Wright [24, 33] and the survey articles of M. Wright, Todd, and Forsgren et al. [32, 26, 27, 5]. Despite their very nice theoretical properties, interior point methods do not deal very gracefully with infeasible or unbounded instances. The simplex method (a finite, combinatorial algorithm) first determines whether a linear programming instance is feasible: if not, it produces a so called ....

....the column vectors u and v. Hence the first component of x i is required to be at least the Euclidean norm of the vector of the remaining components. Both of these classes of optimization problems have nice theory and wide ranging applications: see, e.g. Ben Tal and Nemirovski [2] or Todd [27]. The problem dual to ( P ) is ( D) maximize y#, A # y s = c, s where A # : Y E # is the adjoint transformation to A and K # : s # 0 for all x K is the cone dual to K. In the two cases above, K is self dual, so that K # = K (we have identified E and E # ) Given a ....

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M. J. Todd. Semidefinite optimization. Acta Numer., 10:515--560, 2001.

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Todd, M.J.: Semidefinite optimization. Acta Numer. 10, 515--560 (2001)

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M. Todd, Semidefinite optimization, Acta Numer. 10 (2001), 515--560.

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M. Todd, "Semidefinite Optimization," Acta Numerica, vol. 10, pp. 515--560, 2001.

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M. Todd, "Semidefinite Optimization," Acta Numerica, vol. 10, pp. 515--560, 2001.

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M.J. Todd, Semidefinite optimization, Acta Numerica 10 (2001) 515--560.

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