Z-Q. LUO, J.F. STURM, and S. ZHANG. Duality and self-duality for conic convex programming. Technical report, Erasmus University Rotterdam, The Netherlands, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in Table 6. LMI problems These problems are pure SDP s that arise in control applications. On the problems where the nal complementarity residual is unsatisfactory (see Table 7) the dual iterates indeed get large, indicating that the primal problem may be infeasible or weakly infeasible [11]. 6 Other Computational Issues Several enhancements to the basic interior point iteration of Section 2 have been proposed, and some used with success, in linear programming. We are concerned with extending some of these techniques to SQLP, namely Gondzio s multiple centrality corrections [7] ....

Z.-Q. Luo, J. F. Sturm, and Z. Zhang. Duality and self{duality for conic convex programming. Technical report, Econometric Institute, Erasmus University, Rotterdam (The Netherlands), March 1996.

....be this good predictor point and a corresponding message update to good predictor point is displayed. 3 Homogeneous and self dual algorithms Homogeneous embedding of an SDP in a self dual problem was first developed by Potra and Sheng [22] and subsequently extended independently by Luo et al. [17] and de Klerk et al. 15] The implementation of such homogeneous and self dual algorithms for SDP first appeared in [9] where they are based on those appearing in [32] for linear programming (LP) Our algorithms are also the SDP extensions of those appearing in [32] for LP. However, we use a ....

Z.-Q. Luo, J. F. Sturm, and S. Zhang, Duality and self-duality for conic convex programming, Technical Report 9620/A, Tinbergen Institute, Erasmus University, Rotterdam, 1996.

.... strategies may be found in [58] One of the first infeasible start predictor corrector algorithms was by Potra and Sheng [46] Other references include [31, 37] The idea of embedding the SDP problem in a self dual problem with known feasible starting point was investigated for SDP in [13] and [36]. A solution of the self dual embedding gives information about the solution of the original problem. This analysis was extended in [16] to include pathological cases caused by the weaker duality theory of SDP (as compared to LP) In the latter case the stronger ELSD (extended Lagrange Slater) ....

Z.-Q. Luo, J.F. Sturm, and S. Zhang. Duality and self-duality for conic convex programming. Technical Report 9620/A, Tinbergen Institute, Erasmus University Rotterdam, 1996.

....pair form a set of measure zero. The general framework on the facial structure, nondegeneracy and strict complementarity for general cone programs was described by Pataki in [26] and in the dissertation [27] Strict complementarity was also introduced independently by Luo, Sturm and Zhang [22]. For Section 4 The multiplicity of the critical eigenvalue in eigenvalue optimization was studied in [28] by Pataki. For Section 5 The algorithm to find an extreme point feasible solution of a cone lp was given in [26] and [27] by Pataki. The method for sensitivity analysis is a generalization ....

Z-Q. LUO, J.F. STURM, and S. ZHANG. Duality and self-duality for conic convex programming. Technical report, Erasmus University Rotterdam, The Netherlands, 1996.

....without proofs, and formulate the results in a setting which is general enough for the present purposes, but avoids deeper results from Hilbert space theory and functional analysis. The presentation here follows mostly [14] We also mention similar approaches, contained in [48, 39, 60] In [40], a self dual formulation is investigated. 3.1 Deriving the Dual Problem We formulate a pair of dual problems (P) and (D) as follows. Let S and T be a pair of closed convex cones. A set S is called a cone if s 2 S implies ffs 2 S for all ff 0. We denote by S the dual cone, defined by S = ....

....x 22 = x 12 = x 23 = 0; and therefore x 33 = 1: The primal problem is clearly feasible, and M = 1: The dual is only feasible for y 1 = 0 and y 2 0: Therefore M 0 = 0: In general it is considered a computationally nontrivial task to recognize whether or not the duality gap is zero. Luo et al. [40] investigate this question both from a theoretical and a computational point of view. On the other hand, if SDP arises as a relaxation from an integer problem, it is often possible to give an explicit construction for a (primal) matrix X satisfying a constraint qualification (Slater or ....

Z.Q. LUO, J.F. STURM and S. ZHANG. Duality and self-duality for conic convex programming. Technical Report, Erasmus University Rotterdam, 1996.

....convex programming problems in conic form. In this way, we are able to start an interior point algorithm without any prior knowledge of the feasibility (primal or dual) of the initial problem. Earlier work on this topic includes Nesterov [9] de Klerk, Roos and Terlaky [6] Luo, Sturm and Zhang [7], Potra and Sheng [13] and Andersen and Ye [1] We present several path following and potential reduction primal dual interior point methods, which try to find a recession direction of the feasible set of a projective version of the model. Such a direction usually either constitutes an optimal ....

....(in)feasible problems are stable. Note that Nesterov [9] uses the term strictly ill posed for a different class of problems we will see more of them in Subsection 6.4. The relations between the feasibility cases for nonlinear cones are rather complicated (see [10] Section 4.2. 2, and [7] for a discussion) Our goal now is to describe our possibilities in detecting these cases. This is unfortunately not straightforward. In general, what we can obtain is the following: If the pair is strictly feasible, for all the algorithms described above we can generate an ffl optimal pair of ....

Z.-Q. Luo, J. Sturm, S. Zhang, "Duality and self-duality for conic convex programming, " Report 9620/A, Econometric Institute, Erasmus University, Rotterdam, 1996.

.... embedding of monotone nonlinear complementarity problems is discussed by Andersen and Ye in [4] For semidefinite programming the homogeneous embedding idea was first developed by Potra and Sheng [14] The embedding strategy was extended by De Klerk et al. in [5] and independently by Luo et al. [10] to obtain self dual embedding problems with nonempty interiors. The resulting embedding problem has a known centered starting point, unlike the homogeneous embedding; it can therefore be solved using any feasible path following interior point method. This is an advantage in the SDP case, where ....

....of the paper After some preliminaries in Section 2, a review of recent results concerning the convergence of the central path is given in Section 3, with simplified proofs. The embedding strategy 1 Examples of these effects will be given in Sections 2 and 7, and can also be found in [20] and [10]. is discussed thereafter in Section 4. Solution strategies for solving the embedding problem are given in Section 5. In Section 6 it is shown how to interpret an ffl optimal solution of the embedding problem in order to draw conclusions about the solution of the original problem pair (i.e. to ....

[Article contains additional citation context not shown here]

Z.-Q. Luo, J.F. Sturm, and S. Zhang. Duality and self-duality for conic convex programming. Technical Report 9620/A, Tinbergen Institute, Erasmus University Rotterdam, 1996.

....be this good predictor point and a corresponding message update to good predictor point is displayed. 3 Homogeneous and self dual algorithms Homogeneous embedding of an SDP in a self dual problem was first developed by Potra and Sheng [17] and subsequently extended independently by Luo et al. [13] and de Klerk et al. 11] The implementation of such homogeneous and self dual algorithms for SDP first appeared in [7] where they are based on those appearing in [25] for linear programming (LP) Our algorithms are also the SDP extensions of those appearing in [25] for LP. However, we use a ....

Z.-Q. Luo, J.F. Sturm, and S. Zhang, Duality and self-duality for conic convex programming, Technical Report 9620/A, Tinbergen Institute, Erasmus University, Rotterdam, 1996.

.... as well as semidefinite programming problems with several di#erent directions (see, e.g. 19] Extensions of the first homogeneous self dual model or a related homogeneous feasibility problem were studied by Potra and Sheng [18] by de Klerk, Roos and Terlaky [7, 8] and by Luo, Sturm, and Zhang [10]. The second homogeneous selfdual approach was already considered in a general conic setting in [17] For these more general problems, x and s lie in more general (finite dimensional real vector) spaces and are restricted to a closed convex cone and its dual; the inner product of two vectors in ....

Z.-Q. Luo, J. F. Sturm, and S. Zhang. Duality and self-duality for conic convex programming, Technical Report, Erasmus University Rotterdam, The Netherlands, 1996.

....embedding technique, which was first proposed by Ye, Todd and Mizuno [19] Using this technique a linear program can completely and efficiently be solved without resorting to any type of phase one procedure. Later, this technique was generalized to more general classes of convex optimization; see [10] and the references therein. The homogeneous self dual embedding technique of Ye, Todd and Mizuno was later simplified (and also generalized in a sense) by Xu, Hung and Ye [18] in which no optimization problem is explicitly solved; instead a system of homogeneous linear equations and inequalities ....

Z.-Q. Luo, J.F. Sturm and S. Zhang, Duality and self-duality for conic convex programming, Technical Report 9620/A, Erasmus University Rotterdam, 1996.

....programming which has been the center of recent research activities in the interior point community. Several groups of authors independently extended the self dual embedding technique to semidefinite programming, viz. Potra and Sheng [20] De Klerk, Roos and Terlaky [6] Luo, Sturm and Zhang [10] and Nesterov, Todd and Ye [19] The latter two papers concern the more general case of conic convex programming, each with a different emphasis. Nesterov, Todd and Ye [19] analyzed the application of logarithmically homogeneous barrier techniques to self dual embeddings and considered the ....

....two papers concern the more general case of conic convex programming, each with a different emphasis. Nesterov, Todd and Ye [19] analyzed the application of logarithmically homogeneous barrier techniques to self dual embeddings and considered the associated complexity issues. Luo, Sturm and Zhang [10] were concerned with a general duality theory for conic problems and with the question how to determine the status of the original problem from the sequence of iterates solving the selfdual embedding system. Report [10] was lengthy and covered two virtually separate topics. To make our results ....

[Article contains additional citation context not shown here]

Z.-Q. Luo, J.F. Sturm, and S. Zhang. Duality and self-duality for conic convex programming. Technical Report 9620/A, Econometric Institute, Erasmus University Rotterdam, Rotterdam, The Netherlands, 1996.

....viz. that x and y are almost feasible, whereas c T x Gamma b T y is considerably negative (kxk and or kyk must then obviously be very large) SeDuMi will then report an infinite number of digits in accuracy, according to formula (4) This phenomenon was explained by Luo, Sturm and Zhang [11] and Sturm [18] It is possible that an optimization model has both nonnegativity and quadratic cone constraints. For instance, we may extend the above example with the restriction that y 3 [1] Gamma0:1, where y 3 [1] denotes the first component in the vector y 3 . This restriction can be added ....

Z.-Q. Luo, J.F. Sturm, and S. Zhang. Duality and self-duality for conic convex programming. Technical Report 9620/A, Econometric Institute, Erasmus University Rotterdam, Rotterdam, The Netherlands, 1996.

....[31] interior point methods are well suited for solving conic convex programs. Although interior point methods typically require the existence of primal and dual interior solutions, it is possible to solve conic programs that are not strongly feasible by using the self dual embedding technique [29]. With (P) being a nonlinear program, it is not surprising that the interior point methods (or indeed any other methods) require an infinite number of iterations to obtain an exact solution. Within a finite number of iterations these iterative methods can only compute an approximate solution of ....

....Elaborating on the results of [42] we have also discussed the value of approximate dual solutions. We believe that duality results under no constraint qualifications have not received enough attention in the past. It is our hope that this paper will help popularize these results in future. In [29], we show that this type of duality relation can be used fruitfully in the design of algorithms whose convergence is guaranteed even in the absence of constraint qualifications. Our survey is restricted to conic convex programming in finite dimensional real linear spaces. As such, it includes ....

[Article contains additional citation context not shown here]

Z.-Q. Luo, J.F. Sturm, and S. Zhang. Duality and self-duality for conic convex programming. Technical Report 9620/A, Econometric Institute, Erasmus University Rotterdam, Rotterdam, The Netherlands, 1996.

....X Gamma C 1 2 A and Z Gamma C 2 2 A . It is well known that with the above assumption (a primal dual Slater condition) the pair (2) 3) is equivalent to finding a complementary solution pair, which is by definition a feasible solution pair (X; Z) with X ffl Z = 0 (i.e. no duality gap) see [1, 13, 9] among others. It is convenient to combine (2) 3) into the following formulation: SDP ) min X ffl Z s:t: X Gamma C 1 2 A Z Gamma C 2 2 A X 0; Z 0: Notice that the set of complementary solution pairs for (2) 3) is the optimal solution set of (SDP) Sturm and Zhang: On weighted centers ....

Luo, Z.-Q., Sturm, J.F. and Zhang, S., "Duality and self-duality for conic convex programming, " Report 9620/A, Econometric Institute, Erasmus University Rotterdam, Rotterdam, The Netherlands, 1996.

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Z-Q. LUO, J.F. STURM, and S. ZHANG. Duality and self-duality for conic convex programming. Technical report, Erasmus University Rotterdam, The Netherlands, 1996.

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Z.-Q. Luo, J. F. Sturm, and S. Zhang, "Duality and self-duality for conic convex programming", Technical report, Department of Electrical and Computer Engineering, McMaster University, 1996.

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Luo, Z.Q., J.F. Sturm, S.Z. Zhang (1996b). Duality and self-duality for conic convex programming, manuscript, Econometric Institute, Erasmus University, Rotterdam, The Netherlands.

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Luo, Z.Q., J.F. Sturm, S.Z. Zhang (1996). Duality and self-duality for conic convex programming, manuscript, Econometric Institute, Erasmus University, Rotterdam, The Netherlands.

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