A New Notion of Weighted Centers for Semidefinite Programming

Chua, Chek Beng

doi:10.1137/040613378

Cited by 9 publications

(14 citation statements)

References 22 publications

(27 reference statements)

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“…Concerning the second subject, we have already obtained a partial result in the proof of (iii) of Theorem 5.5. As a related result, we should refer to Chua's work [3] for homogeneous conic programming: The author showed that the paths defined by a class of optimal barriers converge to analytical centers of optimal faces whenever the primal-dual pair of problems has strictly complementarity solutions.…”

Section: A Homogeneous Model For the Cpmentioning

confidence: 96%

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Interior Point Trajectories and a Homogeneous Model for Nonlinear Complementarity Problems over Symmetric Cones

Yoshise¹

2007

SIAM J. Optim.

100

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We study the continuous trajectories for solving monotone nonlinear mixed complementarity problems over symmetric cones. While the analysis in [L. Faybusovich, Positivity, 1 (1997), pp. 331-357] depends on the optimization theory of convex log-barrier functions, our approach is based on the paper of Monteiro and Pang [Math. Oper. Res., 23 (1998), pp. 39-60], where a vast set of conclusions concerning continuous trajectories is shown for monotone complementarity problems over the cone of symmetric positive semidefinite matrices. As an application of the results, we propose a homogeneous model for standard monotone nonlinear complementarity problems over symmetric cones and discuss its theoretical aspects. Consequently, we show the existence of a path having the following properties: (a) The path is bounded and has a trivial starting point without any regularity assumption concerning the existence of feasible or strictly feasible solutions. (b) Any accumulation point of the path is a solution of the homogeneous model. (c) If the original problem is solvable, then every accumulation point of the path gives us a finite solution. (d) If the original problem is strongly infeasible, then, under the assumption of Lipschitz continuity, any accumulation point of the path gives us a finite certificate proving infeasibility.1. Introduction. Let (V, •) be a Euclidean Jordan algebra with an identity element e. We denote by K the symmetric cone of V which is a self-dual closed convex cone such that for any two elements x ∈ intK and y ∈ intK, there exists an invertible map Γ : V → V satisfying Γ(K) = K and Γ(x) = y. It is known that a cone in V is symmetric if and only if it is the cone of squares of V given byFaybusovich [6] studied the linear monotone complementarity problem (LCP) over symmetric cones of the form

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Section: A Homogeneous Model For the Cpmentioning

confidence: 96%

“…It is known that the self-scaled cones associated with the self-scaled barriers are closely related to the symmetric cones [1,10,11,27]. See also [15,7,23,28,25,24,3] for other extensions of primal-dual methods to the positive semidefinite cones, the symmetric cones, the self-scaled cones, or the homogeneous cones.…”

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confidence: 97%

Interior Point Trajectories and a Homogeneous Model for Nonlinear Complementarity Problems over Symmetric Cones

Yoshise¹

2007

SIAM J. Optim.

100

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“…Hence, in this section, we consider the relation between the standard central paths for SDP and weighted central paths given by the weighted barriers. (See [3] for some convergence properties of these weighted central paths. )…”

Section: Neighbourhoods Of the Weighted Central Pathsmentioning

confidence: 99%

Invariance and efficiency of convex representations

Chua

Tunçel

2006

Math. Program.

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We consider two notions for the representations of convex cones: G-representation and lifted-G-representation. The former represents a convex cone as a slice of another; the latter allows in addition, the usage of auxiliary variables in the representation. We first study the basic properties of these representations. We show that some basic properties of convex cones are invariant under one notion of representation but not the other. In particular, we prove that lifted-G-representation is closed under duality when the representing cone is self-dual. We also prove that strict complementarity of a convex optimization problem in conic form is preserved under G-representations. Then we move to study efficiency measures for representations. We evaluate the representations of homogeneous convex cones based on the "smoothness" of the transformations mapping the central path of the representation to the central path of the represented optimization problem.

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“…Such paths have been studied in [11,17,20,22]. The paper [5] also proposes a definition of paths which requires a Cholesky factorization in addition to the algebraic equations (2).…”

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confidence: 99%

Underlying paths in interior point methods for the monotone semidefinite linear complementarity problem

Sim¹,

Zhao

2006

Math. Program.

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An interior point method (IPM) defines a search direction at each interior point of the feasible region. The search directions at all interior points together form a direction field, which gives rise to a system of ordinary differential equations (ODEs). Given an initial point in the interior of the feasible region, the unique solution of the ODE system is a curve passing through the point, with tangents parallel to the search directions along the curve. We call such curves off-central paths. We study off-central paths for the monotone semidefinite linear complementarity problem (SDLCP). We show that each off-central path is a well-defined analytic curve with parameter µ ranging over (0, ∞) and any accumulation point of the off-central path is a solution to SDLCP. Through a simple example we show that the off-central paths are not analytic as a function of √ µ and have first derivatives which are unbounded as a function of µ at µ = 0 in general. On the other hand, for the same example, we can find a subset of off-central paths which are analytic at µ = 0. These "nice" paths are characterized by some algebraic equations.

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A New Notion of Weighted Centers for Semidefinite Programming

Cited by 9 publications

References 22 publications

Interior Point Trajectories and a Homogeneous Model for Nonlinear Complementarity Problems over Symmetric Cones

Interior Point Trajectories and a Homogeneous Model for Nonlinear Complementarity Problems over Symmetric Cones

Invariance and efficiency of convex representations

Underlying paths in interior point methods for the monotone semidefinite linear complementarity problem

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