Interior-point Algorithms For Semidefinite Programming Research Articles

Simplified analysis for full-Newton step infeasible interior-point algorithm for semidefinite programming

We present an analysis of the full-Newton step infeasible interior-point algorithm for semidefinite optimization, which is an extension of the algorithm introduced by Roos [C. Roos, A full-Newton step 𝒪(n) infeasible interior-point algorithm for linear optimization, SIAM J. Optim. 16 (2006), pp. 1110–1136] for the linear optimization case. We use the proximity measure σ(V) = ‖I − V 2‖ to overcome the difficulty of obtaining an upper bound of updated proximity after one full-Newton step, where I is an identity matrix and V is a symmetric positive definite matrix. It turns out that the complexity analysis of the algorithm is simplified and the iteration bound obtained is improved slightly.

New complexity analysis of a Mehrotra-type predictor–corrector algorithm for semidefinite programming

In this paper, we propose a new Mehrotra-type predictor–corrector interior-point algorithm for semidefinite programming. This algorithm is an extension of the variant of Mehrotra-type algorithm that was proposed by Salahi et al. [On Mehrotra-type predictor–corrector algorithms, SIAM J. Optim. 18 (2007), pp. 1377–1397] for linear programming problems. We modify the step sizes lightly in the predictor step of Koulaei and Terlaky [On the complexity analysis of a Mehrotra-type primal–dual feasible algorithm for semidefinite optimization, Optim. Methods Softw. 25 (2010), pp. 467–485]. In such a way, we obtain O(nlog(Tr(X 0 S 0)/ϵ)) iteration complexity of the algorithm, where (X 0, y 0, S 0) is the initial feasible point and ϵ is the required precision.

A new second-order corrector interior-point algorithm for semidefinite programming

In this paper, we propose a second-order corrector interior-point algorithm for semidefinite programming (SDP). This algorithm is based on the wide neighborhood. The complexity bound is \({O(\sqrt{n}L)}\) for the Nesterov-Todd direction, which coincides with the best known complexity results for SDP. To our best knowledge, this is the first wide neighborhood second-order corrector algorithm with the same complexity as small neighborhood interior-point methods for SDP. Some numerical results are provided as well.

A New Full-NT-Step Infeasible Interior-Point Algorithm for SDP Based on a Specific Kernel Function

In this paper, we propose a new infeasible interior-point algorithm with full NesterovTodd (NT) steps for semidefinite programming (SDP). The main iteration consists of a feasibility step and several centrality steps. We used a specific kernel function to induce the feasibility step. The analysis is more simplified. The iteration bound coincides with the currently best known bound for infeasible interior-point methods.

A wide neighbourhood interior-point method with iteration-complexity bound for semidefinite programming

In this article we propose a new class of primal-dual path-following interior-point algorithms for semidefinite programming (SDP). Each iterate always follows the usual wide neighbourhood , not necessarily remaining within it, but must stay within the wider neighbourhood . We show that the algorithm has iteration-complexity bound which is better than that of the usual wide neighbourhood algorithm O(nL). It is the best result in regard to iteration-complexity bound in the context of path-following method for SDP.

An inexact interior-point method for system analysis

In this article, a primal-dual interior-point algorithm for semidefinite programming that can be used for analysing e.g. polytopic linear differential inclusions is tailored in order to be more computationally efficient. The key to the speedup is to allow for inexact search directions in the interior-point algorithm. These are obtained by aborting an iterative solver for computing the search directions prior to convergence. A convergence proof for the algorithm is given. Two different preconditioners for the iterative solver are proposed. The speedup is in many cases more than an order of magnitude. Moreover, the proposed algorithm can be used to analyse much larger problems as compared to what is possible with off-the-shelf interior-point solvers.

The Primal-Dual Second-Order Cone Approximations Algorithm for Symmetric Cone Programming

Given any open convex cone K, a logarithmically homogeneous, self-concordant barrier for K, and any positive real number r < 1, we associate, with each direction x e K, a second-order cone Kr(x) containing K. We show that K is the interior of the intersection of the second-order cones Kr(x), as x ranges over all directions in K. Using these second-order cones as approximations to cones of symmetric, positive definite matrices, we develop a new polynomial-time primal-dual interior-point algorithm for semidefinite programming. The algorithm is extended to symmetric cone programming via the relation between symmetric cones and Euclidean Jordan algebras.

Interior-Point Algorithms for Semidefinite Programming Based on a Nonlinear Formulation

Recently in Burer et al. (Mathematical Programming A, submitted), the authors of this paper introduced a nonlinear transformation to convert the positive definiteness constraint on an n × n matrix-valued function of a certain form into the positivity constraint on n scalar variables while keeping the number of variables unchanged. Based on this transformation, they proposed a first-order interior-point algorithm for solving a special class of linear semidefinite programs. In this paper, we extend this approach and apply the transformation to general linear semidefinite programs, producing nonlinear programs that have not only the n positivity constraints, but also n additional nonlinear inequality constraints. Despite this complication, the transformed problems still retain most of the desirable properties. We propose first-order and second-order interior-point algorithms for this type of nonlinear program and establish their global convergence. Computational results demonstrating the effectiveness of the first-order method are also presented.

Associative and Jordan Algebras, and Polynomial Time Interior-Point Algorithms for Symmetric Cones

We present a general framework whereby analysis of interior-point algorithms for semidefinite programming can be extended verbatim to optimization problems over all classes of symmetric cones derivable from associative algebras. In particular, such analyses are extendible to the cone of positive semidefinite Hermitian matrices with complex and quaternion entries, and to the Lorentz cone. We prove the case of the Lorentz cone by using the embedding of its associated Jordan algebra in the Clifford algebra. As an example of such extensions we take Monterio's polynomial-time complexity analysis of the family of similarly scaled directions—introduced by Monteiro and Zhang (1998)—and generalize it to cone-LP over all representable symmetric cones.

Sdpha: a Matlab implementation of homogeneous interior-point algorithms for semidefinite programming*

Mehrotra type primal-dual predictor-corrector interior-point algorithms for semidefinite programming are implemented, using the homogeneous formulation proposed and analyzed by Potra and Sheng. Several search directions, including the AHO, HKM, NT, Toh, and Gu directions, are used. A rank-2 update technique is employed in our MATLAB code so that the computation of homogeneous directions is only slightly more expensive than in the non-homogeneous case. However, the homogeneous algorithms generally take fewer iterations to compute an approximate solution within a desired accuracy. Numerical results show that the homogeneous algorithms outperform their non-homogeneous counterparts, with improvement of more than 20% in many cases, in terms of total CPU time.

On long-step predictor-corrector interior-point algorithm for semidefinite programming with Monteiro-Zhang unified search directions

We present a long-step predictor-corrector interior-point algorithm for the monotone semidefinite linear complementarity problems using the Monteiro-Zhang unified search directions. Our algorithm is based on the long-step predictor-corrector interior-point algorithm proposed by Kojima, Shida and Shindoh using the Alizadeh-Haeberly-Overton search direction, though the AHO search direction does not belong to the MZ unified search directions in general.

On a general class of interior-point algorithms for semidefinite programming with polynomial complexity and superlinear convergence

On a general class of interior-point algorithms for semidefinite programming with polynomial complexity and superlinear convergence

Polynomial Convergence of Primal-Dual Algorithms for Semidefinite Programming Based on the Monteiro and Zhang Family of Directions

This paper establishes the polynomial convergence of the class of primal-dual feasible interior-point algorithms for semidefinite programming (SDP) based on the Monteiro and Zhang family of search directions. In contrast to Monteiro and Zhang's work [Math. Programming, 81 (1998), pp. 281--299], here no condition is imposed on the scaling matrix that determines the search direction. We show that the polynomial iteration-complexity bounds of two well-known algorithms for linear programming, namely the short-step path-following algorithm of Kojima, Mizuno, and Yoshise [Math. Programming, 44 (1989), pp. 1--26] and Monteiro and Adler [Math. Programming, 44 (1989), pp. 27--41 and pp. 43--66] and the predictor-corrector algorithm of Mizuno, Todd, and Ye [Math. Oper. Res., 18 (1993), pp. 945--981] carry over into the context of SDP. Since the Monteiro and Zhang family of directions includes the Alizadeh, Haeberly, and Overton direction, we establish for the first time the polynomial convergence of algorithms based on this search direction.

A unified analysis for a class of long-step primal-dual path-following interior-point algorithms for semidefinite programming

We present a unified analysis for a class of long-step primal-dual path-following algorithms for semidefinite programming whose search directions are obtained through linearization of the symmetrized equation of the central path Hp(XS) -- [PXSP -~ + (PXSP 1)TI/2 = #I, introduced by Zhang. At an iterate (X, S), we choose a scaling matrix P from the class of nonsingular matrices P such that PXSP -~ is symmetric. This class of matrices includes the three well-known choices, namely: P = S ~/2 and P - X -~/2 proposed by Monteiro, and the matrix P corresponding to the Nesterov Todd direction. We show that within the class of algorithms studied in this paper, the one based on the Nesterov-Todd direction has the lowest possible iteration-complexity bound that can provably be derived from our analysis. More specifically, its iteration-complexity bound is of the same order as that of the corresponding long-step primal-dual path-following algorithm for linear programming introduced by Kojima, Mizuno and Yoshise. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Primal--Dual Path-Following Algorithms for Semidefinite Programming

This paper deals with a class of primal--dual interior-point algorithms for semidefinite programming (SDP) which was recently introduced by Kojima, Shindoh, and Hara [SIAM J. Optim., 7 (1997), pp. 86--125]. These authors proposed a family of primal-dual search directions that generalizes the one used in algorithms for linear programming based on the scaling matrix X1/2S-1/2. They study three primal--dual algorithms based on this family of search directions: a short-step path-following method, a feasible potential-reduction method, and an infeasible potential-reduction method. However, they were not able to provide an algorithm which generalizes the long-step path-following algorithm introduced by Kojima, Mizuno, and Yoshise [Progress in Mathematical Programming: Interior Point and Related Methods, N. Megiddor, ed., Springer-Verlag, Berlin, New York, 1989, pp. 29--47]. In this paper, we characterize two search directions within their family as being (unique) solutions of systems of linear equations in symmetric variables. Based on this characterization, we present a simplified polynomial convergence proof for one of their short-step path-following algorithms and, for the first time, a polynomially convergent long-step path-following algorithm for SDP which requires an extra $\sqrt{n}$ factor in its iteration-complexity order as compared to its linear programming counterpart, where n is the number of rows (or columns) of the matrices involved.