Mehrotra-type Predictor-corrector Algorithm Research Articles

A Mehrotra-type second-order predictor–corrector algorithm for nonlinear complementarity problems over symmetric cones

This paper presents a Mehrotra-type predictor–corrector algorithm for monotone nonlinear complementarity problems over symmetric cones. The proposed algorithm is a second-order predictor–corrector interior point algorithm. Based on a one-norm wide neighbourhood, the iteration complexity of the algorithm is estimated and some numerical results are provided. The numerical results show that the algorithm is efficient and reliable.

On complexity of a new Mehrotra-type interior point algorithm for P_{*}(kappa ) linear complementarity problems

In this paper, a variant of Mehrotra-type predictor–corrector algorithm is proposed for P_{*}(kappa ) linear complementarity problems. In this algorithm, a safeguard step is used to avoid small step sizes and a new corrector direction is adopted. The algorithm has polynomial iteration complexity and the iteration bound is O ((14kappa +11)sqrt{(1+4 kappa )(1+2kappa )}~nlog frac{(x^{0})^{T}s^{0}}{varepsilon } ). Some numerical results are reported as well.

Infeasible Mehrotra-type predictor–corrector algorithm for cartesian P*(κ) nonlinear complementarity problems over symmetric cones

ABSTRACTThis paper establishes a theoretical framework of infeasible Mehrotra-type predictor–corrector algorithm for Cartesian nonlinear complementarity problems over symmetric cones which can be regarded as an extension of Liu et al. [Polynomial convergence of Mehrotra-type predictor–corrector algorithm for the Cartesian -LCP over symmetric cones, Optimization 64(4) (2013), pp. 1–23] for Cartesian linear complementarity problems over symmetric cones to Cartesian nonlinear complementarity problems over symmetric cones. The iteration complexities of the proposed algorithm are estimated and some numerical results are provided. The numerical results show that the proposed algorithm is efficient and reliable.

A New Infeasible Mehrotra-Type Predictor–Corrector Algorithm for Nonlinear Complementarity Problems Over Symmetric Cones

This paper establishes a theoretical framework of infeasible Mehrotra-type predictor–corrector algorithm for nonmonotone nonlinear complementarity problems over symmetric cones which can be regarded as an extension the Mehrotra’s algorithm proposed by Salahi et al. (On Mehrotra-type predictor–corrector algorithms. SIAM J Optim 18(4):1377–1397, 2005) from nonnegative orthant to symmetric cone. The iteration complexity of the algorithm is estimated, and some numerical results are provided. The numerical results show that the algorithm is efficient and reliable.

A new second-order Mehrotra-type predictor-corrector algorithm for SDO

In Zhang’s recent works, a second-order Mehrotra-type predictor-corrector algorithm for linear optimization was extended to semidefinite optimization and derived that the algorithm for semidefinite optimization had O(n 3/2log(X 0)T•S 0/e) iteration complexity based on the NT direction as Newton search direction. In this paper, we extend the second-order Mehrotra-type predictor-corrector algorithm for linear optimization to semidefinite optimization and discuss the polynomial convergence of the algorithm by modifying the corrector direction and new iterates. It is proved that the iteration complexity is reduced to O(nlogX 0•S 0/e), which coincides with the currently best iteration bound of Mehrotra-type predictor-corrector algorithm for semidefinite optimization.

An $$O(\sqrt{n}L)$$ O ( n L ) iteration Mehrotra-type predictor-corrector algorithm for monotone linear complementarity problem

In this paper we propose a new class of Mehrotra-type predictor-corrector algorithm for the monotone linear complementarity problems (LCPs). At each iteration, the method computes a corrector direction in addition to the Ai–Zhang direction (SIAM J Optim 16:400–417, 2005), in an attempt to improve performance. Starting with a feasible point \((x^0, s^0)\) in the wide neighborhood \(\mathcal {N}(\tau ,\beta )\), the algorithm enjoys the low iteration bound of \(O(\sqrt{n}L)\), where \(n\) is the dimension of the problem and \(L=\log \frac{(x^0)^T s^0}{\varepsilon }\) with \(\varepsilon \) the required precision. We also prove that the new algorithm can be specified into an easy implementable variant for solving the monotone LCPs, in such a way that the iteration bound is still \(O(\sqrt{n}L)\). Some preliminary numerical results are provided as well.

Infeasible Mehrotra-Type Predictor-Corrector Interior-Point Algorithm for the Cartesian P *(κ)-LCP Over Symmetric Cones

We propose an infeasible Mehrotra-type predictor-corrector algorithm with a new center parameter updating scheme for Cartesian P *(κ)-linear complementarity problem over symmetric cones. Based on the Nesterov-Todd direction, we show that the iteration-complexity bound of the proposed algorithm is 𝒪((1 + κ)3 r 2log ε−1), where r is the rank of the associated Euclidean Jordan algebras and κ is the handicap of the problem and ε > 0 is the required precision. Some numerical results are reported as well.

A second-order Mehrotra-type predictor-corrector algorithm with a new wide neighbourhood for semi-definite programming

In this paper, we present a new second-order Mehrotra-type predictor–corrector algorithm for semi-definite programming (SDP). The proposed algorithm is based on a new wide neighbourhood. We are particularly concerned with an important inequality. Based on the inequality, the convergence is shown for a specific class of search directions. In particular, the complexity bound is O(√nlogϵ−1) for the Nesterov–Todd search direction and O(n logϵ−1) for the Helmberg-Kojima-Monteiro search direction. The derived complexity bounds coincide with the currently best known theoretical complexity bounds obtained so far for SDP. We provide some preliminary numerical results as well.

A second order Mehrotra-type predictor-corrector algorithm for semidefinite optimization

Mehrotra-type predictor-corrector algorithm is one of the most effective primal-dual interiorpoint methods. This paper presents an extension of the recent variant of second order Mehrotra-type predictor-corrector algorithm that was proposed by Salahi, et al. (2006) for linear optimization. Based on the NT direction as Newton search direction, it is shown that the iteration-complexity bound of the algorithm for semidefinite optimization is {ie1108-1}, which is similar to that of the corresponding algorithm for linear optimization.

Mehrotra-type predictor–corrector algorithms for sufficient linear complementarity problem

Two Mehrotra-type predictor–corrector algorithms are proposed for solving sufficient linear complementarity problems. Both algorithms are independent on the handicap χ of the problems. The first version of the Mehrotra-type algorithm is a generalization of the safeguard based Mehrotra-type algorithm for linear programming, that was proposed by Salahi et al. [M. Salahi, J. Peng, T. Terlaky, On Mehrotra-type predictor–corrector algorithms, SIAM J. Optim. 18 (2007) 1377–1397]. We also present a new variant of Mehrotra-type predictor–corrector algorithm using a new adaptive updating strategy of the centering parameter. We show that both algorithms enjoy O(n(1+χ)log((x0)Ts0/ϵ)) iteration complexity. Some numerical results are reported as well.

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.

On the complexity analysis of a Mehrotra-type primal–dual feasible algorithm for semidefinite optimization

It has been shown in various papers that most interior-point algorithms and their analysis can be generalized to semidefinite optimization. This paper presents an extension of the recent variant of Mehrotra's predictor–corrector algorithm that was proposed by Salahi et al. [M. Salahi, J. Peng, and T. Terlaky, On Mehrotra-type predictor–corrector algorithms, SIAM J. Optim. 18 (2007), pp. 1377–1397] for linear optimization (LO) problems. Based on the NT [Y.E. Nesterov and M.J. Todd, Self-scaled barriers and interior-point methods for convex programming, Math. Oper. Res. 22 (1997), pp. 1–42] directions which are Newton search directions, we show that the iteration-complexity bound of the algorithm is of the same order as that of the corresponding algorithm for LO.

Mehrotra-type predictor-corrector algorithm revisited

Motivated by a numerical example that shows that a feasible version of Mehrotra's original predictor-corrector algorithm might be inefficient in practice, Salahi et al. [M. Salahi, J. Peng and T. Terlaky, On Mehrotra-type predictor-corrector algorithms, to apper in SIAM J. Optim.] proposed a so-called safeguard-based variant of the algorithm that enjoys polynomial iteration complexity, although its practical efficiency is preserved. In this paper, we analyse the same Mehrotra's algorithm from a different perspective. We give a condition on the maximum step size in the predictor direction, the violation of which might imply a very small or zero step size in the corrector direction of the algorithm. This might explain the reason for occasional ill behaviour of the feasible version of Mehrotra's original algorithm. We propose to cut the maximum step size in the predictor direction if it is above a certain threshold. If this cut does not give a desirable step size, then we cut it for the second time that allows us to give a lower bound for the step size in the corrector direction. This enables us to prove an 𝒪(n 5/2log (n/ε)) worst case iteration complexity bound for the new algorithm. By slightly modifying the Newton system in the corrector step, we reduce the iteration complexity to 𝒪 (n 3/2log (n/ε)). Finally, we report some illustrative computational results using the McIPM software package.

A finite termination Mehrotra-type predictor–corrector algorithm

Mehrotra-type predictor–corrector algorithms are the backbone of most Interior Point Methods based software packages. Salahi et al. [M. Salahi, J. Peng, T. Terlaky, On Mehrotra-type predictor–corrector algorithms, Technical Report 2005/4, Advanced Optimization Lab., McMaster University, Hamilton, ON, Canada. http://www.cas.mcmaster.ca/~oplab/publication] in their recent works have shown some ill behaviors of Mehrotra’s original algorithm which motivated them to modify it in order to achieve the polynomial iteration complexity while preserving its practical efficiency. In this paper we analyze the same algorithm from a different perspective and give a condition on the maximum feasible step size in the predictor step, violation of which might lead to a very small or even zero step size in the corrector step. If the maximum step size in the predictor step is above a certain threshold, then we cut it to satisfy the derived condition. This enables us to prove that the algorithm terminates finitely.

A Mehrotra-type predictor-corrector algorithm with polynomiality andQ-subquadratic convergence

Mehrotra's predictor-corrector algorithm [3] is currently considered to be one of the most practically efficient interior-point methods for linear programming. Recently, Zhang and Zhang [18] studied the global convergence properties of the Mehrotra-type predictor-corrector approach and established polynomial complexity bounds for two interior-point algorithms that use the Mehrotra predictor-corrector approach. In this paper, we study the asymptotic convergence rate for the Mehrotra-type predictor-corrector interior-point algorithms. In particular, we construct an infeasible-interior-point algorithm based on the second algorithm proposed in [18] and show that while retaining a complexity bound ofO(n1.5t)-iterations, under certain conditions the algorithm also possesses aQ-subquadratic convergence, i.e., a convergence ofQ-order 2 with an unboundedQ-factor.