Second-order Corrector Research Articles

A long-step second-order corrector interior point algorithm for linear optimization based on the algebraic equivalent transformation

In this paper, we present a new large-step primal-dual second-order corrector interior-point algorithm for linear optimization. This algorithm is based on the Nagy-Varga neighborhood of the central path, which uses the algebraic equivalent transformation strategy of Darvay et al. to determine the search directions. In addition, our method uses a second-order corrector direction in each iteration. The iterates remain in the proposed neighborhood by taking the largest possible step lengths along the search directions. We show that the new long-step algorithm is convergent and has the same complexity as the best short-step algorithms for linear optimization. To the best of our knowledge this is the first primal-dual second-order corrector interior-point algorithm that uses a different neighborhood from those available in the literature. Finally, numerical experiments show that the proposed algorithm is efficient and competitive.

A wide neighbourhood primal-dual second-order corrector interior point algorithm for semidefinite optimization

In this paper, we proposed a new primal-dual second-order corrector interior-point algorithm for semidefinite optimization. The algorithm is based on Darvay–Takács neighbourhood of the central path. In the new algorithm, the search directions are determined by the Darvay–Takács's direction and a second-order corrector direction in each iteration. The iteration complexity bound is for the Nesterov–Todd scaling direction, which coincides with the best-known complexity results for semidefinite optimization. Finally, numerical experiments show that the proposed algorithm is promising.

Homogenized equation of second-order accuracy for conductivity of laminates

The high order homogenization techniques potentially generate the so-called infinite order homogenized equations. Since long ago, the coefficients at higher order derivatives in these equations have been calculated within various refined theories for both periodic composites and thin structures. However, it was not always clear, what is a well-posed mathematical formulation for such equations. In the present paper, we discuss two techniques for constructing a second-order homogenized equation. One of them is concerned with the projection of a weak formulation of the original problem on an ‘ansatz subspace’. The second one corresponds to the traditional two scale asymptotic expansion using the representation of a second-order corrector via the solution of the classical (leading order) homogenized equation.

A second-order corrector wide neighborhood infeasible interior-point method for linear optimization based on a specific kernel function

In this paper, we present a second-order corrector infeasibleinterior-point method for linear optimization in a largeneighborhood of the central path. The innovation of our method is tocalculate the predictor directions using a specific kernel functioninstead of the logarithmic barrier function. We decompose thepredictor direction induced by the kernel function to two orthogonaldirections of the corresponding to the negative and positivecomponent of the right-hand side vector of the centering equation.The method then considers the new point as a linear combination ofthese directions along with a second-order corrector direction. Theconvergence analysis of the proposed method is investigated and itis proved that the complexity bound is Ο(n5/4 log e-1).

A Second-Order Corrector Infeasible Interior-Point Method for Semidefinite Optimization Based on a Wide Neighborhood

This paper proposes a second-order corrector infeasible interior-point method for semidefinite optimization in a large neighborhood of the central path. Our algorithm uses the Nesterov–Todd search directions as a Newton direction. We make use of the scaled Newton directions for symmetric search directions. Based on Ai and Zhang idea, we decompose the Newton directions into two orthogonal directions corresponding to positive and negative parts of the right-hand side of the Newton equation. In such a way, we use different step lengths for each of them and the corrector step is multiplied by the square of the step length of the infeasible directions in the expression of the new iterate. Starting with a point $$(X^0, y^0, S^0)$$ in the neighborhood in terms of Frobenius norm, we show that the algorithm terminates after at most $$\mathcal {O}(n^{\frac{5}{4}}\log \varepsilon ^{-1})$$ iterations, improving by a factor $$n^{\frac{1}{4}}$$ results on these kinds of algorithms. We believe that this is the first infeasible interior-point algorithm in a large neighborhood, which uses a Newton direction decomposition and a second-order corrector step to improve the iteration complexity. The main difference of the proposed method comparing with the existing methods in literature is the strategy for generating corrector directions. Some preliminary numerical results are given to verify the efficiency of the algorithm.

A Second-order Corrector Infeasible Interior-point Method with One-norm wide Neighborhood for Symmetric Optimization

A Second-order Corrector Infeasible Interior-point Method with One-norm wide Neighborhood for Symmetric Optimization

A new wide neighborhood primal-dual second-order corrector algorithm for linear optimization

We propose a new large-step primal-dual second-order corrector interior-point method for linear optimization. At each iteration, our method uses the new wide neighborhood introduced by Darvay and Takacs (Cent Eur J Oper Res 26(3):551–563, 2018. https://doi.org/10.1007/s10100-018-0524-0 ). In this paper we would like to improve the directions proposed by Darvay and Takacs by adding a second-order corrector direction. The corrector step is multiplied by the square of the step length in the expression of the new iterate. To our best knowledge, this is the first primal-dual second-order corrector interior-point algorithm based on Darvay–Takacs’s new wide neighborhood, which has the same complexity as the best short-step algorithms for linear optimization. Finally, numerical experiments show that the proposed algorithm is efficient and reliable.

Stochastic Homogenization of Linear Elliptic Equations: Higher-Order Error Estimates in Weak Norms Via Second-Order Correctors

We are concerned with the homogenization of second-order linear elliptic equations with random coefficient fields. For symmetric coefficient fields with only short-range correlations, quantified through a logarithmic Sobolev inequality for the ensemble, we prove that when measured in weak spatial norms, the solution to the homogenized equation provides a higher-order approximation of the solution to the equation with oscillating coefficients. In the case of nonsymmetric coefficient fields, we provide a higher-order approximation (in weak spatial norms) of the solution to the equation with oscillating coefficients in terms of solutions to constant-coefficient equations. In both settings, we also provide optimal error estimates for the two-scale expansion truncated at second order. Our results rely on novel estimates on the second-order homogenization corrector, which we establish via sensitivity estimates for the second-order corrector and a large-scale $L^p$ theory for elliptic equations with random coefficients. Our results also cover the case of elliptic systems.

A new second-order corrector interior-point algorithm for P*(k)-LCP

In this paper, we propose a second-order corrector interior-point algorithm for solving P*(k)- linear complementarity problems. The method generates a sequence of iterates in a wide neighborhood of the central path introduced by Ai and Zhang. In each iteration, the method computes a corrector direction in addition to the Ai-Zhang direction, in an attempt to improve performance. The algorithm does not depend on the handicap k of the problem, so that it can be used for any P*(k)-linear complementarity problems. It is shown that the iteration complexity bound of the algorithm is O ((1+k)3 ? nL). Some numerical results are provided to illustrate the performance of the algorithm.

Adaptive Adams–Bashforth method for modeling of highly doped fiber amplifiers and fiber lasers

We report the formulation of the adaptive variable coefficient predictor-corrector Adams-Bashforth scheme for modeling the population dynamics in erbium-doped fiber amplifiers and lasers based on highly doped fibers. A modified form of the scheme derived using a Lagrange polynomial is shown to result in 75% reduction of step-size as compared to a conventional adaptive Euler method. Our analysis shows that the second-order predictor and third-order corrector is most suitable for accurate modeling of the above problem. The model has been validated by re-generating the absorption and emission spectrum for doped fibers found in the literature. This modeling approach has been carried further to simulate a filterless fiber laser cavity, in which several gain and saturation dynamics are encountered. Spectral power evolution in the fiber is simulated by using the above method, and the steady-state lasing wavelength is evaluated as a function of cavity attenuation. The experimental results are found to match very well with the simulation.

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.

Local convergence of filter methods for equality constrained non-linear programming

In Gonzaga et al. [A globally convergent filter method for nonlinear programming, SIAM J. Optimiz. 14 (2003), pp. 646–669] we discuss general conditions to ensure global convergence of inexact restoration filter algorithms for non-linear programming. In this article we show how to avoid the Maratos effect by means of a second-order correction. The algorithms are based on feasibility and optimality phases, which can be either independent or not. The optimality phase differs from the original one only when a full Newton step for the tangential minimization of the Lagrangian is efficient but not acceptable by the filter method. In this case a second-order corrector step tries to produce an acceptable point keeping the efficiency of the rejected step. The resulting point is tested by trust region criteria. Under the usual hypotheses, the algorithm inherits the quadratic convergence properties of the feasibility and optimality phases. This article includes a comparison between classical Sequential Quadratic Programming (SQP) and Inexact Restoration (IR) iterations, showing that both methods share the same asymptotic convergence properties.

An $${O(\sqrt{n}L)}$$ iteration primal-dual second-order corrector algorithm for linear programming

In this paper, we propose a primal-dual second-order corrector interior point algorithm for linear programming problems. At each iteration, the method computes a corrector direction in addition to the Ai–Zhang direction [Ai and Zhang in SIAM J Optim 16:400–417 (2005)], in an attempt to improve performance. The corrector is multiplied by the square of the step-size in the expression of the new iterate. We prove that the use of the corrector step does not cause any loss in the worst-case complexity of the algorithm. To our best knowledge, this is the first wide neighborhood second-order corrector algorithm enjoyed the low iteration bound of \({O(\sqrt{n}L)}\), the same as the best known complexity results for interior point methods.

On correctors of symplectic integrators

An intensive discussion is given here with numerical illustration on the symplectic correctors proposed by Wisdom et al. (1996). A simple method is given for deriving the first and second order correctors of any symplectic integrators in terms of Lie series for the general case, in which the Hamiltonian can be separated into a main integrable part and several smaller integrable parts. The method is numerically applied to the 3-body system Sun-Jupiter-Saturn with the Hamiltonian of the form introduced by Duncan et al. (1998). It is shown that the first-order corrector is more precise that the uncorrected symplectic integrator. As to precision, numerical stability and computer efficiency, the first-order corrector is greatly superior to the second-order corrector when a large stepsize is adopted.

FINITE-DIFFERENCE-BASED LATTICE BOLTZMANN METHOD FOR INVISCID COMPRESSIBLE FLOWS

A finite-difference-based lattice Boltzmann model, employing the 2-D, 9-velocity square (D2Q9) lattice for the compressible Euler equations, is presented. The model is constructed by allowing the particles to possess both kinetic and thermal energies. Such a lattice structure can represent both incompressible and compressible flow regimes. In the numerical treatment, to attain desirable accuracy, the total-variation-diminishing (TVD) scheme is adopted with either the minmod function or a second-order corrector as the flux limiter. The model can treat shock/expansion waves as well as contact discontinuity. Both one- and two-dimensional test cases are computed, and the results are compared with the exact as well as other reported numerical solutions, demonstrating that there is consistency between macroscopic and kinetic computations for the compressible flow.