Simplified infeasible interior-point algorithm for linear optimization based on a simple function

Based on a similar kernel function, we present an infeasible version of the interior-point algorithm for linear optimization introduced by Wang et al. (2016). The property of exponential convexity is still important to simplify the analysis of the algorithm. The iteration bound coincides with the currently best iteration bound for infeasible interior-point algorithms.     

Complexity analysis of a full-Newton step interior-point method for linear optimization

This paper concerns a short-update primal-dual interior-point method for linear optimization based on a new search direction. We apply a vector-valued function generated by a univariate function on the nonlinear equation of the system which defines the central path. The common way to obtain the equivalent form of the central path is using the square root function. In this paper we consider a new function formed by the difference of the identity map and the square root function. We apply Newton’s method in order to get the new directions. In spite of the fact that the analysis is more difficult in this case, we prove that the complexity of the algorithm is identical with the one of the best known methods for linear optimization.     

A wide neighborhood interior-point algorithm for linear optimization based on a specific kernel function

This paper presents an interior point algorithm for solving linear optimization problems in a wide neighborhood of the central path introduced by Ai and Zhang (SIAM J Optim 16:400–417, 2005). In each iteration, the algorithm computes the new search directions by using a specific kernel function. The convergence of the algorithm is shown and it is proved that the algorithm has the same iteration bound as the best short-step algorithms. We demonstrate the computational efficiency of the proposed algorithm by testing some Netlib problems in standard form. To best our knowledge, this is the first wide neighborhood path-following interior-point method with the same complexity as the best small neighborhood path-following interior-point methods that uses the kernel function.

     

A large-update interior-point algorithm for convex quadratic semi-definite optimization based on a new kernel function

In this paper, we present a large-update interior-point algorithm for convex quadratic semi-definite optimization based on a new kernel function. The proposed function is strongly convex. It is not self-regular function and also the usual logarithmic function. The goal of this paper is to investigate such a kernel function and show that the algorithm has favorable complexity bound in terms of the elegant analytic properties of the kernel function. The complexity bound is shown to be $O\left( {\sqrt n \left( {\log n} \right)^2 \log \frac{n} {\varepsilon }} \right)$ . This bound is better than that by the classical primal-dual interior-point methods based on logarithmic barrier function and recent kernel functions introduced by some authors in optimization fields. Some computational results have been provided.     

Complexity analysis of a linear complementarity algorithm based on a Lyapunov function

We consider a path following algorithm for solving linear complementarity problems with positive semi-definite matrices. This algorithm can start from any interior solution and attain a linear rate of convergence. Moreover, if the starting solution is appropriately chosen, this algorithm achieves a complexity of O( L}) iterations, wherem is the number of variables andL is the size of the problem encoding in binary. We present a simple complexity analysis for this algorithm, which is based on a new Lyapunov function for measuring the nearness to optimality. This Lyapunov function has itself interesting properties that can be used in a line search to accelerate convergence. We also develop an inexact line search procedure in which the line search stepsize is obtainable in a closed form. Finally, we extended this algorithm to handle directly variables which are unconstrained in sign and whose corresponding matrix is positive definite. The rate of convergence of this extended algorithm is shown to be independent of the number of such variables.This research is partially supported by the U.S. Army Research Office, contract DAAL03-86-K-0171 (Center for Intelligent Control Systems), and by the National Science Foundation, grant NSF-ECS-8519058.     

Primal-dual interior-point algorithm for semidefinite optimization based on a new kernel function with trigonometric barrier term

In this paper we propose primal-dual interior-point algorithms for semidefinite optimization problems based on a new kernel function with a trigonometric barrier term. We show that the iteration bounds are $O(\sqrt{n}\log(\frac{n}{\epsilon}))$ for small-update methods and $O(n^{\frac{3}{4}}\log(\frac{n}{\epsilon}))$ for large-update, respectively. The resulting bound is better than the classical kernel function. For small-update, the iteration complexity is the best known bound for such methods.     

基于指数型核函数的线性规划原始对偶内点算法

给出线性规划原始对偶内点算法的一个单变量指数型核函数.首先研究了这个指数型核函数的性质以及其对应的障碍函数.其次,基于这个指数型核函数,设计了求解线性规划问题的原始对偶内点算法,得到了目前小步算法最好的理论迭代界.最后,通过数值算例比较了基于指数型核函数的原始对偶内点算法和基于对数型核函数的原始对偶内点算法的计算效果.     

A primal-dual interior-point algorithm for symmetric optimization based on a new method for finding search directions

We introduce an interior-point method for symmetric optimization based on a new method for determining search directions. In order to accomplish this, we use a new equivalent algebraic transformation on the centring equation of the system which characterizes the central path. In this way, we obtain a new class of directions. We analyse a special case of this class, which leads to the new interior-point algorithm mentioned before. Another way to find the search directions is using barriers derived from kernel functions. We show that in our case the corresponding direction cannot be deduced from a usual kernel function. In spite of this fact, we prove the polynomial complexity of the proposed algorithm.     

An infeasible interior-point algorithm with full-Newton step for linear optimization

Recently, Roos (SIAM J Optim 16(4):1110–1136, 2006) presented a primal-dual infeasible interior-point algorithm that uses full-Newton steps and whose iteration bound coincides with the best known bound for infeasible interior-point algorithms. In the current paper we use a different feasibility step such that the definition of the feasibility step in Mansouri and Roos (Optim Methods Softw 22(3):519–530, 2007) is a special case of our definition, and show that the same result on the order of iteration complexity can be obtained.      

A new primal-dual path-following interior-point algorithm for semidefinite optimization

In this paper we present a new primal-dual path-following interior-point algorithm for semidefinite optimization. The algorithm is based on a new technique for finding the search direction and the strategy of the central path. At each iteration, we use only full Nesterov-Todd step. Moreover, we obtain the currently best known iteration bound for the algorithm with small-update method, namely, , which is as good as the linear analogue.     

Complexity analysis and numerical implementation of primal-dual interior-point methods for convex quadratic optimization based on a finite barrier

In this paper, we present primal-dual interior-point methods for convex quadratic optimization based on a finite barrier, which has been investigated earlier for the case of linear optimization by Bai et al. (SIAM J Optim 13(3):766–782, 2003). By means of the feature of the finite kernel function, we study the complexity analysis of primal-dual interior-point methods based on the finite barrier and derive the iteration bounds that match the currently best known iteration bounds for large- and small-update methods, namely, $O(\sqrt{n}\log{n}\log{\frac{n}{\varepsilon}})$ and $O(\sqrt{n}\log{\frac{n}{\varepsilon}})$ , respectively, which are as good as the linear optimization analogue. Numerical tests demonstrate the behavior of the algorithms with different parameters.     

A primal-dual interior-point method for linear programming based on a weighted barrier function

One motivation for the standard primal-dual direction used in interior-point methods is that it can be obtained by solving a least-squares problem. In this paper, we propose a primal-dual interior-point method derived through a modified least-squares problem. The direction used is equivalent to the Newton direction for a weighted barrier function method with the weights determined by the current primal-dual iterate. We demonstrate that the Newton direction for the usual, unweighted barrier function method can be derived through a weighted modified least-squares problem. The algorithm requires a polynomial number of iterations. It enjoys quadratic convergence if the optimal vertex is nondegenerate.The research of the second author was supported in part by ONR Grants N00014-90-J-1714 and N00014-94-1-0391.     

CurveLP-A MATLAB implementation of an infeasible interior-point algorithm for linear programming

Mehrotra’s algorithm has been the most successful infeasible interior-point algorithm for linear programming since 1990. Most popular interior-point software packages for linear programming are based on Mehrotra’s algorithm. This paper describes a proposal and implementation of an alternative algorithm, an arc-search infeasible interior-point algorithm. We will demonstrate, by testing Netlib problems and comparing the test results obtained by the arc-search infeasible interior-point algorithm and Mehrotra’s algorithm, that the proposed arc-search infeasible interior-point algorithm is a more reliable and efficient algorithm than Mehrotra’s algorithm.     

An interior-point trust-funnel algorithm for nonlinear optimization

We present an interior-point trust-funnel algorithm for solving large-scale nonlinear optimization problems. The method is based on an approach proposed by Gould and Toint (Math Prog 122(1):155–196, 2010) that focused on solving equality constrained problems. Our method is similar in that it achieves global convergence guarantees by combining a trust-region methodology with a funnel mechanism, but has the additional capability of being able to solve problems with both equality and inequality constraints. The prominent features of our algorithm are that (i) the subproblems that define each search direction may be solved with matrix-free methods so that derivative matrices need not be formed or factorized so long as matrix-vector products with them can be performed; (ii) the subproblems may be solved approximately in all iterations; (iii) in certain situations, the computed search directions represent inexact sequential quadratic optimization steps, which may be desirable for fast local convergence; (iv) criticality measures for feasibility and optimality aid in determining whether only a subset of computations need to be performed during a given iteration; and (v) no merit function or filter is needed to ensure global convergence.     

A full-Newton step infeasible interior-point algorithm for monotone LCP based on a locally-kernel function

We propose a new full-Newton step infeasible interior-point algorithm for monotone linear complementarity problems based on a simple locally-kernel function. The algorithm uses the simple locally-kernel function to determine the search directions and define the neighborhood of central path. Two types of full-Newton steps are used, feasibility step and centering step. The algorithm starts from strictly feasible iterates of a perturbed problem, on its central path, and feasibility steps find strictly feasible iterates for the next perturbed problem. By using centering steps for the new perturbed problem, we obtain strictly feasible iterates close enough to the central path of the new perturbed problem. The procedure is repeated until an ?-approximate solution is found. We analyze the algorithm and obtain the complexity bound, which coincides with the best-known result for monotone linear complementarity problems.     

A QMR-based interior-point algorithm for solving linear programs

A new approach for the implementation of interior-point methods for solving linear programs is proposed. Its main feature is the iterative solution of the symmetric, but highly indefinite 2×2-block systems of linear equations that arise within the interior-point algorithm. These linear systems are solved by a symmetric variant of the quasi-minimal residual (QMR) algorithm, which is an iterative solver for general linear systems. The symmetric QMR algorithm can be combined with indefinite preconditioners, which is crucial for the efficient solution of highly indefinite linear systems, yet it still fully exploits the symmetry of the linear systems to be solved. To support the use of the symmetric QMR iteration, a novel stable reduction of the original unsymmetric 3×3-block systems to symmetric 2×2-block systems is introduced, and a measure for a low relative accuracy for the solution of these linear systems within the interior-point algorithm is proposed. Some indefinite preconditioners are discussed. Finally, we report results of a few preliminary numerical experiments to illustrate the features of the new approach.     

A modified layered-step interior-point algorithm for linear programming

The layered-step interior-point algorithm was introduced by Vavasis and Ye. The algorithm accelerates the path following interior-point algorithm and its arithmetic complexity depends only on the coefficient matrixA. The main drawback of the algorithm is the use of an unknown big constant in computing the search direction and to initiate the algorithm. We propose a modified layered-step interior-point algorithm which does not use the big constant in computing the search direction. The constant is required only for initialization when a well-centered feasible solution is not available, and it is not required if an upper bound on the norm of a primal—dual optimal solution is known in advance. The complexity of the simplified algorithm is the same as that of Vavasis and Ye. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Research supported in part by ONR contract N00014-94-C-0007 and the Grant-in-Aid for Scientific Research (C) 08680478 and the Grant-in-Aid for Encouragement of Young Scientists (A) 08780227 of the Ministry of Science, Education and Culture of Japan. This research was partially done while S. Mizuno and T. Tsuchiya were visiting IBM Almaden Research Center in the summer of 1995.