Large-update Interior-point Method Research Articles

Complexity analysis and numerical implementation of large-update interior-point methods for SDLCP based on a new parametric barrier kernel function

In this paper, we deal with the complexity analysis and the numerical implementation of primal–dual interior-point methods for monotone semidefinite linear complementarity problems based on a new parametric kernel function. The proposed kernel function is neither a self-regular and nor the usual logarithmic barrier function. By means of the feature of the parametric kernel function, we study the complexity analysis of primal–dual IPMs and derive the currently best known iteration bound for the large-update algorithm, namely, which is as good as the linear and the semidefinite optimization analogue. Finally, we report some numerical results to show the practical performance of the proposed algorithm with different parameters.

New complexity analysis for primal-dual interior-point methods for self-scaled optimization problems

A linear optimization problem over a symmetric cone, defined in a Euclidean Jordan algebra and called a self-scaled optimization problem (SOP), is considered. We formulate an algorithm for a large-update primal-dual interior-point method (IPM) for the SOP by using a proximity function defined by a new kernel function, and we obtain the best known complexity results of the large-update IPM for the SOP by using the Euclidean Jordan algebra techniques.

A new primal-dual interior-point algorithm for convex quadratic optimization

In this paper, a new primal-dual interior-point algorithm for convex quadratic optimization (CQO) based on a kernel function is presented. The proposed function has some properties that are easy for checking. These properties enable us to improve the polynomial complexity bound of a large-update interior-point method (IPM) to O(√n log n log n/ɛ), which is the currently best known polynomial complexity bound for the algorithm with the large-update method. Numerical tests were conducted to investigate the behavior of the algorithm with different parameters p, q and θ, where p is the growth degree parameter, q is the barrier degree of the kernel function and θ is the barrier update parameter.

A polynomial-time algorithm for linear optimization based on a new class of kernel functions

In this paper we present a class of polynomial primal-dual interior-point algorithms for linear optimization based on a new class of kernel functions. This class is fairly general and includes the class of finite kernel functions by Y.Q. Bai, M.El Ghami and C. Roos [Y.Q. Bai, M. El Ghami, and C. Roos. A new efficient large-update primal-dual interior-point method based on a finite barrier, SIAM Journal on Optimization, 13 (3) (2003) 766–782]. The proposed functions have a finite value at the boundary of the feasible region. They are not exponentially convex and also not strongly convex like the usual barrier functions. The goal of this paper is to investigate such a class of kernel functions and to show that the interior-point methods based on these functions have favorable complexity results. In order to achieve these complexity results, several new arguments had to be used for the analysis. The iteration bound of large-update interior-point methods based on these functions and analyzed in this paper, is shown to be O ( n log n log n ϵ ) . For small-update interior-point methods the iteration bound is O ( n log n ϵ ) , which is currently the best-known bound for primal-dual IPMs. We also present some numerical results which show that by using a new kernel function, the best iteration numbers were achieved in most of the test problems.

Complexity analysis of interior-point methods for linear optimization based on some conditions on kernel function

Interior point methods have shown their powers in solving linear optimization problems and large classes of other optimization problems. However, at present there is still a gap between the practical behavior of these algorithms and their theoretical worst case complexity. The so-called large update interior point methods perform in practice much better than the small update methods which have the best known theoretical complexity. Recently, this gap has been reduced by Peng, Roos and Terlaky by introducing new self regular kernel functions. In this paper, by focusing on linear optimization problem and motivated by the self regular family of kernel functions, we impose some mild condition on the kernel functions and we give a new class of kernel functions. We also give a simple complexity analysis for large-update interior point methods based on the kernel functions in this new class. Finally, we apply our analysis to two family of kernel functions in our new class. We also explore the complexity of the algorithm and we show that the so far worst case O ( n log n log n ϵ ) iteration bound can be achieved in special case.

An interior point method for linear programming based on a class of Kernel functions

Interior point methods are not only the most effective methods for solving optimisation problems in practice but they also have polynomial time complexity. However, there is still a gap between the practical behavior of the interior point method algorithms and their theoretical complexity results. In this paper, by focusing on linear programming problems, we introduce a new family of kernel functions that have some simple and easy to check properties. We present a simplified analysis to obtain the complexity of generic interior point methods based on the proximity functions induced by these kernel functions. Finally, we prove that this family of kernel functions leads to improved iteration bounds of the large-update interior point methods.

On self-regular IPMs

Primal-dual interior-point methods (IPMs) have shown their power in solving large classes of optimization problems. However, at present there is still a gap between the practical behavior of these algorithms and their theoretical worst-case complexity results, with respect to the strategies of updating the duality gap parameter in the algorithm. The so-called small-update IPMs enjoy the best known theoretical worst-case iteration bound, but work very poorly in practice. To the contrary, the so-called large-update IPMs have superior practical performance but with relatively weaker theoretical results. In this paper we discuss the new algorithmic variants and improved complexity results with respect to the new family of Self-Regular proximity based IPMs for Linear Optimization problems, and their generalizations to Conic and Semidefinite Optimization

A dynamic large-update primal‐dual interior-point method for linear optimization

Primal-dual interior-point methods (IPMs) have shown their power in solving large classes of optimization problems. However, at present there is still a gap between the practical behavior of these algorithms and their theoretical worst-case complexity results, with respect to the strategies of updating the duality gap parameter in the algorithm. The so-called small-update IPMs enjoy the best known theoretical worst-case iteration bound but work very poorly in practice, while the so-called large-update IPMs have superior practical performance but with relatively weaker theoretical results. In this article, by restricting us to linear optimization (LO), we first exploit some interesting properties of a self-regular proximity function, proposed recently by the authors of this work and Roos, and use it to define a neighborhood of the central path. These simple but interesting properties of the proximity function indicate that, when the current iterate is in a large neighborhood of the central path, then the large-update IPM emerges to be the only natural choice. Then, we apply these results to design a specific self-regularity based IPM. Among others, we show that this self-regularity based IPM can also predict precisely the change of the duality gap as the standard IPM does. Therefore, we can directly apply the modified IPM to the simplified self-dual homogeneous model for LO. This provides a remedy for an implementation issue of the new self-regular IPMs. A dynamic large-update IPM in large neighborhood is proposed. Different from traditional large-update IPMs, the new dynamic IPM always takes large-updates and does not utilize any inner iteration to get centered. An worst-case iteration bound of the algorithm is established.

A New Efficient Large-Update Primal-Dual Interior-Point Method Based on a Finite Barrier

We introduce a new barrier-type function which is not a barrier function in the usual sense: it has finite value at the boundary of the feasible region. Despite this, the iteration bound of a large-update interior-point method based on this function is shown to be $O({\sqrt{n}\,({\rm log}\,n)\,{\rm log}\,\frac{n}{\varepsilon}})$, which is as good as the currently best known bound for large-update methods. The recently introduced property of \emph{exponential convexity} for the kernel function underlying the barrier function, as well as the strong convexity of the kernel function, are crucial in the analysis.