Large-update Primal-dual Interior-point Method Research Articles

An efficient primal-dual interior point method for linear programming problems based on a new kernel function with a finite exponential-trigonometric barrier term

In this paper, we first propose a new finite exponential-trigonometric kernel function that has finite value at the boundary of the feasible region. Then by using some simple analysis tools, we show that the new kernel function has exponential convexity property. We prove that the large-update primal-dual interior-point method based on this kernel function for solving linear optimization problems has $$O\left( \sqrt{n}\log n\log \frac{n}{\epsilon }\right)$$ iteration bound in the worst case when the barrier parameter is taken large enough. Moreover, the numerical results reveal that the new finite exponential-trigonometric kernel function has better results than the other kernel functions.

Complexity of Interior Point Methods for a Class of Linear Complementarity Problems Using a Kernel Function with Trigonometric Growth Term

In this paper, we propose a large-update primal-dual interior point method for solving a class of linear complementarity problems based on a new kernel function. The main aspects distinguishing our proposed kernel function from the others are as follows: Firstly, it incorporates a specific trigonometric function in its growth term, and secondly, the corresponding barrier term takes finite values at the boundary of the feasible region. We show that, by resorting to relatively simple techniques, the primal-dual interior point methods designed for a specific class of linear complementarity problems enjoy the so-called best-known iteration complexity for the large-update methods.

A unified complexity analysis of interior point methods for semidefinite problems based on trigonometric kernel functions

In this paper, we present an interior-point algorithm for semidefinite optimization (SDO) problems based on a new generic trigonometric kernel function, which is constructed by introducing some new conditions on the kernel function. Based on these conditions, we propose a new trigonometric kernel function and present some properties of this function. We present some complexity results for the generic kernel function and prove that, a large-update primal–dual interior-point method for solving SDO problems with this new kernel function enjoys as worst case iteration complexity bound which matches the currently best known complexity bound for large update methods. Moreover, some numerical results show that the new proposed kernel function has better results than the other trigonometric kernel functions.

A large-update primal–dual interior-point algorithm for second-order cone optimization based on a new proximity function

In this paper, we propose a large-update primal–dual interior-point algorithm for second-order cone optimization (SOCO) based on a class of kernel functions consisting of a trigonometric barrier term. The algorithm starts from a strictly feasible point and generates a sequence of points converging to an optimal solution of the problem. Using a simple analysis, we show that the algorithm has worst case iteration complexity for large-update primal–dual interior point methods which coincides with the so far best-known iteration bound for SOCO.

A Parametric Kernel Function Yielding the Best Known Iteration Bound of Interior-Point Methods for Semidefinite Optimization

In this paper, a class of large-update primal-dual interior-point methods for semidefinite optimization based on a parametric kernel function are presented. The proposed kernel function is not only used for determining the search directions but also for measuring the distance between the given iterate and the center for the algorithms. By means of the Nesterov and Todd scaling scheme, the currently best known iteration bounds for large-update methods is established.

Kernel function-based primal-dual interior-point methods for symmetric cones optimization

In this paper, we present a large-update primal-dual interior-point method for symmetric cone optimization (SCO) based on a new kernel function, which determines both search directions and the proximity measure between the iterate and the center path. The kernel function is neither a self-regular function nor the usual logarithmic kernel function. Besides, by using Euclidean Jordan algebraic techniques, we achieve the favorable iteration complexity \(O(\sqrt r (\log r)^2 \log (r/\varepsilon ))\), which is as good as the convex quadratic semi-definite optimization analogue.

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.

New parameterized kernel functions for linear optimization

Recent studies on the kernel function-based primal-dual interior-point algorithms indicate that a kernel function not only represents a measure of the distance between the iteration and the central path, but also plays a critical role in improving the computational complexity of an interior-point algorithm. In this paper, we propose a new class of parameterized kernel functions for the development of primal-dual interior-point algorithms for solving linear programming problems. The properties of the proposed kernel functions and corresponding parameters are investigated. The results lead to a complexity bounds of $${O\left(\sqrt{n}\,{\rm log}\,n\,{\rm log}\,\frac{n}{\epsilon}\right)}$$ for the large-update primal-dual interior point methods. To the best of our knowledge, this is the best known bound achieved.

A new kind of simple kennel function yielding good iteration bounds for primal-dual interior-point methods

We introduce a new kind of kernel function, which yields efficient large-update primal-dual interior-point methods. We conclude that in some situations its iteration bounds are O ( m 3 m + 1 2 m n m + 1 2 m log n ϵ ) , which are at least as good as the best known bounds so far, O ( n log n log n ϵ ) , for large-update primal-dual interior-point methods. The result decreases the gap between the practical behavior of the large-update algorithms and their theoretical performance results, which is an open problem. Numerical results show that the algorithms are feasible.

ON COMPLEXITY ANALYSIS OF THE PRIMAL-DUAL INTERIOR-POINT METHOD FOR SECOND-ORDER CONE OPTIMIZATION PROBLEM

The purpose of this paper is to obtain new complexity results for a second-order cone optimization (SOCO) problem. We define a proximity function for the SOCO by a kernel function. Furthermore we formulate an algorithm for a large-update primal-dual interior-point method (IPM) for the SOCO by using the proximity function and give its complexity analysis, and then we show that the new worst-case iteration bound for the IPM is O(q√N(logN)q+1/q log N/e), where q?1.

A LARGE-UPDATE INTERIOR POINT ALGORITHM FOR $P_*(\kappa)$ LCP BASED ON A NEW KERNEL FUNCTION

In this paper we generalize large-update primal-dual interior point methods for linear optimization problems in [2] to the linear complementarity problems based on a new kernel function which includes the kernel function in [2] as a special case. The kernel function is neither self-regular nor eligible. Furthermore, we improve the complexity result in [2] from to .

Exploring complexity of large update interior-point methods for [formula omitted] linear complementarity problem based on Kernel function

Interior-point methods not only are the most effective methods in practice but also have polynomial-time complexity. The large update interior-point methods perform in practice much better than the small update methods which have the best known theoretical complexity. In this paper, motivated by the complexity results for linear optimization based on kernel functions, we extend a generic primal-dual interior-point algorithm based on a new kernel function to solve P ∗ ( κ ) linear complementarity problems. By using some elegant and simple tools and having interior-point condition, we show that the large update primal-dual interior-point methods for solving P ∗ ( κ ) linear complementarity problems enjoys O q ( 1 + 2 κ ) n ( log n ) q + 1 q log n ε iteration bound which becomes O ( 1 + 2 κ ) n log n log ( log n ) log n ε with special choices of the parameter q. This bound is much better than the classical primal-dual interior-point methods based on logarithmic barrier function and recent kernel functions introduced by some authors in optimization field. Some computational results have been provided.

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.