Large-update Primal-dual Interior Point Algorithm Research Articles

A Large-update Primal-dual Interior-point Algorithm for Convex Quadratic Optimization Based on a New Bi-parameterized Bi-hyperbolic Kernel Function

A Large-update Primal-dual Interior-point Algorithm for Convex Quadratic Optimization Based on a New Bi-parameterized Bi-hyperbolic Kernel Function

A Parametric Kernel Function Generating the best Known Iteration Bound for Large-Update Methods for CQSDO

In this paper, we propose a large-update primal-dual interior point algorithm for convex quadratic semidefiniteoptimization (CQSDO) based on a new parametric kernel function. This kernel function is a parameterized version of the kernel function introduced by M.W. Zhang (Acta Mathematica Sinica. 28: 2313-2328, 2012) for CQSDO. The investigation according to it generating the best known iteration bound O for large-update methods. Thus improves the iteration bound obtained by Zhang for large-update methods. Finally, we present few numerical results to show the efficiency of the proposed algorithm.

A new parameterized logarithmic kernel function for linear optimization with a double barrier term yielding the best known iteration bound

Abstract In this paper, we propose a large-update primal-dual interior point algorithm for linear optimization. The method is based on a new class of kernel functions which differs from the existing kernel functions in which it has a double barrier term. The investigation according to it yields the best known iteration bound O n log ( n ) log ( n ∈ ) \sqrt n \log (n)\log \left( {{n \over \in }} \right) for large-update algorithm with the special choice of its parameter m and thus improves the iteration bound obtained in Bai et al. [2] for large-update algorithm.

Primal–dual interior-point method for linear optimization based on a kernel function with trigonometric growth term

In this paper, we propose a large-update primal–dual interior-point algorithm for linear optimization problems based on a new kernel function with a trigonometric growth term. By simple analysis, we prove that in the large neighbourhood of the central path, the worst case iteration complexity of the new algorithm is bounded above by , which matches the currently best known iteration bound for large-update methods. Moreover, we show that, most of the so far proposed kernel functions can be rewritten as a kernel function with trigonometric growth term. Finally, numerical experiments on some test problems confirm that the new kernel function is well promising in practice in comparison with some existing kernel functions in the literature.

A primal-dual large-update interior-point algorithm for P*(κ)-LCP based on a new class of kernel functions

In this paper, we propose a large-update primal-dual interior point algorithm for P*(κ)-linear complementarity problem. The method is based on a new class of kernel functions which is neither classical logarithmic function nor self-regular functions. It is determines both search directions and the proximity measure between the iterate and the center path. We show that if a strictly feasible starting point is available, then the new algorithm has \(o\left( {(1 + 2k)p\sqrt n {{\left( {\frac{1}{p}\log n + 1} \right)}^2}\log \frac{n}{\varepsilon }} \right)\) iteration complexity which becomes \(o\left( {(1 + 2k)\sqrt n log{\kern 1pt} {\kern 1pt} n\log \frac{n}{\varepsilon }} \right)\) with special choice of the parameter p. It is matches the currently best known iteration bound for P*(κ)-linear complementarity problem. Some computational results have been provided.

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.

Large-update interior point algorithm for P ∗ -linear complementarity problem

It is well known that each barrier function defines an interior point algorithm and each barrier function is determined by its univariate kernel function. In this paper we present a new large-update primal-dual interior point algorithm for solving P ∗ -linear complementarity problem (LCP) based on a parametric version of the kernel function in (Bai et al. in SIAM J. Optim. 13:766-782, 2003). We show that the algorithm has O((1+2κ) ( log p ) 3 n (logn)log n μ 0 ϵ ) iteration complexity, where p is a barrier function parameter and κ is the handicap of the matrix. This is the best known complexity result for such a method.MSC:90C33, 90C51.

A Large-Update Feasible Interior-Point Algorithm for Convex Quadratic Semi-definite Optimization Based on a New Kernel Function

In this paper we present a large-update primal-dual interior-point algorithm for convex quadratic semi-definite optimization problems based on a new parametric kernel function. The goal of this paper is to investigate such a kernel function and show that the algorithm has the best complexity bound. The complexity bound is shown to be \(\mathcal{O}(\sqrt{n}\log n\log\frac{n}{\epsilon})\).

Interior point algorithm for [formula omitted] nonlinear complementarity problems

In this paper, we propose a new large-update primal–dual interior point algorithm for P ∗ complementarity problems (CPs). Different from most interior point methods which are based on the logarithmic kernel function, the new method is based on a class of kernel functions ψ ( t ) = ( t p + 1 − 1 ) / ( p + 1 ) + ( t − q − 1 ) / q , p ∈ [ 0 , 1 ] , q > 0 . We show that if a strictly feasible starting point is available and the undertaken problem satisfies some conditions, then the new large-update primal–dual interior point algorithm for P ∗ CPs has O ( ( 1 + 2 κ ) n log n log ( n μ 0 / ε ) ) iteration complexity which is currently the best known result for such methods with p = 1 and q = ( log n ) / 2 − 1 .

An adaptive-step primal-dual interior point algorithm for linear optimization

A common feature shared by most practical algorithms in interior point methods is the use of Mehrotra’s predictor–corrector algorithm in [S. Mehrotra, On the implementation of a (primal-dual) interior point method, SIAM Journal on Optimization 2 (1992) 575–601.] where the predictor step is never performed but it is used only to calculate an adaptive update, and thus instead of a predictor and a corrector centering step, a single Newton step is made toward the adaptively chosen target. In this paper we propose a new adaptive single-step large-update primal-dual interior point algorithm with wide neighborhood for linear optimization(LO) problems based on the simple kernel function which is first defined in [Y.Q. Bai, C. Roos, A primal-dual interior-point method based on a new kernel function with linear growth rate, in: Proceedings of Industrial Optimization Symposium and Optimization Day, Nov., 2002]. We show that the algorithm has O ( q n τ log ( n / ε ) ) complexity which is similar to the one in the above-mentioned reference.

A new large-update interior point algorithm for [formula omitted] linear complementarity problems

In this paper we propose a new large-update primal-dual interior point algorithm for P * ( κ ) linear complementarity problems (LCPs). We generalize Bai et al.'s [A primal-dual interior-point method for linear optimization based on a new proximity function, Optim. Methods Software 17(2002) 985–1008] primal-dual interior point algorithm for linear optimization (LO) problem to P * ( κ ) LCPs. New search directions and proximity measures are proposed based on a kernel function which is not logarithmic barrier nor self-regular for P * ( κ ) LCPs. We showed that if a strictly feasible starting point is available, then the new large-update primal-dual interior point algorithm for solving P * ( κ ) LCPs has the polynomial complexity O ( ( 1 + 2 κ ) n 3 / 4 log ( n / ε ) ) and gives a simple complexity analysis. This proximity function has not been used in the complexity analysis of interior point method (IPM) for P * ( κ ) LCPs before.

Complexity of large-update interior point algorithm for [formula omitted] linear complementarity problems

In this paper we propose a new large-update primal–dual interior point algorithm for P ∗ ( κ ) linear complementarity problems (LCPs). We extend Bai et al.’s primal–dual interior point algorithm for linear optimization (LO) problems to P ∗ ( κ ) LCPs with generalized kernel functions. New search directions and proximity functions are proposed based on a simple kernel function which is neither a logarithmic barrier nor a self-regular. We show that if a strictly feasible starting point is available, then the new large-update primal–dual interior point algorithms for solving P ∗ ( κ ) LCPs have O ( ( 1 + 2 κ ) n log n μ 0 ε ) polynomial complexity which is similar to the polynomial complexity obtained for LO and give a simple complexity analysis. This proximity function has not been used in the complexity analysis of interior point method (IPM) for P ∗ ( κ ) LCPs before.

A new large-update interior point algorithm for P∗( κ) LCPs based on kernel functions

In this paper we propose a new large-update primal-dual interior point algorithm for P ∗( κ) linear complementarity problems (LCPs). Recently, Peng et al. introduced self-regular barrier functions for primal-dual interior point methods (IPMs) for linear optimization (LO) problems and reduced the gap between the practical behavior of the algorithm and its theoretical worst case complexity. We introduce a new class of kernel functions which is not logarithmic barrier nor self-regular in the complexity analysis of interior point method (IPM) for P ∗( κ) linear complementarity problem (LCP). New search directions and proximity measures are proposed based on the kernel function. We showed that if a strictly feasible starting point is available, then the new large-update primal-dual interior point algorithms for solving P ∗( κ) LCPs have the polynomial complexity O q 3 2 ( 1 + 2 κ ) n ( log n ) q + 1 q log n ϵ which is better than the classical large-update primal-dual algorithm based on the classical logarithmic barrier function.