Large-update Primal-dual Interior Point Research Articles

Complexity analysis of primal-dual interior-point methods for linear optimization based on a new efficient Bi-parameterized kernel function with a trigonometric barrier term

In this paper we are generalizing the efficient kernel function with trigonometric barrier term given by (M. Bouafia, D. Benterki and A. Yassine, J. Optim. Theory Appl. 170 (2016) 528–545). Using an elegant and simple analysis and under some easy to check conditions, we explore the best complexity result for the large update primal-dual interior point methods for linear optimization. This complexity estimate improves results obtained in (X. Li and M. Zhang, Oper. Res. Lett. 43 (2015) 471–475; M.R. Peyghami and S.F. Hafshejani, Numer. Algo. 67 (2014) 33–48; M. Bouafia, D. Benterki and A. Yassine, J. Optim. Theory Appl. 170 (2016) 528–545). Our comparative numerical experiments on some test problems consolidate and confirm our theoretical results according to which the new kernel function has promising applications compared to the kernel function given by (M. Bouafia and A. Yassine, Optim. Eng. 21 (2020) 651–672). Moreover, the comparative numerical study that we have established favors our new kernel function better than other best trigonometric kernel functions (M. Bouafia, D. Benterki and A. Yassine, J. Optim. Theory Appl. 170 (2016) 528–545; M. Bouafia and A. Yassine, Optim. Eng. 21 (2020) 651–672).

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.

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 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.

An interior-point method for $$P_*(\kappa )$$ P ∗ ( κ ) -linear complementarity problem based on a trigonometric kernel function

Recently, El Ghami (Optim Theory Decis Mak Oper Res Appl 31:331–349, 2013) proposed a primal dual interior point method for $$P_*(\kappa )$$ -Linear Complementarity Problem (LCP) based on a trigonometric barrier term and obtained the worst case iteration complexity as $$O\left( (1+2\kappa )n^{\frac{3}{4}}\log \frac{n}{\epsilon }\right) $$ for large-update methods. In this paper, we present a large update primal–dual interior point algorithm for $$P_{*}(\kappa )$$ -LCP based on a new trigonometric kernel function. By a simple analysis, we show that our algorithm based on the new kernel function enjoys the worst case $$O\left( (1+2\kappa )\sqrt{n}\log n\log \frac{n}{\epsilon }\right) $$ iteration bound for solving $$P_*(\kappa )$$ -LCP. This result improves the worst case iteration bound obtained by El Ghami for $$P_*(\kappa )$$ -LCP based on trigonometric kernel functions significantly.

Complexity of interior-point methods for linear optimization based on a new trigonometric kernel function

In this paper, we propose a new kernel function with trigonometric barrier term for primal–dual interior point methods in linear optimization. Using an elegant and simple analysis and under some easy to check conditions, we explore the worst case complexity result for the large update primal–dual interior point methods. We obtain the worst case iteration bound for the large update primal–dual interior point methods as O(n23lognϵ) which improves the so far obtained complexity results for the trigonometric kernel function in [M. El Ghami, Z.A. Guennoun, S. Boula, T. Steihaug, Interior-point methods for linear optimization based on a kernel function with a trigonometric barrier term, Journal of Computational and Applied Mathematics 236 (2012) 3613–3623] significantly.

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 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 .

NEW COMPLEXITY ANALYSIS OF PRIMAL-DUAL IMPS FOR P*LAPS BASED ON LARGE UPDATES

In this paper we present new large-update primal-dual interior point algorithms for <TEX>$P_*$</TEX> linear complementarity problems(LAPS) based on a class of kernel functions, <TEX>${\psi}(t)={\frac{t^{p+1}-1}{p+1}}+{\frac{1}{\sigma}}(e^{{\sigma}(1-t)}-1)$</TEX>, p <TEX>$\in$</TEX> [0, 1], <TEX>${\sigma}{\geq}1$</TEX>. It is the first to use this class of kernel functions in the complexity analysis of interior point method(IPM) for <TEX>$P_*$</TEX> LAPS. We showed that if a strictly feasible starting point is available, then new large-update primal-dual interior point algorithms for <TEX>$P_*$</TEX> LAPS have <TEX>$O((1+2+\kappa)n^{{\frac{1}{p+1}}}lognlog{\frac{n}{\varepsilon}})$</TEX> complexity bound. When p = 1, we have <TEX>$O((1+2\kappa)\sqrt{n}lognlog\frac{n}{\varepsilon})$</TEX> complexity which is so far the best known complexity for large-update methods.

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.

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.