Complexity of primal-dual interior-point algorithm for linear programming based on a new class of kernel functions

summary:In this paper, we first present a polynomial-time primal-dual interior-point method (IPM) for solving linear programming (LP) problems, based on a new kernel function (KF) with a hyperbolic-logarithmic barrier term. To improve the iteration bound, we propose a parameterized version of this function. We show that the complexity result meets the currently best iteration bound for large-update methods by choosing a special value of the parameter. Numerical experiments reveal that the new KFs have better results comparing with the existing KFs including $\log t$ in their barrier term. To the best of our knowledge, this is the first IPM based on a parameterized hyperbolic-logarithmic KF. Moreover, it contains the first hyperbolic-logarithmic KF (Touil and Chikouche in Filomat 34:3957-3969, 2020) as a special case up to a multiplicative constant, and improves significantly both its theoretical and practical results

Coefficients of singularities of the biharmonic problem of Neumann type: case of the crack

This paper concerns the biharmonic problem of Neumann type in a sector V. We give a representation of the solution u of the problem in a form of a series u=∑α∈ECα rα ϕα, and the functions ϕα are solutions of an auxiliary problem obtained by the separation of variables

Real interpolation spaces between the domain of the Laplace operator with transmission conditions and L^p on a polygonal domain

We provide a description of the real interpolation spaces between the domain of the Laplace operator (with transmission conditions in a polygonal domain $Omega$) and $L^p(Omega)$ as interpolation spaces between $mathcal{W}^{2,p}(Omega)$ (possibly augmented with singular solutions) and $L^p(Omega)$. This result relies essentially on estimates on the resolvent and the reiteration theorem

Lᵖ-регулярность решения уравнения теплопроводности с разрывными коэффициентами

In this paper, we consider the transmission problem for the heat equation on a bounded plane sector in Lᵖ-Sobolev spaces. By Applying the theory of the sums of operators of Da Prato-Grisvard and Dore-Venni, we prove that the solution can be splited into a regular part in Lp-Sobolev space and an explicit singular partВ этой статье мы рассмотрим задачу прохождения для уравнения теплопроводности на ограниченном плоском секторе в пространствах Lᵖ-Соболева. Применяя теорию сумм операто- ров Да Прато-Грисварда и Доре-Венни, мы доказываем, что решение можно разбить на регулярную часть в пространстве Lp-Соболева и явную особую част