We present a differential calculus on the extension of the quantum plane obtained considering that the (bosonic) generator x is invertible and furthermore working polynomials in ln x instead of polynomials in x. We call quantum Lie algebra to this extension and we obtain its Hopf algebra structure and its dual Hopf algebra.

We propose a family of search directions based on primal-dual entropy in the context of interior point methods for linear programming. This new family contains previously proposed search directions in the context of primal-dual entropy. We analyze the new family of search directions by studying their primal-dual affine-scaling and constant-gap centering components. We then design primal-dual in...

In this paper, we consider a primal-dual interior point method for solving nonlinear semidefinite programming problems. We propose primal-dual interior point methods based on the unscaled and scaled Newton methods, which correspond to the AHO, HRVW/KSH/M and NT search directions in linear SDP problems. We analyze local behavior of our proposed methods and show their local and superlinear conver...

In this paper, we present a new primal-dual interior-point algorithm for linear optimization based on a trigonometric kernel function. By simple analysis, we derive the worst case complexity for a large-update primal-dual interior-point method based on this kernel function. This complexity estimate improves a result from [1] and matches the one obtained in [2].

Many recent convergence results obtained for primal-dual interior-point methods for nonlinear programming, use assumptions of the boundedness of generated iterates. In this paper we replace such assumptions by new assumptions on the NLP problem, develop a modification of a primal-dual interior-point method implemented in software package loqo and analyze convergence of the new method from any i...

In this paper, primal-dual methods for general nonconvex nonlinear optimization problems are considered. The proposed methods are exterior point type methods that permit primal variables to violate inequality constraints during the iterations. The methods are based on the exact penalty type transformation of inequality constraints and use a smooth approximation of the problem to form primal-dua...

In this paper we propose a new large-update primal-dual interior point algorithm for P∗(κ) linear complementarity problems (LCPs). We generalize the analysis of BER’s primal-dual interior point algorithm for LP to P∗(κ) LCPs. New search directions and proximity measures are proposed based on a new kernel function which has linear growth term. We showed that if a strictly feasible starting point...

A critical disadvantage of primal-dual interior-point methods against dual interior-point methods for large scale SDPs (semidenite programs) has been that the primal positive semidenite variable matrix becomes fully dense in general even when all data matrices are sparse. Based on some fundamental results about positive semidenite matrix completion, this article proposes a general method of exp...

A critical disadvantage of primal-dual interior-point methods compared to dual interior-point methods for large scale semidefinite programs (SDPs) has been that the primal positive semidefinite matrix variable becomes fully dense in general even when all data matrices are sparse. Based on some fundamental results about positive semidefinite matrix completion, this article proposes a general met...

In this paper we present a primal-dual interior-point algorithm to solve a class of multi-objective network flow problems. More precisely, our algorithm is an extension of the single-objective primal-dual infeasible and inexact interior point method for multi-objective linear network flow problems. A comparison with standard interior point methods is provided and experimental results on bi-obje...