In this paper, we show that the relative algebraic INTERIOR is a suitable replacement for both of the topological INTERIOR and the algebraic INTERIOR for the cases where these are empty. Also, we present some properties of (relative) algebraic INTERIOR and some fixed POINT theorems for increasing mapping. The obtained results can be viewed as an extension and improvement of the known corresponding results. Also, some examples are to support our conclusions considered.

This paper proposes a new linear program for finding a relative INTERIOR POINT of a polyhedral set. Based on characterizing the relative INTERIOR of a polyhedral set through its polyhedral representing sets, two main contributions are made. First, we complete the existing results in the literature that require the non-negativity of the given polyhedral set. Then, we deal with the general case where this requirement may not be met.

In this paper, we present a new primal-dual INTERIOR-POINT algorithm based on a new large neighbourhood ( ,  )  N for semidefinite optimization. This large neighbourhood is based on the infinity norm. It is larger than the N( ,  ) large neighborhood of the central path, which is popular wide neighborhood. We demonstraite the convergence of the proposed algorithm and show that the algorithm has ( log ) 1 2 1    p O n iteration complexity bound for the Nesterov-Todd direction.

We study the two-stage stochastic infinity norm optimization problem with recourse based on a commutative algebra. First, we explore and develop the algebraic structure of the infinity norm cone, and utilize it to compute the derivatives of the barrier recourse functions. Then, we prove that the barrier recourse functions and the composite barrier functions for this optimization problem are self-concordant families with reference to barrier parameters. These findings are used to develop INTERIOR-POINT algorithms based on primal decomposition for this class of stochastic programming problems. Our complexity results for the short- and long-step algorithms show that the dominant complexity terms are linear in the rank of the underlying cone. Despite the asymmetry of the infinity norm cone, we also show that the obtained complexity results match (in terms of rank) the best known results in the literature for other well-studied stochastic symmetric cone programs. Finally, we demonstrate the efficiency of the proposed algorithm by presenting some numerical experiments on both stochastic uniform facility location problems and randomly-generated problems.

We consider the application of the ABS procedure to the linear system arising in the primal-dual INTERIOR POINT METHOD where Newton METHOD is used to compute the path to the solution. When approaching the solution the linear system, which has the form of normal equations of the second kind, becomes more and more ill conditioned. We show how the use of the Huang algorithm in the ABS class can reduce the ill conditioning. Preliminary numerical experiments show that the proposed approach can provide a residual in the computed solution up to sixteen orders lower.