Primal-Dual Interior-Point Methods (IPMs) have shown their power in solving large classes of optimization problems. However, there is still a gap between the practical behavior of these algorithms and their theoretical worst-case complexity results with respect to the update strategies of the duality gap parameter in the algorithm. The so-called small-update IPMs enjoy the best … Read more
We provide two new, simple proofs of Afriat’s celebrated theorem stating that a finite set of price-quantity observations is consistent with utility maximization if, and only if, the observations satisfy a variation of the Strong Axiom of Revealed Preference known as the Generalized Axiom of Revealed Preference. CitationTechnical Report No. 1381, School of Operations Research … Read more
In this paper, we consider a modified version of a well-known long-step primal-dual infeasible IP algorithm for solving the linear program $\min\{c^T x : Ax=b, \, x \ge 0\}$, $A \in \Re^{m \times n}$, where the search directions are computed by means of an iterative linear solver applied to a preconditioned normal system of equations. … Read more
In this paper we study the distribution tails and the moments of a condition number which arises in the study of homogeneous systems of linear inequalities. We consider the case where this system is defined by a Gaussian random matrix and characterise the exact decay rates of the distribution tails, improve the existing moment estimates, … Read more
Recently, so-called self-regular barrier functions for primal-dual interior-point methods (IPMs) for linear optimization were introduced. Each such barrier function is determined by its (univariate) self-regular kernel function. We introduce a new class of kernel functions. The class is defined by some simple conditions on the kernel function and its derivatives. These properties enable us to … Read more
Primal-Dual Interior-Point Methods (IPMs) have shown their power in solving large classes of optimization problems. In this paper a self-regular proximity based Infeasible Interior Point Method (IIPM) is proposed for linear optimization problems. First we mention some interesting perties of a specific self-regular proximity function, studied recently by Peng and Terlaky, and use it to … Read more
It is well known that the so-called first-order predictor-corrector methods working in a large neighborhood of the central path are among the most efficient interior-point methods (IPMs) for linear optimization (LO) problems. However, the best known iteration complexity of this type of methods is $O\br{n \log\frac{(x^0)^Ts^0}{\varepsilon}}$. It is of interests to investigate whether the complexity … Read more
This article considers continuous trajectories of the vector fields induced by primal-dual potential-reduction algorithms for solving linear programming problems. It is known that these trajectories converge to the analytic center of the primal-dual optimal face. We establish that this convergence may be tangential to the central path, tangential to the optimal face, or in between, … Read more
Solving systems of linear equations with “normal” matrices of the form $A D^2 A^T$ is a key ingredient in the computation of search directions for interior-point algorithms. In this article, we establish that a well-known basis preconditioner for such systems of linear equations produces scaled matrices with uniformly bounded condition numbers as $D$ varies over … Read more
We perform a smoothed analysis of the termination phase of an interior-point method. By combining this analysis with the smoothed analysis of Renegar’s interior-point algorithm by Dunagan, Spielman and Teng, we show that the smoothed complexity of an interior-point algorithm for linear programming is $O (m^{3} \log (m/\sigma ))$. In contrast, the best known bound … Read more