It is well known that any feasible arc-flow solution to a network problem defined on a graph $G = (N, A)$, where $N$ is the set of nodes whereas $A$ is the set of arcs, can be expressed using at most $|A| + |N|$ paths and cycles having nonzero flow, out of these, at most … Read more
This paper describes three recent tools for dealing with primal degeneracy in linear programming. The first one is the Improved Primal Simplex (IPS) algorithm which turns degeneracy into a possible advantage. The constraints of the original problem are dynamically partitioned based on the numerical values of the current basic variables. The idea is to work … Read more
We present a simple transformation of any linear program or semidefinite program into an equivalent convex optimization problem whose only constraints are linear equations. The objective function is defined on the whole space, making virtually all subgradient methods be immediately applicable. We observe, moreover, that the objective function is naturally “smoothed,” thereby allowing most first-order … Read more
It is shown that a recently proposed method by Chubanov for solving linear homogeneous systems with positive variables can be improved, both theoretically and computationally. CitationTU Delft, September 2014ArticleDownload View PDF
In the mid-eighties Tardos proposed a strongly polynomial algorithm for solving linear programming problems for which the size of the coefficient matrix is polynomially bounded by the dimension. Combining Orlin’s primal-based modification and Mizuno’s use of the simplex method, we introduce a modification of Tardos’ algorithm considering only the primal problem and using simplex method … Read more
Motivated by Bland’s linear-programming generalization of the renowned Edmonds-Karp efficient refinement of the Ford-Fulkerson maximum-flow algorithm, we discuss three closely-related natural augmentation rules for linear and integer-linear optimization. In several nice situations, we show that polynomially-many augmentation steps suffice to reach an optimum. In particular, when using “discrete steepest-descent augmentations” (i.e., directions with the best … Read more
It is an open question whether there is an interior-point algorithm for linear optimization problems with a lower iteration-complexity than the classical bound $\mathcal{O}(\sqrt{n} \log(\frac{\mu_1}{\mu_0}))$. This paper provides a negative answer to that question for a variant of the Mizuno-Todd-Ye predictor-corrector algorithm. In fact, we prove that for any $\epsilon >0$, there is a redundant … Read more
Kitahara and Mizuno get new bounds for the number of distinct solutions generated by the simplex method for linear programming (LP). In this paper, we combine results of Kitahara and Mizuno and Tardos’s strongly polynomial algorithm, and propose an algorithm for solving a standard form LP problem. The algorithm solves polynomial number of artificial LP … Read more
Although it is easy to prove the sufficient conditions for optimality of a linear program, the necessary conditions pose a pedagogical challenge. A widespread practice in deriving the necessary conditions is to invoke Farkas’ lemma, but proofs of Farkas’ lemma typically involve “nonlinear” topics such as separating hyperplanes between disjoint convex sets, or else more … Read more
This paper provides operative point-based formulas (only involving the nominal data, and not data in a neighborhood) for computing or estimating the calmness modulus of the optimal set (argmin) mapping in linear optimization under uniqueness of nominal optimal solutions. Our analysis is developed in two different parametric settings. First, in the framework of canonical perturbations … Read more