The leading approach to the simplex method, a widely used technique for balancing complex logistical constraints, can't get any better. In 1939, upon arriving late to his statistics course at UC Berkeley, George Dantzig--a first-year graduate student--copied two problems off the blackboard, thinking they were a homework assignment. He found the homework "harder to do than usual," he would later recount, and apologized to the professor for taking some extra days to complete it. A few weeks later, his professor told him that he had solved two famous open problems in statistics. Dantzig's work would provide the basis for his doctoral dissertation and, decades later, inspiration for the film .

We then provide sufficient conditions under which PSM always outputs sparse solutions such that its computational performance can be significantly boosted.

Over the past decades, Linear Programming (LP) has been widely used in different areas and considered as one of the mature technologies in numerical optimization. However, the complexity offered by state-of-the-art algorithms (i.e.

Neural Information Processing Systems http://nips.cc/

Optimal transport is a fundamental topic that has attracted a great amount of attention from the optimization community in the past decades. In this paper, we consider an interesting discrete dynamic optimal transport problem: can we efficiently update the optimal transport plan when the weights or the locations of the data points change? This problem is naturally motivated by several applications in machine learning. For example, we often need to compute the optimal transport cost between two different data sets; if some changes happen to a few data points, should we re-compute the high complexity cost function or update the cost by some efficient dynamic data structure? We are aware that several dynamic maximum flow algorithms have been proposed before, however, the research on dynamic minimum cost flow problem is still quite limited, to the best of our knowledge. We propose a novel 2D Skip Orthogonal List together with some dynamic tree techniques. Although our algorithm is based on the conventional simplex method, it can efficiently find the variable to pivot within expected $O(1)$ time, and complete each pivoting operation within expected $O(|V|)$ time where $V$ is the set of all supply and demand nodes. Since dynamic modifications typically do not introduce significant changes, our algorithm requires only a few simplex iterations in practice. So our algorithm is more efficient than re-computing the optimal transport cost that needs at least one traversal over all $|E| = O(|V|^2)$ variables, where $|E|$ denotes the number of edges in the network. Our experiments demonstrate that our algorithm significantly outperforms existing algorithms in the dynamic scenarios.

One of the primary goals of the mathematical analysis of algorithms is to provide guidance about which algorithm is the "best" for solving a given computational problem. Worst-case analysis summarizes the performance profile of an algorithm by its worst performance on any input of a given size, implicitly advocating for the algorithm with the best-possible worst-case performance. Strong worst-case guarantees are the holy grail of algorithm design, providing an application-agnostic certification of an algorithm's robustly good performance. However, for many fundamental problems and performance measures, such guarantees are impossible and a more nuanced analysis approach is called for. This chapter surveys several alternatives to worst-case analysis that are discussed in detail later in the book.

An algorithm is an unambiguous rule of action to solve a problem or a class of problems. Algorithms consist of a finite number of well-defined individual steps. Thus, they can be implemented in a computer program for execution, but can also be formulated in human language. When solving a problem, a specific input is converted into a particular output. In the following, five algorithms are listed that have significantly influenced our world.

Over the past decades, Linear Programming (LP) has been widely used in different areas and considered as one of the mature technologies in numerical optimization. However, the complexity offered by state-of-the-art algorithms (i.e. In this paper, we investigate a general LP algorithm based on the combination of Augmented Lagrangian and Coordinate Descent (AL-CD), giving an iteration complexity of $O((\log(1/\epsilon)) 2)$ with $O(nnz(A))$ cost per iteration, where $nnz(A)$ is the number of non-zeros in the $m\times n$ constraint matrix $A$, and in practice, one can further reduce cost per iteration to the order of non-zeros in columns (rows) corresponding to the active primal (dual) variables through an active-set strategy. The algorithm thus yields a tractable alternative to standard LP methods for large-scale problems of sparse solutions and $nnz(A)\ll mn$. We conduct experiments on large-scale LP instances from $\ell_1$-regularized multi-class SVM, Sparse Inverse Covariance Estimation, and Nonnegative Matrix Factorization, where the proposed approach finds solutions of $10 {-3}$ precision orders of magnitude faster than state-of-the-art implementations of interior-point and simplex methods. Papers published at the Neural Information Processing Systems Conference.

An algorithm is an unambiguous rule of action to solve a problem or a class of problems. Algorithms consist of a finite number of well-defined individual steps. Thus, they can be implemented in a computer program for execution, but can also be formulated in human language. When solving a problem, a specific input is converted into a particular output. In the following, five algorithms are listed that have significantly influenced our world.

An algorithm is an unambiguous rule of action to solve a problem or a class of problems. Algorithms consist of a finite number of well-defined individual steps. Thus, they can be implemented in a computer program for execution, but can also be formulated in human language. When solving a problem, a specific input is converted into a particular output. In the following, five algorithms are listed that have significantly influenced our world.