Interior Point Methods with a Gradient Oracle

We provide an interior point method based on quasi-Newton iterations, which only requires first-order access to a strongly self-concordant barrier function. To achieve this, we extend the techniques of Dunagan-Harvey [STOC '07] to maintain a preconditioner, while using only first-order information. We measure the quality of this preconditioner in terms of its relative excentricity to the unknown Hessian matrix, and we generalize these techniques to convex functions with a slowly-changing Hessian. We combine this with an interior point method to show that, given first-order access to an appropriate barrier function for a convex set $K$, we can solve well-conditioned linear optimization problems over $K$ to $\varepsilon$ precision in time $\widetilde{O}\left(\left(\mathcal{T}+n^{2}\right)\sqrt{n\nu}\log\left(1/\varepsilon\right)\right)$, where $\nu$ is the self-concordance parameter of the barrier function, and $\mathcal{T}$ is the time required to make a gradient query. As a consequence we show that: $\bullet$ Linear optimization over $n$-dimensional convex sets can be solved in time $\widetilde{O}\left(\left(\mathcal{T}n+n^{3}\right)\log\left(1/\varepsilon\right)\right)$. This parallels the running time achieved by state of the art algorithms for cutting plane methods, when replacing separation oracles with first-order oracles for an appropriate barrier function. $\bullet$ We can solve semidefinite programs involving $m\geq n$ matrices in $\mathbb{R}^{n\times n}$ in time $\widetilde{O}\left(mn^{4}+m^{1.25}n^{3.5}\log\left(1/\varepsilon\right)\right)$, improving over the state of the art algorithms, in the case where $m=\Omega\left(n^{\frac{3.5}{\omega-1.25}}\right)$. Along the way we develop a host of tools allowing us to control the evolution of our potential functions, using techniques from matrix analysis and Schur convexity.Comment: STOC 202

SOME INSIGHTS REGARDING CREATIVE ACCOUNTING IN ROMANIAN ACCOUNTING ENVIRONMENT - REGULATORS, FINANCIAL AUDITORS AND PROFESSIONAL BODIES OPINION

This empirical study reports the results of a survey designed to explore the existence and magnitude of creative accounting practices in the Romanian accounting environment using as a reference point the opinion of some of the top representative accounting professionals like: financial auditors, regulators and professional bodies representatives of the Chamber of Financial Auditors of Romania and also representative of the Body of Expert and Licensed Accountants of Romania. \\r\\nSince the existence of creative accounting practices are connected in accounting with issues of vulnerability and in some cases panic getting to know its magnitude can be regarded of higher importance. In this respect we were interested to see if our respondents can document its existence and magnitude based on their experience. In order to achieve this goal our methodology employed neutral and direct interviews based on closed questions questionnaire. \\r\\nThe results of our empirical study documented that the credibility of accounting profession in the Romanian accounting environment is not affected by items like creative accounting since all our respondents asserted that is not facile to employ creative accounting schemes in the practice of accounting. One particular question was concerned about the ease of detection of creative accounting practices. In this respect we interrogated our respondents and all had similar opinions that in order to detect those practice skilled professionals are needed and more than that the desire to engage in this demarche since it is not specified particularly in the law.\\r\\nWhen it comes to creative accounting schemes that our respondents could identify in their day to day work they shared similar views: items like profit overstatement and profit undervaluation, income tax and leasing can be included frequently in those schemes. On the other hand practices of creative accounting that include goodwill, provisions and developments costs are not found in a significant proportion in the Romanian economic environment. \\r\\n \\r\\ncreative accounting, creative accounting practices, credibility, existence, magnitude

Cura aquarum and curator aquarum – the Head of Rome’s Water Supply Administration

It is obvious the importance of water supply to any human community. At the end of the 1st century A.D. Rome developed an impressive water infrastructure consisting of nine aqueducts. This huge network of pipes that distributed water on almost the whole surface of the ancient city could not function without rigorous maintenance. This work fell under the responsibility of the aquarii, a team that formed the familia aquaria, a component part of cura aquarum, an office led by the curator aquarum. The description of the structure of the cura aquarum and of the tasks of the team that represented it, as well as the activity of curator aquarum, are the subjects of this paper

Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods

In this paper, we study matrix scaling and balancing, which are fundamental problems in scientific computing, with a long line of work on them that dates back to the 1960s. We provide algorithms for both these problems that, ignoring logarithmic factors involving the dimension of the input matrix and the size of its entries, both run in time $\widetilde{O}\left(m\log \kappa \log^2 (1/\epsilon)\right)$ where $\epsilon$ is the amount of error we are willing to tolerate. Here, $\kappa$ represents the ratio between the largest and the smallest entries of the optimal scalings. This implies that our algorithms run in nearly-linear time whenever $\kappa$ is quasi-polynomial, which includes, in particular, the case of strictly positive matrices. We complement our results by providing a separate algorithm that uses an interior-point method and runs in time $\widetilde{O}(m^{3/2} \log (1/\epsilon))$. In order to establish these results, we develop a new second-order optimization framework that enables us to treat both problems in a unified and principled manner. This framework identifies a certain generalization of linear system solving that we can use to efficiently minimize a broad class of functions, which we call second-order robust. We then show that in the context of the specific functions capturing matrix scaling and balancing, we can leverage and generalize the work on Laplacian system solving to make the algorithms obtained via this framework very efficient.Comment: To appear in FOCS 201

Improved Parallel Algorithms for Spanners and Hopsets

We use exponential start time clustering to design faster and more work-efficient parallel graph algorithms involving distances. Previous algorithms usually rely on graph decomposition routines with strict restrictions on the diameters of the decomposed pieces. We weaken these bounds in favor of stronger local probabilistic guarantees. This allows more direct analyses of the overall process, giving: * Linear work parallel algorithms that construct spanners with $O(k)$ stretch and size $O(n^{1+1/k})$ in unweighted graphs, and size $O(n^{1+1/k} \log k)$ in weighted graphs. * Hopsets that lead to the first parallel algorithm for approximating shortest paths in undirected graphs with $O(m\;\mathrm{polylog}\;n)$ work