Smoothed Analysis of Selected Optimization Problems and Algorithms

Optimization problems arise in almost every field of economics, engineering, and science. Many of these problems are well-understood in theory and sophisticated algorithms exist to solve them efficiently in practice. Unfortunately, in many cases the theoretically most efficient algorithms perform poorly in practice. On the other hand, some algorithms are much faster than theory predicts. This discrepancy is a consequence of the pessimism inherent in the framework of worst-case analysis, the predominant analysis concept in theoretical computer science. We study selected optimization problems and algorithms in the framework of smoothed analysis in order to narrow the gap between theory and practice. In smoothed analysis, an adversary specifies the input, which is subsequently slightly perturbed at random. As one example we consider the successive shortest path algorithm for the minimumcost flow problem. While in the worst case the successive shortest path algorithm takes exponentially many steps to compute a minimum-cost flow, we show that its running time is polynomial in the smoothed setting. Another problem studied in this thesis is makespan minimization for scheduling with related machines. It seems to be unlikely that there exist fast algorithms to solve this problem exactly. This is why we consider three approximation algorithms: the jump algorithm, the lex-jump algorithm, and the list scheduling algorithm. In the worst case, the approximation guarantees of these algorithms depend on the number of machines. We show that there is no such dependence in smoothed analysis. We also apply smoothed analysis to multicriteria optimization problems. In particular, we consider integer optimization problems with several linear objectives that have to be simultaneously minimized. We derive a polynomial upper bound for the size of the set of Pareto-optimal solutions contrasting the exponential worst-case lower bound. As the icing on the cake we find that the insights gained from our smoothed analysis of the running time of the successive shortest path algorithm lead to the design of a randomized algorithm for finding short paths between two given vertices of a polyhedron. We see this result as an indication that, in future, smoothed analysis might also result in the development of fast algorithms.Optimierungsprobleme treten in allen wirtschaftlichen, naturwissenschaftlichen und technischen Gebieten auf. Viele dieser Probleme sind ausführlich untersucht und aus praktischer Sicht effizient lösbar. Leider erweisen sich in vielen Fällen die theoretisch effizientesten Algorithmen in der Praxis als ungeeignet. Auf der anderen Seite sind einige Algorithmen viel schneller als die Theorie vorhersagt. Dieser scheinbare Widerspruch resultiert aus dem Pessimismus, der dem in der theoretischen Informatik vorherrschenden Analysekonzept, der Worst-Case-Analyse, innewohnt. Um die Lücke zwischen Theorie und Praxis zu verkleinern, untersuchen wir ausgewählte Optimierungsprobleme und Algorithmen auf gegnerisch vorgegebenen Instanzen, die durch ein leichtes Zufallsrauschen gestört werden. Solche perturbierten Instanzen bezeichnen wir als semi-zufällige Eingaben. Als Beispiel betrachten wir den Successive- Shortest-Path-Algorithmus für das Minimum-Cost-Flow-Problem. Während dieser Algorithmus imWorst Case exponentiell viele Schritte benötigt, um einen Minimum-Cost-Flow zu berechnen, zeigen wir, dass seine Laufzeit auf semi-zufälligen Eingaben polynomiell ist. Ein weiteres Problem, das wir in dieser Arbeit untersuchen, ist die Minimierung des Makespans für Scheduling auf unterschiedlich schnellen Maschinen. Es scheint, dass dieses Problem nicht effizient gelöst werden kann. Daher betrachten wir drei Approximationsalgorithmen: den Jump-, den Lex-Jump- und den List-Scheduling-Algorithmus. Im Worst Case hängt die Approximationsgüte dieser Algorithmen von der Anzahl der Maschinen ab. Wir zeigen, dass das auf semi-zufälligen Eingaben nicht der Fall ist. Des Weiteren betrachten wir ganzzahlige Optimierungsprobleme mit mehreren linearen Zielfunktionen, die simultan minimiert werden sollen. Wir leiten eine polynomielle obere Schranke für die Größe der Pareto-Menge auf semi-zufälligen Eingaben her, die im Gegensatz zu der exponentiellen unteren Worst-Case-Schranke steht. Mit den Erkenntnissen aus der Laufzeitanalyse des Successive-Shortest-Path-Algorithmus entwerfen wir einen randomisierten Algorithmus zur Bestimmung eines kurzen Pfades zwischen zwei gegebenen Ecken eines Polyeders. Wir betrachten dieses Ergebnis als ein Indiz dafür, dass in Zukunft Analysen auf semi-zufälligen Eingaben auch zu der Entwicklung schneller Algorithmen führen könnten

A Smooth Primal-Dual Optimization Framework for Nonsmooth Composite Convex Minimization

We propose a new first-order primal-dual optimization framework for a convex optimization template with broad applications. Our optimization algorithms feature optimal convergence guarantees under a variety of common structure assumptions on the problem template. Our analysis relies on a novel combination of three classic ideas applied to the primal-dual gap function: smoothing, acceleration, and homotopy. The algorithms due to the new approach achieve the best known convergence rate results, in particular when the template consists of only non-smooth functions. We also outline a restart strategy for the acceleration to significantly enhance the practical performance. We demonstrate relations with the augmented Lagrangian method and show how to exploit the strongly convex objectives with rigorous convergence rate guarantees. We provide numerical evidence with two examples and illustrate that the new methods can outperform the state-of-the-art, including Chambolle-Pock, and the alternating direction method-of-multipliers algorithms.Comment: 35 pages, accepted for publication on SIAM J. Optimization. Tech. Report, Oct. 2015 (last update Sept. 2016

Templates for Convex Cone Problems with Applications to Sparse Signal Recovery

This paper develops a general framework for solving a variety of convex cone problems that frequently arise in signal processing, machine learning, statistics, and other fields. The approach works as follows: first, determine a conic formulation of the problem; second, determine its dual; third, apply smoothing; and fourth, solve using an optimal first-order method. A merit of this approach is its flexibility: for example, all compressed sensing problems can be solved via this approach. These include models with objective functionals such as the total-variation norm, ||Wx||_1 where W is arbitrary, or a combination thereof. In addition, the paper also introduces a number of technical contributions such as a novel continuation scheme, a novel approach for controlling the step size, and some new results showing that the smooth and unsmoothed problems are sometimes formally equivalent. Combined with our framework, these lead to novel, stable and computationally efficient algorithms. For instance, our general implementation is competitive with state-of-the-art methods for solving intensively studied problems such as the LASSO. Further, numerical experiments show that one can solve the Dantzig selector problem, for which no efficient large-scale solvers exist, in a few hundred iterations. Finally, the paper is accompanied with a software release. This software is not a single, monolithic solver; rather, it is a suite of programs and routines designed to serve as building blocks for constructing complete algorithms.Comment: The TFOCS software is available at http://tfocs.stanford.edu This version has updated reference

Continuation of Nesterov's Smoothing for Regression with Structured Sparsity in High-Dimensional Neuroimaging

Predictive models can be used on high-dimensional brain images for diagnosis of a clinical condition. Spatial regularization through structured sparsity offers new perspectives in this context and reduces the risk of overfitting the model while providing interpretable neuroimaging signatures by forcing the solution to adhere to domain-specific constraints. Total Variation (TV) enforces spatial smoothness of the solution while segmenting predictive regions from the background. We consider the problem of minimizing the sum of a smooth convex loss, a non-smooth convex penalty (whose proximal operator is known) and a wide range of possible complex, non-smooth convex structured penalties such as TV or overlapping group Lasso. Existing solvers are either limited in the functions they can minimize or in their practical capacity to scale to high-dimensional imaging data. Nesterov's smoothing technique can be used to minimize a large number of non-smooth convex structured penalties but reasonable precision requires a small smoothing parameter, which slows down the convergence speed. To benefit from the versatility of Nesterov's smoothing technique, we propose a first order continuation algorithm, CONESTA, which automatically generates a sequence of decreasing smoothing parameters. The generated sequence maintains the optimal convergence speed towards any globally desired precision. Our main contributions are: To propose an expression of the duality gap to probe the current distance to the global optimum in order to adapt the smoothing parameter and the convergence speed. We provide a convergence rate, which is an improvement over classical proximal gradient smoothing methods. We demonstrate on both simulated and high-dimensional structural neuroimaging data that CONESTA significantly outperforms many state-of-the-art solvers in regard to convergence speed and precision.Comment: 11 pages, 6 figures, accepted in IEEE TMI, IEEE Transactions on Medical Imaging 201

Optic nerve head segmentation

Reliable and efficient optic disk localization and segmentation are important tasks in automated retinal screening. General-purpose edge detection algorithms often fail to segment the optic disk due to fuzzy boundaries, inconsistent image contrast or missing edge features. This paper presents an algorithm for the localization and segmentation of the optic nerve head boundary in low-resolution images (about 20 /spl mu//pixel). Optic disk localization is achieved using specialized template matching, and segmentation by a deformable contour model. The latter uses a global elliptical model and a local deformable model with variable edge-strength dependent stiffness. The algorithm is evaluated against a randomly selected database of 100 images from a diabetic screening programme. Ten images were classified as unusable; the others were of variable quality. The localization algorithm succeeded on all bar one usable image; the contour estimation algorithm was qualitatively assessed by an ophthalmologist as having Excellent-Fair performance in 83% of cases, and performs well even on blurred image