Recovering convex boundaries from blurred and noisy observations

We consider the problem of estimating convex boundaries from blurred and noisy observations. In our model, the convolution of an intensity function $f$ is observed with additive Gaussian white noise. The function $f$ is assumed to have convex support $G$ whose boundary is to be recovered. Rather than directly estimating the intensity function, we develop a procedure which is based on estimating the support function of the set $G$. This approach is closely related to the method of geometric hyperplane probing, a well-known technique in computer vision applications. We establish bounds that reveal how the estimation accuracy depends on the ill-posedness of the convolution operator and the behavior of the intensity function near the boundary.Comment: Published at http://dx.doi.org/10.1214/009053606000000326 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Approximation of Multiobjective Optimization Problems

We study optimization problems with multiple objectives. Such problems are pervasive across many diverse disciplines -- in economics, engineering, healthcare, biology, to name but a few -- and heuristic approaches to solve them have already been deployed in several areas, in both academia and industry. Hence, there is a real need for a rigorous investigation of the relevant questions. In such problems we are interested not in a single optimal solution, but in the tradeoff between the different objectives. This is captured by the tradeoff or Pareto curve, the set of all feasible solutions whose vector of the various objectives is not dominated by any other solution. Typically, we have a small number of objectives and we wish to plot the tradeoff curve to get a sense of the design space. Unfortunately, typically the tradeoff curve has exponential size for discrete optimization problems even for two objectives (and is typically infinite for continuous problems). Hence, a natural goal in this setting is, given an instance of a multiobjective problem, to efficiently obtain a ``good'' approximation to the entire solution space with ``few'' solutions. This has been the underlying goal in much of the research in the multiobjective area, with many heuristics proposed for this purpose, typically however without any performance guarantees or complexity analysis. We develop efficient algorithms for the succinct approximation of the Pareto set for a large class of multiobjective problems. First, we investigate the problem of computing a minimum set of solutions that approximates within a specified accuracy the Pareto curve of a multiobjective optimization problem. We provide approximation algorithms with tight performance guarantees for bi-objective problems and make progress for the more challenging case of three and more objectives. Subsequently, we propose and study the notion of the approximate convex Pareto set; a novel notion of approximation to the Pareto set, as the appropriate one for the convex setting. We characterize when such an approximation can be efficiently constructed and investigate the problem of computing minimum size approximate convex Pareto sets, both for discrete and convex problems. Next, we turn to the problem of approximating the Pareto set as efficiently as possible. To this end, we analyze the Chord algorithm, a popular, simple method for the succinct approximation of curves, which is widely used, under different names, in a variety of areas, such as, multiobjective and parametric optimization, computational geometry, and graphics