Let P be a convex polytope in R, d = 3 or 2, with n vertices. We present linear time algorithms for approximating P by simpler polytopes. For instance, one such algorithm selects k < n vertices of P whose convex hull is the approximating polytope. The rate of approximation, in the Hausdorff distance sense, is best possible in the worst case. An analogous algorithm, where the role of vertices is...

Generalizing both mixed-integer linear optimization and convex optimization, mixed-integer convex optimization possesses broad modeling power but has seen relatively few advances in general-purpose solvers in recent years. In this paper, we intend to provide a broadly accessible introduction to our recent work in developing algorithms and software for this problem class. Our approach is based o...

Let X be a topological vector space, Y = IR n , n 2 IN, A a continuous linear map from X to Y , C X, B a convex set dense in C, and d 2 Y a data point. We derive conditions which guarantee that the set B \ A ?1 (d) is nonempty and dense in C \ A ?1 (d). Some applications to shape preserving interpolation and approximation are described.

Inspired by recent work on convex formulations of clustering (Lashkari & Golland, 2008; Nowozin & Bakir, 2008) we investigate a new formulation of the Sparse Coding Problem (Olshausen & Field, 1997). In sparse coding we attempt to simultaneously represent a sequence of data-vectors sparsely (i.e. sparse approximation (Tropp et al., 2006)) in terms of a “code” defined by a set of basis elements,...

In this paper, an effective method with time complexity of O(K3/2N2 log K ǫ0 ) is introduced to find an approximation of the convex hull for N points in dimension n, where K is close to the number of vertices of the approximation. Since the time complexity is independent of dimension, this method is highly suitable for the data in high dimensions. Utilizing a greedy approach, the proposed metho...

In this paper we consider the classical problem of finding a low rank approximation of a given matrix. In a least squares sense a closed form solution is available via factorization. However, with additional constraints, or in the presence of missing data, the problem becomes much more difficult. In this paper we show how to efficiently compute the convex envelopes of a class of rank minimizati...

We prove that for every convex body K with the center of mass at the origin and every ε ∈ ( 0, 12 ) , there exists a convex polytope P with at most eO(d)ε− d−1 2 vertices such that (1− ε)K ⊂ P ⊂ K.

The process of learning the shape of an unknown convex planar object through an adaptive process of simple measurements called Line probings, which reveal tangent lines to the object, is considered. A systematic probing strategy is suggested and an upper bound on the number of probings it requires for achieving an approximation with a pre-specified precision to the unknown object is derived. A ...

This papers presents a convex approximation method for the solution of nonconvex optimal control problems involving input-affine dynamic models. The method relies in the availability of full reference state trajectories. By using these states references as real states trajectories, the dynamic model is approximated such that the resulting problem becomes convex. The convexified problem is solve...