Approximating orthogonal matrices by permutation matrices

Motivated in part by a problem of combinatorial optimization and in part by analogies with quantum computations, we consider approximations of orthogonal matrices U by ``non-commutative convex combinations'' A of permutation matrices of the type A=sum A_sigma sigma, where sigma are permutation matrices and A_sigma are positive semidefinite nxn matrices summing up to the identity matrix. We prove that for every nxn orthogonal matrix U there is a non-commutative convex combination A of permutation matrices which approximates U entry-wise within an error of c n^{-1/2}ln n and in the Frobenius norm within an error of c ln n. The proof uses a certain procedure of randomized rounding of an orthogonal matrix to a permutation matrix.Comment: 18 page

Convex Hull of Points Lying on Lines in o(n log n) Time after Preprocessing

Motivated by the desire to cope with data imprecision, we study methods for taking advantage of preliminary information about point sets in order to speed up the computation of certain structures associated with them. In particular, we study the following problem: given a set L of n lines in the plane, we wish to preprocess L such that later, upon receiving a set P of n points, each of which lies on a distinct line of L, we can construct the convex hull of P efficiently. We show that in quadratic time and space it is possible to construct a data structure on L that enables us to compute the convex hull of any such point set P in O(n alpha(n) log* n) expected time. If we further assume that the points are "oblivious" with respect to the data structure, the running time improves to O(n alpha(n)). The analysis applies almost verbatim when L is a set of line-segments, and yields similar asymptotic bounds. We present several extensions, including a trade-off between space and query time and an output-sensitive algorithm. We also study the "dual problem" where we show how to efficiently compute the (<= k)-level of n lines in the plane, each of which lies on a distinct point (given in advance). We complement our results by Omega(n log n) lower bounds under the algebraic computation tree model for several related problems, including sorting a set of points (according to, say, their x-order), each of which lies on a given line known in advance. Therefore, the convex hull problem under our setting is easier than sorting, contrary to the "standard" convex hull and sorting problems, in which the two problems require Theta(n log n) steps in the worst case (under the algebraic computation tree model).Comment: 26 pages, 5 figures, 1 appendix; a preliminary version appeared at SoCG 201

A scenario approach for non-convex control design

Randomized optimization is an established tool for control design with modulated robustness. While for uncertain convex programs there exist randomized approaches with efficient sampling, this is not the case for non-convex problems. Approaches based on statistical learning theory are applicable to non-convex problems, but they usually are conservative in terms of performance and require high sample complexity to achieve the desired probabilistic guarantees. In this paper, we derive a novel scenario approach for a wide class of random non-convex programs, with a sample complexity similar to that of uncertain convex programs and with probabilistic guarantees that hold not only for the optimal solution of the scenario program, but for all feasible solutions inside a set of a-priori chosen complexity. We also address measure-theoretic issues for uncertain convex and non-convex programs. Among the family of non-convex control- design problems that can be addressed via randomization, we apply our scenario approach to randomized Model Predictive Control for chance-constrained nonlinear control-affine systems.Comment: Submitted to IEEE Transactions on Automatic Contro

Randomized Rounding for the Largest Simplex Problem

The maximum volume $j$-simplex problem asks to compute the $j$-dimensional simplex of maximum volume inside the convex hull of a given set of $n$ points in $\mathbb{Q}^d$. We give a deterministic approximation algorithm for this problem which achieves an approximation ratio of $e^{j/2 + o(j)}$. The problem is known to be $\mathrm{NP}$-hard to approximate within a factor of $c^{j}$ for some constant $c > 1$. Our algorithm also gives a factor $e^{j + o(j)}$ approximation for the problem of finding the principal $j\times j$ submatrix of a rank $d$ positive semidefinite matrix with the largest determinant. We achieve our approximation by rounding solutions to a generalization of the $D$-optimal design problem, or, equivalently, the dual of an appropriate smallest enclosing ellipsoid problem. Our arguments give a short and simple proof of a restricted invertibility principle for determinants

Dynamic Product Assembly and Inventory Control for Maximum Profit

We consider a manufacturing plant that purchases raw materials for product assembly and then sells the final products to customers. There are M types of raw materials and K types of products, and each product uses a certain subset of raw materials for assembly. The plant operates in slotted time, and every slot it makes decisions about re-stocking materials and pricing the existing products in reaction to (possibly time-varying) material costs and consumer demands. We develop a dynamic purchasing and pricing policy that yields time average profit within epsilon of optimality, for any given epsilon>0, with a worst case storage buffer requirement that is O(1/epsilon). The policy can be implemented easily for large M, K, yields fast convergence times, and is robust to non-ergodic system dynamics.Comment: 32 page