We consider an optimization problem of minimization of a linear function subject to the chance constraint Prob{G(x, ξ) ∈ C} ≥ 1 − ε, where C is a convex set, G(x, ξ) is bi-affine mapping and ξ is a vector of random perturbations with known distribution. When C is multi-dimensional and ε is small, like 10−6 or 10−10, this problem is, generically, a problem of minimizing under a nonconvex and dif...

We consider the questions of efficient mixing and un-mixing by incompressible flows which satisfy periodic, no-flow, or no-slip boundary conditions on a square. Under the uniform-in-time constraint ‖∇u(·, t)‖p ≤ 1 we show that any function can be mixed to scale ε in time O(| log ε|1+νp), with νp = 0 for p < 3+ √ 5 2 and νp ≤ 13 for p ≥ 3+ √ 5 2 . Known lower bounds show that this rate is optima...

We present approximation algorithms for balanced partitioning problems. These problems are notoriously hard and we present new bicriteria approximation algorithms, that approximate the optimal cost and relax the balance constraint. In the first scenario, we consider Min-Max k-Partitioning, the problem of dividing a graph into k equal-sized parts while minimizing the maximum cost of edges cut by...

We consider maximizing a monotone submodular function under cardinality constraint or knapsack in the streaming setting. In particular, elements arrive sequentially and at any point of time, algorithm has access to only small fraction data stored primary memory. propose following algorithms taking O(ε− 1) passes: (1) (1 − e− 1 ε)-approximation for cardinality-constrained problem, (2) (0.5 knaps...

This paper is concerned with the numerical approximation of the isothermal Euler equations for charged particles subject to the Lorentz force (the ’Euler-Lorentz’ system). When the magnetic field is large, or equivalently, when the parameter ε representing the non-dimensional ion cyclotron frequency tends to zero, the so-called drift-fluid (or gyrofluid) approximation is obtained. In this limit...

We present a Lagrangian decomposition algorithmwhich uses logarithmic potential reduction to compute an ε-approximate solution of the general max-min resource sharing problem with M nonnegative concave constraints on a convex set B. We show that this algorithm runs in O(M(ε+lnM)) iterations, a data independent bound which is optimal up to polylogarithmic factors for any fixed relative accuracy ...

The firstand second-order optimum achievable exponents in the simple hypothesis testing problem are investigated. The optimum achievable exponent for type II error probability, under the constraint that the type I error probability is allowed asymptotically up to ε, is called the ε-optimum exponent. In this paper, we first give the second-order ε-optimum exponent in the case where the null hypo...

a f(x) dx (which one would use for instance to compute the work required to move a particle from a to b). For simplicity we shall restrict attention here to functions f : R → R which are continuous on the entire real line (and similarly, when we come to differential forms, we shall only discuss forms which are continuous on the entire domain). We shall also informally use terminology such as “i...