We prove a general result demonstrating the power of Lagrangian relaxation in solving constrained maximization problems with arbitrary objective functions. This yields a unified approach for solving a wide class of subset selection problems with linear constraints. Given a problem in this class and some small ε ∈ (0, 1), we show that if there exists a ρ-approximation algorithm for the Lagrangia...

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...

We prove a general result demonstrating the power of Lagrangian relaxation in solving constrained maximization problems with arbitrary objective functions. This yields a unified approach for solving a wide class of subset selection problems with linear constraints. Given a problem in this class and some small ε ∈ (0, 1), we show that if there exists an r-approximation algorithm for the Lagrangi...

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...

We give a proximal bundle method for minimizing a convex function f over R+. It requires evaluating f and its subgradients with a possibly unknown accuracy ε ≥ 0, and maintains a set of free variables I to simplify its prox subproblems. The method asymptotically finds points that are ε-optimal. In Lagrangian relaxation of convex programs, it allows for ε-accurate solutions of Lagrangian subprob...

Submodular-function maximization is a central problem in combinatorial optimization, generalizing many important NP-hard problems including Max Cut in digraphs, graphs and hypergraphs, certain constraint satisfaction problems, maximum-entropy sampling, and maximum facility-location problems. Our main result is that for any k ≥ 2 and any ε > 0, there is a natural local-search algorithm which has...