A multiplicative weights update algorithm for packing and covering semi-infinite linear programs

Abstract

We consider the following semi-infinite linear programming problems: max (resp., min) cT x s.t. yT Aix + (di)Tx ≤ bi (resp., yT Aix + (di)T x ≥ bi), for all y ∈ Yi, for i = 1, …, N, where Yi ⊆ ℝ+mi + are given compact convex sets and Ai ∈ ℝ+mi×n +, b = (b1, …, bN) ∈ ℝN+, di ∈ ℝn+, and c ∈ Rn + are given non-negative matrices and vectors. This general framework is useful in modeling many interesting problems. For example, it can be used to represent a sub-class of Robust optimization in which the coefficients of the constraints are drawn from convex uncertainty sets Yi, and the goal is to optimize the objective function for the worst-case choice in each Yi. When the uncertainty sets Yi are ellipsoids, we obtain a sub-class of Second-Order Cone Programming. We show how to extend the multiplicative weights update method to derive approximation schemes for the above packing and covering problems. When the sets Yi are simple, such as ellipsoids or boxes, this yields substantial improvements in the running time over general convex programming solvers.