A Multiplicative Weight Updates Algorithm for Packing and Covering Semi-infinite Linear Programs

Abstract

We consider the following semi-infinite linear programming problems: $$\max $$ (resp., $$\min $$ ) $$c^Tx$$ s.t. $$y^TA_ix+(d^i)^Tx \le b_i$$ (resp., $$y^TA_ix+(d^i)^Tx \ge b_i)$$ , for all $$y \in {{\mathcal {Y}}}_i$$ , for $$i=1,\ldots ,N$$ , where $${{\mathcal {Y}}}_i\subseteq {\mathbb {R}}^m_+$$ are given compact convex sets and $$A_i\in {\mathbb {R}}^{m_i\times n}_+$$ , $$b=(b_1,\ldots b_N)\in {\mathbb {R}}_+^N$$ , $$d^i\in {\mathbb {R}}_+^n$$ , and $$c\in {\mathbb {R}}_+^n$$ 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 $${{\mathcal {Y}}}_i$$ , and the goal is to optimize the objective function for the worst case choice in each $${{\mathcal {Y}}}_i$$ . When the uncertainty sets $${{\mathcal {Y}}}_i$$ 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 $${{\mathcal {Y}}}_i$$ are simple, such as ellipsoids or boxes, this yields substantial improvements in running time over general convex programming solvers. We also consider the mixed packing/covering problem, in which both packing and covering constraints are given, and the objective is to find an approximately feasible solution.