We consider computationally tractable methods for the experimental design problem, where k out of n design points of dimension p are selected so that certain optimality criteria are approximately satisfied. Our algorithm finds a (1+eps)-approximate optimal design when k is a linear function of p; in contrast, existing results require k to be super-linear in p. Our algorithm also handles all popular optimality criteria, while existing ones only handle one or two such criteria. Numerical results on synthetic and real-world design problems verify the practical effectiveness of the proposed algorithm.