Surrogate Optimization for p-Norms

In this paper, we study the effect of surrogate objective functions in optimization problems. We introduce surrogate ratio as a measure of such effect, where the surrogate ratio is the ratio between the optimal values of the original and surrogate objective functions. We prove that the surrogate ratio is at most μ|1/p−1/q| when the objective functions are pand q-norms, and the feasible region is a μ-dimensional space (i.e., a subspace of R), a μ-intersection of matroids, or a μ-extendible system. We also show that this is the best possible bound. In addition, for μ-systems, we demonstrate that the ratio becomes μ1/p when p < q and unbounded if p > q. Here, a μ-system is an independence system such that for any subset of ground set the ratio of the cardinality of the largest to the smallest maximal independent subset of it is at most μ. We further extend our results to the surrogate ratios for approximate solutions. 1998 ACM Subject Classification G.2.0 General