We consider the problem of hedging a contingent claim with a "semistatic" strategy composed of a dynamic position in one asset and static (buy-and-hold) positions in other assets. We give general representations of the optimal strategy and the hedging error under the criterion of variance optimality and provide tractable formulas using Fourier integration in case of the Heston model. We also consider the problem of optimally selecting a sparse semistatic hedging strategy, i.e., a strategy that only uses a small subset of available hedging assets and discuss parallels to the variable-selection problem in linear regression. The methods developed are illustrated in an extended numerical example where we compute a sparse semistatic hedge for a variance swap using European options as static hedging assets.