A major challenge faced by the marketers attempting to optimize their advertising campaigns is to deal with budget constraints. The problem is even harder in the face of multidimensional budget constraints, particularly in the presence of many decision variables involved, and the interplay among the decision variables through these such constraints. Concise bidding strategies help advertisers deal with this challenge by introducing fewer variables to act on. In this paper, we study the problem of finding optimal concise bidding strategies for advertising campaigns with multiple budget constraints. Given bid landscapes—i.e., predicted value (e.g., number of clicks) and cost per click for any bid—that are typically provided by ad-serving systems, we optimize the value given budget constraints. In particular, we consider bidding strategies that consist of no more than k different bids for all keywords. For constant k, we provide a PTAS to optimize the profit, whereas for arbitrary k we show how constant-factor approximation can be obtained via a combination of solution enumeration and dependent LP-rounding techniques. Finally, we evaluate the performance of our algorithms on real datasets in two regimes with 1- and 3-dimensional budget constraint. In the former case where uniform bidding has provable performance guarantee, our algorithm beats the state of the art by an increase of 1% to 6% in the expected number of clicks. This is achieved by only two or three clusters—contrast with the single cluster permitted in uniform bidding. With only three dimensions in the budget constraint (one for total consumption, and another two for enforcing minimal diversity), the gap between the performance of our algorithm and an enhanced version of uniform bidding grows to an average of 5% to 6% (sometimes as high as 9%). Although the details of experiments for the multidimensional budget constraint to the full version of the paper are deferred to the full version of the paper, we report some highlights from the results.