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This paper proposes a new evolutionary algorithm, called lower-dimensional-search evolutionary algorithm (LDSEA). The crossover operator of the new algorithm searches a lower-dimensional neighbor of the parent points where the neighbor center is the barycenter of the parents therefore the new algorithm converges fast, especially for high-dimensional constrained optimization problems. The niche-impaction operator and the mutation operator preserve the diversity of the population to make the LDSEA algorithm not to be trapped in local optima as much as possible. What's more is that the LDSEA algorithm is simple and easy to be implemented. We have used the 24 constrained benchmark problems  to test the LDSEA algorithm. The experimental results show it works better than or competitive to a known effective algorithm  for higher-dimensional constrained optimization problems.