One of the fundamental difficulties in engineering design is the multiplicity of local solutions. This has triggered great efforts to develop global search algorithms. Globality, however, often has a prohibitively high numerical cost for real problems. A fixed cost local search, which sequentially becomes global is developed. Globalization is achieved by probabilistic restart. A spatial probability of starting a local search is built based on past searches. An improved Nelder-Mead algorithm makes the local optimizer. It accounts for variable bounds. It is additionally made more robust by reinitializing degenerated simplexes. The resulting method, called Globalized Bounded Nelder-Mead (GBNM) algorithm, is particularly adapted to tackle multimodal, discontinuous optimization problems, for which it is uncertain that a global optimization can be afforded. Different strategies for restarting the local search are discussed. Numerical experiments are given on analytical test functions and composite laminate design problems. The GBNM method compares favorably to an evolutionary algorithm, both in terms of numerical cost and accuracy.