One of the fundamental difficulties in engineering design is the multiplicity of local solutions. This has triggered much effort in the development of 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 in this work. Globalization is achieved by probabilistic restarts. A spacial probability of starting a local search is built based on past searches. An improved Nelder-Mead algorithm is the local optimizer. It accounts for variable bounds and nonlinear inequality constraints. It is additionally made more robust by reinitializing degenerated simplexes. The resulting method, called the Globalized Bounded Nelder-Mead (GBNM) algorithm, is particularly adapted to tackling multimodal, discontinuous, constrained optimization problems, for which it is uncertain that a global optimization can be afforded. Numerical experiments are given on two analytical test functions and two composite laminate design problems. The GBNM method compares favorably with an evolutionary algorithm, both in terms of numerical cost and accuracy.