The de facto standard for state-of-the-art real and integer linear reasoning within Satisfiability Modulo Theories (SMT) solvers is the Simplex for DPLL(T) algorithm given by Dutertre and de Moura. This algorithm works by performing a sequence of local optimization operations. While the algorithm is generally efficient in practice, its local pivoting heuristics lead to slow convergence on some problems. More traditional Simplex algorithms minimize a global criterion to determine the feasibility of the input constraints. We present a novel Simplex-based decision procedure for use in SMT that minimizes the sum of infeasibilities of the constraints. Experimental results show that this new algorithm is comparable with or outperforms Simplex for DPLL(T) on a broad set of benchmarks.