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Improving perturbation theory via a variational optimization has generallyproduced in higher orders an embarrassingly large set of solutions, most ofthem unphysical (complex). We introduce an extension of the optimizedperturbation method which leads to a drastic reduction of the number ofacceptable solutions. The properties of this new method are studied and it isthen applied to the calculation of relevant quantities in different $\phi^4$models, such as the anharmonic oscillator energy levels and the criticalBose-Einstein Condensation temperature shift $\Delta T_c$ recently investigatedby various authors. Our present estimates of $\Delta T_c$, incorporating themost recently available six and seven loop perturbative information, are inexcellent agreement with all the available lattice numerical simulations. Thisrepresents a very substantial improvement over previous treatments.Comment: 9 pages, no figures. v2: minor wording changes in title/abstract, to  appear in Phys.Rev.

Improved optimization of perturbation theory: Applications to the oscillator energy levels and Bose-Einstein condensate critical temperature