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A central problem in the power semiconductor industry is to design semiconductor devices with high breakdown voltage and low on-state resistance. This problem is usually addressed computationally by developing a physics-based model for the power device and adjusting the device structure until the required parameters (i.e. breakdown voltage and on-state resistance) are optimized. In the case of finite element models, the optimization of power devices requires computing the optimum value of the doping concentration at each vertex of the finite element mesh, which is equivalent to solving an optimization problem with a large number of control parameters (usually over 105 parameters). This problem cannot be solved using traditional heuristic techniques because such techniques require many device simulations, which result in unpractically long computation times even on dedicated computer clusters. In this dissertation we develop a non-heuristic technique to optimize power semiconductor devices and apply this method to the optimization of Insulated Gate Bipolar Transistors (IGBTs) modeled by finite elements. The method is based on computing the doping sensitivity functions of the breakdown voltage and on-state resistance using an adjoint method that we developed specifically for this task, and it uses these functions in a gradient based optimization algorithm to compute the optimum doping profile inside the device. As numerical examples, we present two optimization cases. First, we optimize the doping profile of a standard IGBT structure in order to maximize the breakdown voltage while keeping the on-state resistance constant; then, we minimize the on-state resistance while keeping the breakdown voltage constant. In both cases, we show that the proposed method can be used successfully to the optimization of the doping profiles that maximizes the breakdown voltage of the IGBT.