It is wel known that the weld bead width becomes wider and the weld pool hangs down as the circumferential welding of small diameter pipes progresses, if the constant welding conditions are maintained over the full joint length and if the compressed backing gas is not supplied. In order to obtain a uniform and no hang-down weld bead over the entire circumference of the pipe, the welding conditions such as the welding current, welding velocity and backing gas pressure should be adjusted and optimized as the welding proceeds. One of the important problems to be solved in the welding engineering is to develop the mathematical method for the determination of optimum process parameters. Thus a mathematical modeling of the welding process for determining the temperature distribution in the weld metal and the surface profile of the resultant weld is indispensable to optimize the process. First, in order to estimate the optimal process parameters such as the welding current and welding velocity in circumferential GTA welding of thin pipes, the objective function was chosen to maintain a uniform bead width over the full circumferential joint, while the constraints consist of the capacity limit of the welding power source and equipment used. The linear complementary problem(LCP) with Lemke``s pivoting algorithm and Powell``s unconstrained search method with the sequential unconstrained minimization technique (SUMT) was applied to evaluate the optimal welding current with a given welding velocity and the optimal welding velocity with a given welding current for a required bead width respectively. Here, the analytical solution of heat conduction equation with a Gaussian heat source was derived and applied to calculate the temperature filed and consequently to determine the weld width in circumferential welding of pipe workpieces. An efficient parameter optimization model for the numerical heat conduction flow was proposed to evaluate the optimal welding current with a give...