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We present a structured algorithm for solving constrained non-linear least squares problems, and establish its local two-step Q-superlinear convergence. The approach is based on an adaptive structured scheme due to Mahdavi-Amiri and Bartels of the exact penalty method of Coleman and Conn for nonlinearly constrained optimization problems. The structured adaptation also makes use of the ideas of Nocedal and Overton for handling quasi-Newton updates of projected Hessians. We discuss the comparative results of the test-ing of our programs and three nonlinear programming codes from KNITRO on some randomly generated test problems due to Bartels and Mahdavi-Amiri. The results indeed confirm the practical significance of our special considerations for the inherent structure of the least squares. [ABSTRACT FROM AUTHOR]