Complete theory for E47 and 94LVI algorithms solving inequality-and-bound constrained quadratic program efficiently

This paper presents the relatively complete theory for the two new numerical algorithms, i.e., E47 algorithm and 94LVI algorithm, for solving the quadratic program (QP) subject to inequality and bound constraints efficiently. Specifically, via the important “Bridge” theorems and with their proofs provided, the QP problem is converted equivalently into a piecewise-linear projection equation (PLPE). The E47 and 94LVI algorithms are thus developed to solve the PLPE and also the QP. Note that, compared with the previous achievements with only the sufficiency proved, strict proofs (i.e., both the sufficiency proofs and the necessity proofs) of the “Bridge” theorems are firstly presented. Besides, the convergence property of the two algorithms is analyzed in the paper, and theoretical proofs ensure the global linear convergence of the two algorithms about the final decision variable vector x, instead of the primal-dual decision variable vector y.