Quartic Formulation of Standard Quadratic Optimization Problems

Standard bi-quadratic optimization problems and unconstrained polynomial reformulations

“…As mentioned in [5] for the single ball constraint case, in our proof of Theorem 4.2, the fact thatᾱ ≥ 0 andβ ≥ 0 is essential. For the general bi-homogeneous optimization over the product of two spheres, we have the following result.…”

“…Based upon this, we discuss the one-to-one correspondence between the global/local solutions of (1.1) and the global/local solutions of the formulated biquartic optimization problem. Our main technique used here is similar to that developed in [5].…”

Unconstrained formulation of standard quadratic optimization problems

“…In [6] a quartic formulation of the StQP has been proposed, which uses the substitution y i = x 2 i , to get rid of the sign constraints y i ≥ 0. Then the condition e y = 1 reads x 2 = 1, and we get the following ball constrained problem (BQP) min{Φ(x) = 1 2 x XAXx :…”

“…Using the special structure of the constraint in (2), in [6] an simple unconstrained formulation (UQP)of (2) has been proposed that in turn can be used to find a local solution of the StQP (1).…”

“…where ε can be freely choosen within (0,ε], and the upper boundε is easily calculated (see [6] for details).…”