This paper describes a method for unconstrained optimization that asso
ciates quasi-Newton methods with conic functions. The derivation is ba
sed upon the construction of a conic function so that a local nonquadr
atic model can interpolate two function and one gradient values of the
objective function at the last two interates as a natural extension o
f existing quasi-Newton methods. The new method is shown to have Q-sup
erlinear rate of convergence under standard assumptions on the objecti
ve function, and to decrease the number of line searches for good choi
ce of parameters. Numerical experiments verify that the new method is
very successful.