We introduce a new notion of generalized monotonicity named (strong) quasar-monot onicity. We establish the relation between the strong quasar-convexity of a differentiable function with the (strong) quasar-monotonicity of its gradient. We then study the well-known Polyak’s projected heavy-ball method when applying to strongly quasar-monotone variational inequalities. The geometric convergence of the iterations is obtained under suitable conditions on the stepsize and momentum. We apply the convergence results to the constrained optimization problem of strongly quasi-(star)-convex and differentiable functions. Numerical examples are given to demonstrate the advantage of the projected heavy-ball method comparing with the classical projection method.
